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Nonlinear Dynamics and Entropy of Complex Systems with Hidden and Self-Excited Attractors II

A special issue of Entropy (ISSN 1099-4300). This special issue belongs to the section "Complexity".

Deadline for manuscript submissions: closed (15 May 2020) | Viewed by 62813

Special Issue Editors

Dr. Christos Volos

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Guest Editor
Laboratory of Nonlinear Systems, Circuits & Coplexity (LaNSCom), Department of Physics, Aristotle University of Thessaloniki, GR-54124 Thessaloniki, Greece
Interests: electrical and electronics engineering; mathematical modeling; control theory; engineering, applied and computational mathematics; numerical analysis; mathematical analysis; numerical modeling; modeling and simulation; robotics
Special Issues, Collections and Topics in MDPI journals
Dr. Sajad Jafari

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Guest Editor
Nonlinear Systems and Applications, Faculty of Electrical and Electronics Engineering, Ton Duc Thang University, Ho Chi Minh City, Vietnam
Interests: chaos; nonlinear dynamics; optimization
Special Issues, Collections and Topics in MDPI journals
Prof. Dr. Jesus M. Munoz-Pacheco

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Guest Editor
Faculty of Electronics Sciences, Benemérita Universidad Autónoma de Puebla, Av. San Claudio y 18 Sur, Puebla 72570, Mexico
Interests: chaos theory; chaotic dynamics and applications; nonlinear circuits and systems; mathematical modeling; electronics; fractional-order chaotic systems; fractional-order calculus
Special Issues, Collections and Topics in MDPI journals
Dr. Jacques Kengne

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Department of Electrical Engineering, University of Dschang, Dschang P.O. Box 134, Cameroon
Interests: chaos theory; nonlinear phenomena; nonlinear circuits; hidden attractors; synchronization
Special Issues, Collections and Topics in MDPI journals
Dr. Karthikeyan Rajagopal

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Center for Nonlinear Systems, Chennai Institute of Technology, Tamil Nadu 600069, India
Interests: optimal control theory; artificial intelligence; adaptive control; neural networks
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

Entropy is a basic and important concept in information theory. It is also often used as a measure of the degree of chaos in dynamical systems, for example, Lyapunov exponents, fractal dimension, and entropy are usually used to describe the complexity of chaotic systems. Thus, it would be interesting to collect the latest advances in the field of studying entropy in nonlinear systems.

Additionally, in the last few years, there has been increasing interest in a new classification of nonlinear dynamical systems, including two kinds of attractors, namely: self-excited attractors and hidden attractors. Self-excited attractors can be localized straight forwardly by applying a standard computational procedure. Some interesting examples of systems with self-excited attractors are chaotic systems with different kinds of symmetry, with multi-scroll attractors, multiple attractors, and extreme multistability. On the other hand, in systems with hidden attractors, we have to develop a specific computational procedure to identify the hidden attractors because of the fact that the equilibrium points do not help in their localization. Some examples of these kinds of systems are chaotic dynamical systems with no equilibrium points, with only stable equilibria, curves of equilibria, surfaces of equilibria, and non-hyperbolic equilibria. There is evidence that hidden attractors play an important role in the various fields, ranging from phase-locked loops, oscillators describing a convective fluid motion, models of drilling systems, information theory, and cryptography to multilevel DC/DC converters. Furthermore, hidden attractors may lead to unexpected and disastrous responses. So, it is very useful to find new tools in order to study entropy for hidden attractors.

This Special Issue is dedicated to the presentation and discussion of the advanced topics of complex systems with hidden attractors and self-excited attractors. The contribution to the Special Issue should focus on the aspects of nonlinear dynamics, entropy, and applications of nonlinear systems with hidden and self-excited attractors.

Potential topics include, but are not limited to, the following:

  • Analytical–numerical methods for investing hidden oscillations
  • Bifurcation and chaos in complex systems
  • Chimera states, spiral waves, and pattern formation in networks of oscillators
  • Self-organization
  • Designing new nonlinear systems with desired features
  • Experimental study of nonlinear systems
  • Extreme multistability
  • Complex networks
  • Fractional order dynamical systems
  • Hidden attractors in complex systems
  • Entropy of hidden attractors
  • Networks of nonlinear oscillators (like neurons)
  • New methods of control and synchronization nonlinear systems
  • Information theory
  • Nonlinear dynamics and chaos in engineering applications
  • Nonlinear systems with an infinite number of equilibrium points
  • Nonlinear systems with a stable equilibrium
  • Nonlinear systems without equilibrium
  • Entropy-based cryptography
  • Novel computation algorithms for studying nonlinear systems
  • Oscillations and chaos in dynamic economic models
  • Quantum chaos
  • Related engineering applications
  • Self-excited attractors

Dr. Christos Volos
Dr. Sajad Jafari
Dr. Jesus M. Munoz-Pacheco
Dr. Jacques Kengne
Dr. Karthikeyan Rajagopal
Guest Editors

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All submissions that pass pre-check are peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 100 words) can be sent to the Editorial Office for announcement on this website.

Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Entropy is an international peer-reviewed open access monthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 2600 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

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Editorial

Jump to: Research

5 pages, 194 KiB  
Editorial
Nonlinear Dynamics and Entropy of Complex Systems with Hidden and Self-Excited Attractors II
by Christos K. Volos, Sajad Jafari, Jesus M. Munoz-Pacheco, Jacques Kengne and Karthikeyan Rajagopal
Entropy 2020, 22(12), 1428; https://doi.org/10.3390/e22121428 - 18 Dec 2020
Cited by 3 | Viewed by 1982
Abstract
According to the pioneering work of Leonov and Kuznetsov [...] Full article
(This article belongs to the Special Issue Nonlinear Dynamics and Entropy of Complex Systems with Hidden and Self-Excited Attractors II)

Research

Jump to: Editorial

16 pages, 1602 KiB  
Article
Chaos Control and Synchronization of a Complex Rikitake Dynamo Model
by Wenkai Pang, Zekang Wu, Yu Xiao and Cuimei Jiang
Entropy 2020, 22(6), 671; https://doi.org/10.3390/e22060671 - 17 Jun 2020
Cited by 5 | Viewed by 2798
Abstract
A novel chaotic system called complex Rikitake system is proposed. Dynamical properties, including symmetry, dissipation, stability of equilibria, Lyapunov exponents and bifurcation, are analyzed on the basis of theoretical analysis and numerical simulation. Further, based on feedback control method, the complex Rikitake system [...] Read more.
A novel chaotic system called complex Rikitake system is proposed. Dynamical properties, including symmetry, dissipation, stability of equilibria, Lyapunov exponents and bifurcation, are analyzed on the basis of theoretical analysis and numerical simulation. Further, based on feedback control method, the complex Rikitake system can be controlled to any equilibrium points. Additionally, this paper not only proves the existence of two types of synchronization schemes in the complex Rikitake system but also designs adaptive controllers to realize them. The proposed results are verified by numerical simulations. Full article
(This article belongs to the Special Issue Nonlinear Dynamics and Entropy of Complex Systems with Hidden and Self-Excited Attractors II)
►▼ Show Figures

Figure 1

Figure 1
<p>The projection of chaotic attractor for the Rikitake dynamo system (<a href="#FD1-entropy-22-00671" class="html-disp-formula">1</a>). (<b>a</b>) in the z-y-x space; (<b>b</b>) in x-y space.</p> Full article ">Figure 2
<p>Chaotic attractors of system (<a href="#FD3-entropy-22-00671" class="html-disp-formula">3</a>) in different spaces. (<b>a</b>) in <math display="inline"><semantics> <mrow> <msub> <mi>x</mi> <mn>3</mn> </msub> <mo>−</mo> <msub> <mi>x</mi> <mn>1</mn> </msub> <mo>−</mo> <msub> <mi>x</mi> <mn>5</mn> </msub> </mrow> </semantics></math> space; (<b>b</b>) in <math display="inline"><semantics> <mrow> <msub> <mi>x</mi> <mn>4</mn> </msub> <mo>−</mo> <msub> <mi>x</mi> <mn>5</mn> </msub> <mo>−</mo> <msub> <mi>x</mi> <mn>2</mn> </msub> </mrow> </semantics></math> space; (<b>c</b>) in <math display="inline"><semantics> <mrow> <msub> <mi>x</mi> <mn>3</mn> </msub> <mo>−</mo> <msub> <mi>x</mi> <mn>5</mn> </msub> </mrow> </semantics></math> space; (<b>d</b>) in <math display="inline"><semantics> <mrow> <msub> <mi>x</mi> <mn>1</mn> </msub> <mo>−</mo> <msub> <mi>x</mi> <mn>3</mn> </msub> </mrow> </semantics></math> space.</p> Full article ">Figure 2 Cont.
<p>Chaotic attractors of system (<a href="#FD3-entropy-22-00671" class="html-disp-formula">3</a>) in different spaces. (<b>a</b>) in <math display="inline"><semantics> <mrow> <msub> <mi>x</mi> <mn>3</mn> </msub> <mo>−</mo> <msub> <mi>x</mi> <mn>1</mn> </msub> <mo>−</mo> <msub> <mi>x</mi> <mn>5</mn> </msub> </mrow> </semantics></math> space; (<b>b</b>) in <math display="inline"><semantics> <mrow> <msub> <mi>x</mi> <mn>4</mn> </msub> <mo>−</mo> <msub> <mi>x</mi> <mn>5</mn> </msub> <mo>−</mo> <msub> <mi>x</mi> <mn>2</mn> </msub> </mrow> </semantics></math> space; (<b>c</b>) in <math display="inline"><semantics> <mrow> <msub> <mi>x</mi> <mn>3</mn> </msub> <mo>−</mo> <msub> <mi>x</mi> <mn>5</mn> </msub> </mrow> </semantics></math> space; (<b>d</b>) in <math display="inline"><semantics> <mrow> <msub> <mi>x</mi> <mn>1</mn> </msub> <mo>−</mo> <msub> <mi>x</mi> <mn>3</mn> </msub> </mrow> </semantics></math> space.</p> Full article ">Figure 3
<p>Poincaré map of system (<a href="#FD3-entropy-22-00671" class="html-disp-formula">3</a>) with <math display="inline"><semantics> <mrow> <mi>α</mi> <mo>=</mo> <mn>5</mn> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mi>β</mi> <mo>=</mo> <mn>2</mn> </mrow> </semantics></math>. (<b>a</b>) in <math display="inline"><semantics> <mrow> <msub> <mi>x</mi> <mn>1</mn> </msub> <mo>−</mo> <msub> <mi>x</mi> <mn>3</mn> </msub> <mo>−</mo> <msub> <mi>x</mi> <mn>5</mn> </msub> </mrow> </semantics></math> space; (<b>b</b>) in <math display="inline"><semantics> <mrow> <msub> <mi>x</mi> <mn>1</mn> </msub> <mo>−</mo> <msub> <mi>x</mi> <mn>3</mn> </msub> </mrow> </semantics></math> space.</p> Full article ">Figure 4
<p>(<b>a</b>) Bifurcation diagram of system (<a href="#FD3-entropy-22-00671" class="html-disp-formula">3</a>) with <math display="inline"><semantics> <mrow> <mi>α</mi> <mo>=</mo> <mn>5</mn> </mrow> </semantics></math>; (<b>b</b>) State variable under different initial values.</p> Full article ">Figure 5
<p>(<b>a</b>) Control the complex Rikitake system (<a href="#FD3-entropy-22-00671" class="html-disp-formula">3</a>) to <math display="inline"><semantics> <msub> <mi>S</mi> <mn>1</mn> </msub> </semantics></math>; (<b>b</b>) <span class="html-italic">k</span> tends to a negative constant.</p> Full article ">Figure 6
<p>(<b>a</b>) Control the complex Rikitake system (<a href="#FD3-entropy-22-00671" class="html-disp-formula">3</a>) to <math display="inline"><semantics> <msub> <mi>S</mi> <mn>2</mn> </msub> </semantics></math>; (<b>b</b>) <span class="html-italic">k</span> approaches to a negative constant.</p> Full article ">Figure 7
<p>(<b>a</b>) CS error system is regulated to the zero equilibrium point; (<b>b</b>) <span class="html-italic">k</span> approaches to a negative constant.</p> Full article ">Figure 8
<p>State variables of the complex Rikitake systems (<a href="#FD10-entropy-22-00671" class="html-disp-formula">10</a>) and (<a href="#FD11-entropy-22-00671" class="html-disp-formula">11</a>) varying time. (<b>a</b>) Trajectories of <math display="inline"><semantics> <mrow> <msubsup> <mi>y</mi> <mn>1</mn> <mi>r</mi> </msubsup> </mrow> </semantics></math> and <math display="inline"><semantics> <msubsup> <mi>z</mi> <mn>1</mn> <mi>r</mi> </msubsup> </semantics></math>; (<b>b</b>) Trajectories of <math display="inline"><semantics> <msubsup> <mi>y</mi> <mn>1</mn> <mi>i</mi> </msubsup> </semantics></math> and <math display="inline"><semantics> <msubsup> <mi>z</mi> <mn>1</mn> <mi>i</mi> </msubsup> </semantics></math>; (<b>c</b>) Trajectories of <math display="inline"><semantics> <msubsup> <mi>y</mi> <mn>2</mn> <mi>r</mi> </msubsup> </semantics></math> and <math display="inline"><semantics> <msubsup> <mi>z</mi> <mn>2</mn> <mi>r</mi> </msubsup> </semantics></math>; (<b>d</b>) Trajectories of <math display="inline"><semantics> <msubsup> <mi>y</mi> <mn>2</mn> <mi>i</mi> </msubsup> </semantics></math> and <math display="inline"><semantics> <msubsup> <mi>z</mi> <mn>2</mn> <mi>i</mi> </msubsup> </semantics></math>; (<b>e</b>) Trajectories of <math display="inline"><semantics> <msub> <mi>y</mi> <mn>3</mn> </msub> </semantics></math> and <math display="inline"><semantics> <msub> <mi>z</mi> <mn>3</mn> </msub> </semantics></math>.</p> Full article ">Figure 9
<p>(<b>a</b>) The synchronization error system is regulated to the zero equilibrium point; (<b>b</b>) <span class="html-italic">k</span> is estimated to a negative constant.</p> Full article ">Figure 10
<p>State variables of two identical complex Rikitake systems (<a href="#FD10-entropy-22-00671" class="html-disp-formula">10</a>) and (<a href="#FD11-entropy-22-00671" class="html-disp-formula">11</a>) varying time. (<b>a</b>) Trajectories of <math display="inline"><semantics> <msubsup> <mi>y</mi> <mn>1</mn> <mi>r</mi> </msubsup> </semantics></math> and <math display="inline"><semantics> <msubsup> <mi>z</mi> <mn>1</mn> <mi>r</mi> </msubsup> </semantics></math>; (<b>b</b>) Trajectories of <math display="inline"><semantics> <msubsup> <mi>y</mi> <mn>1</mn> <mi>i</mi> </msubsup> </semantics></math> and <math display="inline"><semantics> <msubsup> <mi>z</mi> <mn>1</mn> <mi>i</mi> </msubsup> </semantics></math>; (<b>c</b>) Trajectories of <math display="inline"><semantics> <msubsup> <mi>y</mi> <mn>2</mn> <mi>r</mi> </msubsup> </semantics></math> and <math display="inline"><semantics> <msubsup> <mi>z</mi> <mn>2</mn> <mi>r</mi> </msubsup> </semantics></math>; (<b>d</b>) Trajectories of <math display="inline"><semantics> <msubsup> <mi>y</mi> <mn>2</mn> <mi>i</mi> </msubsup> </semantics></math> and <math display="inline"><semantics> <msubsup> <mi>z</mi> <mn>2</mn> <mi>i</mi> </msubsup> </semantics></math>; (<b>e</b>) Trajectories of <math display="inline"><semantics> <msub> <mi>y</mi> <mn>3</mn> </msub> </semantics></math> and <math display="inline"><semantics> <msub> <mi>z</mi> <mn>3</mn> </msub> </semantics></math>.</p> Full article ">
15 pages, 20271 KiB  
Article
Birhythmic Analog Circuit Maze: A Nonlinear Neurostimulation Testbed
by Ian D. Jordan and Il Memming Park
Entropy 2020, 22(5), 537; https://doi.org/10.3390/e22050537 - 11 May 2020
Cited by 6 | Viewed by 3270
Abstract
Brain dynamics can exhibit narrow-band nonlinear oscillations and multistability. For a subset of disorders of consciousness and motor control, we hypothesized that some symptoms originate from the inability to spontaneously transition from one attractor to another. Using external perturbations, such as electrical pulses [...] Read more.
Brain dynamics can exhibit narrow-band nonlinear oscillations and multistability. For a subset of disorders of consciousness and motor control, we hypothesized that some symptoms originate from the inability to spontaneously transition from one attractor to another. Using external perturbations, such as electrical pulses delivered by deep brain stimulation devices, it may be possible to induce such transition out of the pathological attractors. However, the induction of transition may be non-trivial, rendering the current open-loop stimulation strategies insufficient. In order to develop next-generation neural stimulators that can intelligently learn to induce attractor transitions, we require a platform to test the efficacy of such systems. To this end, we designed an analog circuit as a model for the multistable brain dynamics. The circuit spontaneously oscillates stably on two periods as an instantiation of a 3-dimensional continuous-time gated recurrent neural network. To discourage simple perturbation strategies, such as constant or random stimulation patterns from easily inducing transition between the stable limit cycles, we designed a state-dependent nonlinear circuit interface for external perturbation. We demonstrate the existence of nontrivial solutions to the transition problem in our circuit implementation. Full article
(This article belongs to the Special Issue Nonlinear Dynamics and Entropy of Complex Systems with Hidden and Self-Excited Attractors II)
►▼ Show Figures

Figure 1

Figure 1
<p>Planar Limit Cycle with 2D continuous-time gated recurrent unit (ct-GRU) depicted in phase space: The red dot indicates an unstable fixed point at the origin unstable, while orange and pink lines represent the x and y nullclines, respectively. Purple lines indicate various trajectories of the hidden state. Direction of the flow is determined by the black arrows, where the colormap underlying the figure depicts the magnitude of the velocity of the flow in log scale.</p> Full article ">Figure 2
<p>Birhythmicy in 3-dimensions: (<b>A</b>): light blue manifold on the <math display="inline"><semantics> <mrow> <mi>x</mi> <mo>−</mo> <mi>y</mi> </mrow> </semantics></math> plane separates the basins of attraction of the upper and lower limit cycles. Trajectories are colored either dark blue or purple, depending on which basin of attraction they are initialized in. Red dots indicate fixed points, and black arrows depict the direction of flow. (<b>B</b>,<b>C</b>): <span class="html-italic">x</span>, <span class="html-italic">y</span>, and <span class="html-italic">z</span> components of trajectories initialized in the basins of attractions for the top and bottom limit cycles, respectively. Solid colored lines indicate <math display="inline"><semantics> <mrow> <mi>x</mi> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </semantics></math>, dashed lines indicate <math display="inline"><semantics> <mrow> <mi>y</mi> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </semantics></math>, and black lines indicate <math display="inline"><semantics> <mrow> <mi>z</mi> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </semantics></math>.</p> Full article ">Figure 3
<p>Electronic circuit realization of the hyperbolic tangent function, as implemented in Reference [<a href="#B30-entropy-22-00537" class="html-bibr">30</a>]. <math display="inline"><semantics> <msub> <mi>V</mi> <mrow> <mi>i</mi> <mi>n</mi> </mrow> </msub> </semantics></math> and <math display="inline"><semantics> <msub> <mi>V</mi> <mi>o</mi> </msub> </semantics></math> represent the input and output signals, respectively.</p> Full article ">Figure 4
<p>Circuit schematic of <math display="inline"><semantics> <mover accent="true"> <mi>z</mi> <mo>˙</mo> </mover> </semantics></math> for the birhythmic system. The system block labeled <tt>-tanh</tt> represents the circuit depicted in <a href="#entropy-22-00537-f003" class="html-fig">Figure 3</a>, where <tt>I01</tt> and <tt>I02</tt> correspond to <math display="inline"><semantics> <msub> <mi>V</mi> <mrow> <mi>i</mi> <mi>n</mi> </mrow> </msub> </semantics></math> and <math display="inline"><semantics> <msub> <mi>V</mi> <mi>o</mi> </msub> </semantics></math>, respectively. The terminal labeled <tt>stim_out</tt> represents the output to the stimulator circuit, as discussed in <a href="#sec4-entropy-22-00537" class="html-sec">Section 4</a>.</p> Full article ">Figure 5
<p>Circuit schematic of <math display="inline"><semantics> <mover accent="true"> <mi>x</mi> <mo>˙</mo> </mover> </semantics></math> and <math display="inline"><semantics> <mover accent="true"> <mi>y</mi> <mo>˙</mo> </mover> </semantics></math> for the birhythmic system. The system blocks labeled <tt>-tanh1</tt> and <tt>-tanh2</tt> represent the circuit depicted in <a href="#entropy-22-00537-f003" class="html-fig">Figure 3</a>, where <tt>I01</tt> and <tt>I02</tt> correspond to <math display="inline"><semantics> <msub> <mi>V</mi> <mrow> <mi>i</mi> <mi>n</mi> </mrow> </msub> </semantics></math> and <math display="inline"><semantics> <msub> <mi>V</mi> <mi>o</mi> </msub> </semantics></math>, respectively, for both blocks. The two multiplier chips, M1 and M2, are assumed to operate with unity gain.</p> Full article ">Figure 6
<p>Physical birhythmic circuit constructed on a breadboard. Blue boxes represent hyperbolic tangent units. The magenta box indicates the subsection of the circuit generating <math display="inline"><semantics> <mover accent="true"> <mi>z</mi> <mo>˙</mo> </mover> </semantics></math>, and the green box indicates the analog multipliers.</p> Full article ">Figure 7
<p>Experimental recordings of the birhythmic circuit: (<b>A</b>,<b>C</b>): <span class="html-italic">x</span> (yellow), <span class="html-italic">y</span> (blue), and <span class="html-italic">z</span> (pink) with respect to time of trajectories within the basin of attraction of the fast and slow limit cycles, respectively. (<b>B</b>,<b>D</b>): Projection of the corresponding trajectories in (<b>A</b>,<b>C</b>) onto the x-y plane, respectively.</p> Full article ">Figure 8
<p>Schematic for nonlinear <span class="html-italic">stimulator circuit</span>, with input labeled as <span class="html-italic">Stimulus</span>. The output, labeled <tt>stim_out</tt>, is fed into the terminal with the same name presented in the circuit diagram shown in <a href="#entropy-22-00537-f004" class="html-fig">Figure 4</a>. The two multiplier chips, M1 and M2, are assumed to operate with unity gain, and the <span class="html-italic">x</span> and <span class="html-italic">y</span> terminals are fed into the equivalently named terminals depicted in <a href="#entropy-22-00537-f005" class="html-fig">Figure 5</a>.</p> Full article ">Figure 9
<p>A time window of <math display="inline"><semantics> <mrow> <mi>z</mi> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </semantics></math> for all experimental trials: Red and blue trajectories demonstrate resultant behavior from stimulation patterns designed to transition states from the slow and fast limit cycles, respectively. Turquoise trajectories depict trials of constant stimulation, and orange trajectories show trials of random stimulation.</p> Full article ">Figure 10
<p>x (yellow), y (blue), z (pink), and <span class="html-italic">stim_out</span> (green) with respect to time of trajectories within the basin of attraction of the slow (<b>A</b>) and fast (<b>B</b>) limit cycles, under constant stimulation. Note that this stimulation regime does not successfully transition between the two attracting states in either direction.</p> Full article ">Figure 11
<p>x (yellow), y (blue), z (pink), and <span class="html-italic">stim_out</span> (green) with respect to time of trajectories within the basin of attraction of the slow (<b>A</b>) and fast (<b>B</b>) limit cycles, with random stimulation. Note that this stimulation regime does not successfully transition between the two attracting states in either direction.</p> Full article ">Figure 12
<p>Examples of stimulation patterns capable of inducing transition between states: x (yellow), y (blue), z (pink), and <span class="html-italic">stim_out</span> (green) with respect to time of trajectories initialized within the basin of attraction of the slow (<b>A</b>) and fast (<b>B</b>) limit cycles. As <math display="inline"><semantics> <mrow> <mi>z</mi> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </semantics></math> changes with stimulation so does the frequency of oscillation. As such, the time window to stimulate shifts continuously.</p> Full article ">Figure A1
<p>Input/output relation of three physically realized hyperbolic tangent circuits, interpolated through 21 points, compared with the analytic hyperbolic tangent function.</p> Full article ">
15 pages, 3506 KiB  
Article
Neural Computing Enhanced Parameter Estimation for Multi-Input and Multi-Output Total Non-Linear Dynamic Models
by Longlong Liu, Di Ma, Ahmad Taher Azar and Quanmin Zhu
Entropy 2020, 22(5), 510; https://doi.org/10.3390/e22050510 - 30 Apr 2020
Cited by 18 | Viewed by 3070
Abstract
In this paper, a gradient descent algorithm is proposed for the parameter estimation of multi-input and multi-output (MIMO) total non-linear dynamic models. Firstly, the MIMO total non-linear model is mapped to a non-completely connected feedforward neural network, that is, the parameters of the [...] Read more.
In this paper, a gradient descent algorithm is proposed for the parameter estimation of multi-input and multi-output (MIMO) total non-linear dynamic models. Firstly, the MIMO total non-linear model is mapped to a non-completely connected feedforward neural network, that is, the parameters of the total non-linear model are mapped to the connection weights of the neural network. Then, based on the minimization of network error, a weight-updating algorithm, that is, an estimation algorithm of model parameters, is proposed with the convergence conditions of a non-completely connected feedforward network. In further determining the variables of the model set, a method of model structure detection is proposed for selecting a group of important items from the whole variable candidate set. In order to verify the usefulness of the parameter identification process, we provide a virtual bench test example for the numerical analysis and user-friendly instructions for potential applications. Full article
(This article belongs to the Special Issue Nonlinear Dynamics and Entropy of Complex Systems with Hidden and Self-Excited Attractors II)
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Figure 1

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<p>Structure of a neural network corresponding to a total non-linear model.</p> Full article ">Figure 2
<p>Error of <math display="inline"><semantics> <mrow> <msub> <mi>y</mi> <mn>1</mn> </msub> <mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </semantics></math> with sine–sine input.</p> Full article ">Figure 3
<p>Error of <math display="inline"><semantics> <mrow> <msub> <mi>y</mi> <mn>2</mn> </msub> <mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </semantics></math> with sine–sine input.</p> Full article ">Figure 4
<p>Error of <math display="inline"><semantics> <mrow> <msub> <mi>y</mi> <mn>1</mn> </msub> <mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </semantics></math> with sine–square input.</p> Full article ">Figure 5
<p>Error of <math display="inline"><semantics> <mrow> <msub> <mi>y</mi> <mn>2</mn> </msub> <mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </semantics></math> with sine–square input.</p> Full article ">Figure 6
<p>Error of <math display="inline"><semantics> <mrow> <msub> <mi>y</mi> <mn>1</mn> </msub> <mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </semantics></math> with square–square input.</p> Full article ">Figure 7
<p>Error of <math display="inline"><semantics> <mrow> <msub> <mi>y</mi> <mn>2</mn> </msub> <mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </semantics></math> with square–square input.</p> Full article ">Figure 8
<p>Error of <math display="inline"><semantics> <mrow> <msub> <mi>y</mi> <mn>1</mn> </msub> <mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </semantics></math> with noise and sine–sine input.</p> Full article ">Figure 9
<p>Error of <math display="inline"><semantics> <mrow> <msub> <mi>y</mi> <mn>2</mn> </msub> <mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </semantics></math> with noise and sine–sine input.</p> Full article ">Figure 10
<p>Error of <math display="inline"><semantics> <mrow> <msub> <mi>y</mi> <mn>1</mn> </msub> <mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </semantics></math> with noise and sine–square input.</p> Full article ">Figure 11
<p>Error of <math display="inline"><semantics> <mrow> <msub> <mi>y</mi> <mn>2</mn> </msub> <mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </semantics></math> with noise and sine–square input.</p> Full article ">Figure 12
<p>Error of <math display="inline"><semantics> <mrow> <msub> <mi>y</mi> <mn>1</mn> </msub> <mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </semantics></math> with noise and square–square input.</p> Full article ">Figure 13
<p>Error of <math display="inline"><semantics> <mrow> <msub> <mi>y</mi> <mn>2</mn> </msub> <mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mo> </mo> </mrow> </semantics></math>with noise and square–square input.</p> Full article ">
20 pages, 32574 KiB  
Article
Modification of the Logistic Map Using Fuzzy Numbers with Application to Pseudorandom Number Generation and Image Encryption
by Lazaros Moysis, Christos Volos, Sajad Jafari, Jesus M. Munoz-Pacheco, Jacques Kengne, Karthikeyan Rajagopal and Ioannis Stouboulos
Entropy 2020, 22(4), 474; https://doi.org/10.3390/e22040474 - 20 Apr 2020
Cited by 42 | Viewed by 4242
Abstract
A modification of the classic logistic map is proposed, using fuzzy triangular numbers. The resulting map is analysed through its Lyapunov exponent (LE) and bifurcation diagrams. It shows higher complexity compared to the classic logistic map and showcases phenomena, like antimonotonicity and crisis. [...] Read more.
A modification of the classic logistic map is proposed, using fuzzy triangular numbers. The resulting map is analysed through its Lyapunov exponent (LE) and bifurcation diagrams. It shows higher complexity compared to the classic logistic map and showcases phenomena, like antimonotonicity and crisis. The map is then applied to the problem of pseudo random bit generation, using a simple rule to generate the bit sequence. The resulting random bit generator (RBG) successfully passes the National Institute of Standards and Technology (NIST) statistical tests, and it is then successfully applied to the problem of image encryption. Full article
(This article belongs to the Special Issue Nonlinear Dynamics and Entropy of Complex Systems with Hidden and Self-Excited Attractors II)
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<p>Examples of fuzzy trigonometric numbers for <math display="inline"><semantics> <mrow> <mi>z</mi> <mo>=</mo> <mn>0</mn> <mo>,</mo> <mn>0.3</mn> <mo>,</mo> <mn>0.5</mn> <mo>,</mo> <mn>1</mn> </mrow> </semantics></math>.</p> Full article ">Figure 2
<p>Bifurcation diagram of the logistic map.</p> Full article ">Figure 3
<p>Diagram of Lyapunov exponent of the Logistics map.</p> Full article ">Figure 4
<p>Bifurcation diagram of <span class="html-italic">x</span> versus the bifurcation parameter <span class="html-italic">r</span> and corresponding diagram of Lyapunov exponent for <math display="inline"><semantics> <mrow> <mi>z</mi> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math> (<b>left</b>) and <math display="inline"><semantics> <mrow> <mi>z</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math> (<b>right</b>).</p> Full article ">Figure 5
<p>Bifurcation diagram of <span class="html-italic">x</span> versus the bifurcation parameter <span class="html-italic">r</span> and corresponding diagram of Lyapunov exponent for <math display="inline"><semantics> <mrow> <mi>z</mi> <mo>=</mo> <mn>0.1</mn> </mrow> </semantics></math>.</p> Full article ">Figure 6
<p>Bifurcation diagram of <span class="html-italic">x</span> versus the bifurcation parameter <span class="html-italic">r</span> and corresponding diagram of Lyapunov exponent for <math display="inline"><semantics> <mrow> <mi>z</mi> <mo>=</mo> <mn>0.3</mn> </mrow> </semantics></math>.</p> Full article ">Figure 7
<p>Bifurcation diagram of <span class="html-italic">x</span> versus the bifurcation parameter <span class="html-italic">r</span> and corresponding diagram of Lyapunov exponent for <math display="inline"><semantics> <mrow> <mi>z</mi> <mo>=</mo> <mn>0.5</mn> </mrow> </semantics></math>.</p> Full article ">Figure 8
<p>Bifurcation diagram of <span class="html-italic">x</span> versus the bifurcation parameter <span class="html-italic">r</span> and corresponding diagram of Lyapunov exponent for <math display="inline"><semantics> <mrow> <mi>z</mi> <mo>=</mo> <mn>0.7</mn> </mrow> </semantics></math>.</p> Full article ">Figure 9
<p>Diagram of Lyapunov exponents.</p> Full article ">Figure 10
<p>Bifurcation diagram of <span class="html-italic">x</span> versus the bifurcation parameter <span class="html-italic">z</span> and corresponding diagram of Lyapunov exponent for <math display="inline"><semantics> <mrow> <mi>r</mi> <mo>=</mo> <mn>1.4</mn> </mrow> </semantics></math>.</p> Full article ">Figure 11
<p>Bifurcation diagram of <span class="html-italic">x</span> versus the bifurcation parameter <span class="html-italic">z</span> and corresponding diagram of Lyapunov exponent for <math display="inline"><semantics> <mrow> <mi>r</mi> <mo>=</mo> <mn>2.5</mn> </mrow> </semantics></math>.</p> Full article ">Figure 12
<p>Bifurcation diagram of <span class="html-italic">x</span> versus the bifurcation parameter <span class="html-italic">z</span> and corresponding diagram of Lyapunov exponent for <math display="inline"><semantics> <mrow> <mi>r</mi> <mo>=</mo> <mn>3</mn> </mrow> </semantics></math>.</p> Full article ">Figure 13
<p>Bifurcation diagram of <span class="html-italic">x</span> versus the bifurcation parameter <span class="html-italic">z</span> and corresponding diagram of Lyapunov exponent for <math display="inline"><semantics> <mrow> <mi>r</mi> <mo>=</mo> <mn>3.4</mn> </mrow> </semantics></math>.</p> Full article ">Figure 14
<p>Bifurcation diagram of <span class="html-italic">x</span> versus the bifurcation parameter <span class="html-italic">z</span> and corresponding diagram of Lyapunov exponent for <math display="inline"><semantics> <mrow> <mi>r</mi> <mo>=</mo> <mn>3.87</mn> </mrow> </semantics></math>.</p> Full article ">Figure 15
<p>Bifurcation diagram of <span class="html-italic">x</span> versus the bifurcation parameter <span class="html-italic">z</span> and corresponding diagram of Lyapunov exponent for <math display="inline"><semantics> <mrow> <mi>r</mi> <mo>=</mo> <mn>4</mn> </mrow> </semantics></math>.</p> Full article ">Figure 16
<p>Bifurcation diagram of (<a href="#FD4-entropy-22-00474" class="html-disp-formula">4</a>) with respect to parameter <span class="html-italic">z</span> for <math display="inline"><semantics> <mrow> <mi>r</mi> <mo>=</mo> <mn>3.98</mn> </mrow> </semantics></math>.</p> Full article ">Figure 17
<p>Bifurcation diagram of (<a href="#FD4-entropy-22-00474" class="html-disp-formula">4</a>) with respect to parameter <span class="html-italic">z</span> for <math display="inline"><semantics> <mrow> <mi>r</mi> <mo>=</mo> <mn>1.4</mn> </mrow> </semantics></math>.</p> Full article ">Figure 18
<p>Sensitivity to initial conditions and parameter changes for (<b>a</b>) different initial conditions <math display="inline"><semantics> <mrow> <mo>(</mo> <mi>r</mi> <mo>=</mo> <mn>4</mn> <mo>,</mo> <mi>z</mi> <mo>=</mo> <mn>0.5</mn> <mo>)</mo> </mrow> </semantics></math>, (<b>b</b>) different <span class="html-italic">z</span>, <math display="inline"><semantics> <mrow> <mo>(</mo> <msub> <mi>x</mi> <mn>0</mn> </msub> <mo>=</mo> <msub> <mi>y</mi> <mn>0</mn> </msub> <mo>=</mo> <mn>0.1</mn> <mo>,</mo> <mi>r</mi> <mo>=</mo> <mn>4</mn> <mo>)</mo> </mrow> </semantics></math>, and (<b>c</b>) different <span class="html-italic">r</span>, <math display="inline"><semantics> <mrow> <mo>(</mo> <msub> <mi>x</mi> <mn>0</mn> </msub> <mo>=</mo> <msub> <mi>y</mi> <mn>0</mn> </msub> <mo>=</mo> <mn>0.1</mn> <mo>,</mo> <mi>z</mi> <mo>=</mo> <mn>0.5</mn> <mo>)</mo> </mrow> </semantics></math>.</p> Full article ">Figure 19
<p>Auto-correlation and cross-correlation of the proposed RBG, for <math display="inline"><semantics> <mrow> <mo>(</mo> <mi>r</mi> <mo>=</mo> <mn>4</mn> <mo>,</mo> <mi>z</mi> <mo>=</mo> <mn>0.5</mn> <mo>)</mo> </mrow> </semantics></math>.</p> Full article ">Figure 20
<p>Occurrence of 1’s in the sequence the proposed PRBG, for <math display="inline"><semantics> <mrow> <mo>(</mo> <mi>r</mi> <mo>=</mo> <mn>4</mn> <mo>,</mo> <mi>z</mi> <mo>=</mo> <mn>0.5</mn> <mo>)</mo> </mrow> </semantics></math>.</p> Full article ">Figure 21
<p>(<b>a</b>) Original image, (<b>b</b>) encrypted, and (<b>c</b>) decrypted.</p> Full article ">Figure 22
<p>Histograms of the plain and encrypted image.</p> Full article ">Figure 23
<p>Correlation analysis of two (<b>a</b>) horizontal, (<b>b</b>) vertical and (<b>c</b>) diagonal adjacent pixels for the original (<b>left</b>) and encrypted (<b>right</b>) image.</p> Full article ">Figure 23 Cont.
<p>Correlation analysis of two (<b>a</b>) horizontal, (<b>b</b>) vertical and (<b>c</b>) diagonal adjacent pixels for the original (<b>left</b>) and encrypted (<b>right</b>) image.</p> Full article ">
32 pages, 16399 KiB  
Article
Fractional-Order Chaotic Memory with Wideband Constant Phase Elements
by Jiri Petrzela
Entropy 2020, 22(4), 422; https://doi.org/10.3390/e22040422 - 9 Apr 2020
Cited by 16 | Viewed by 3166
Abstract
This paper provides readers with three partial results that are mutually connected. Firstly, the gallery of the so-called constant phase elements (CPE) dedicated for the wideband applications is presented. CPEs are calculated for 9° (decimal orders) and 10° phase steps including ¼, ½, [...] Read more.
This paper provides readers with three partial results that are mutually connected. Firstly, the gallery of the so-called constant phase elements (CPE) dedicated for the wideband applications is presented. CPEs are calculated for 9° (decimal orders) and 10° phase steps including ¼, ½, and ¾ orders, which are the most used mathematical orders between zero and one in practice. For each phase shift, all necessary numerical values to design fully passive RC ladder two-terminal circuits are provided. Individual CPEs are easily distinguishable because of a very high accuracy; maximal phase error is less than 1.5° in wide frequency range beginning with 3 Hz and ending with 1 MHz. Secondly, dynamics of ternary memory composed by a series connection of two resonant tunneling diodes is investigated and, consequently, a robust chaotic behavior is discovered and reported. Finally, CPEs are directly used for realization of fractional-order (FO) ternary memory as lumped chaotic oscillator. Existence of structurally stable strange attractors for different orders is proved, both by numerical analyzed and experimental measurement. Full article
(This article belongs to the Special Issue Nonlinear Dynamics and Entropy of Complex Systems with Hidden and Self-Excited Attractors II)
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Figure 1

Figure 1
<p>Basic network structures of fully passive ladder circuits dedicated for approximation of CPE: (<b>a</b>) series-parallel RC, (<b>b</b>) parallel-series RC, (<b>c</b>) parallel-series RL, (<b>d</b>) series-parallel RL.</p> Full article ">Figure 2
<p>Numerical verification of designed wideband CPEs for all orders considered in this paper. Locations of zeroes and poles on frequency axis of CPE considered as admittance two-terminal device, module (red and blue) and phase (brown and green) frequency response, absolute error of first (red) and second (blue) type of RC approximation circuit: (<b>a</b>) α = 1/10, (<b>b</b>) α = 1/9, (<b>c</b>) α = 1/5, and (<b>d</b>) α = 2/9.</p> Full article ">Figure 3
<p>Numerical verification of designed wideband CPEs for all orders considered in this paper, continuation of the previous figure: (<b>a</b>) α = 1/4, (<b>b</b>) α = 3/10, (<b>c</b>) α = 1/3, (<b>d</b>) α = 2/5, (<b>e</b>) α = 4/9, (<b>f</b>) α = 1/2, (<b>g</b>) α = 5/9, (<b>h</b>) α = 3/5, (<b>i</b>) α = 2/3, and (<b>j</b>) α = 7/10.</p> Full article ">Figure 4
<p>Numerical verification of designed wideband CPEs for all orders considered in this paper, continuation of the previous figure: (<b>a</b>) α = 3/4, (<b>b</b>) α = 7/9, (<b>c</b>) α = 4/5, (<b>d</b>) α = 8/9, and (<b>e</b>) α = 9/10.</p> Full article ">Figure 5
<p>Polar plots of complex frequency responses of designed CPEs; series-parallel RC structures.</p> Full article ">Figure 6
<p>Polar plots of complex frequency responses of designed CPEs; parallel-series RC structures.</p> Full article ">Figure 7
<p>Different structures of the analyzed memory: (<b>a</b>) principal concept, (<b>b</b>) high frequency model, (<b>c</b>) electronic circuit after transformation of AVC of both RTDs toward origin.</p> Full article ">Figure 8
<p>Localization of attractors, fixed points (black dots) and PWL functions for memory having different values of transconductance slope <span class="html-italic">g</span><sup>2</sup><sub>outer</sub>: <span class="html-italic">g</span><sup>2</sup><sub>outer</sub> = 14 S (upper row), <span class="html-italic">g</span><sup>2</sup><sub>outer</sub> = 15 S (middle row) and <span class="html-italic">g</span><sup>2</sup><sub>outer</sub> = 18 S (lower row), initial conditions are set to points: <b>x</b><sub>0</sub> = (0, 0.1, 0)<sup>T</sup> (red), <b>x</b><sub>0</sub> = (0, −0.1, 0)<sup>T</sup> (blue), <b>x</b><sub>0</sub> = (−0.5, 0.1, 0)<sup>T</sup> (green) and <b>x</b><sub>0</sub> = (0.5, −0.1, 0)<sup>T</sup> (brown).</p> Full article ">Figure 9
<p>3D visualization of the mutual geometrical relations between calculated chaotic attractors: mirrored single-spirals (blue and green), mirrored funnels (orange and red), double-scroll (brown). Individual plots: (<b>a</b>) Poincaré section defined by plane <span class="html-italic">z</span> = 0, (<b>b</b>) perspective views on strange attractors, (<b>c</b>) state space rotation used for the best visualization of presented strange attractors and its separation into segments, (<b>d</b>) sensitivity of both single-scroll attractors to tiny changes of the initial conditions—black dots represent fixed points. See text for further clarification.</p> Full article ">Figure 10
<p>BA of analyzed memory with basic set of parameters (<span class="html-italic">g</span><sup>2</sup><sub>outer</sub> = 18 S) leading to the separated single-spiral attractors, sandwiched horizontal slices of state space defined by the following planes: (<b>a</b>) <span class="html-italic">z</span><sub>0</sub> = 0, (<b>b</b>) <span class="html-italic">z</span><sub>0</sub> = 1, (<b>c</b>) <span class="html-italic">z</span><sub>0</sub> = 1.5, (<b>d</b>) <span class="html-italic">z</span><sub>0</sub> = 2, (<b>e</b>) <span class="html-italic">z</span><sub>0</sub> = 2.5, (<b>f</b>) <span class="html-italic">z</span><sub>0</sub> = 4, (<b>g</b>) <span class="html-italic">z</span><sub>0</sub> = 5, and (<b>h</b>) <span class="html-italic">z</span><sub>0</sub> = 6.</p> Full article ">Figure 11
<p>BA of analyzed memory with basic set of parameters (<span class="html-italic">g</span><sup>2</sup><sub>outer</sub> = 20 S) leading to the separated single-spiral attractors, horizontal slices of state space defined by the planes: (<b>a</b>) <span class="html-italic">z</span><sub>0</sub> = 0, (<b>b</b>) <span class="html-italic">z</span><sub>0</sub> = 1, (<b>c</b>) <span class="html-italic">z</span><sub>0</sub> = 1.5, (<b>d</b>) <span class="html-italic">z</span><sub>0</sub> = 2, (<b>e</b>) <span class="html-italic">z</span><sub>0</sub> = 3, (<b>f</b>) <span class="html-italic">z</span><sub>0</sub> = 4, (<b>g</b>) <span class="html-italic">z</span><sub>0</sub> = 5, and (<b>h</b>) <span class="html-italic">z</span><sub>0</sub> = 6.</p> Full article ">Figure 12
<p>Complete analog circuitry realization of ternary memory with real CPEs approximated by passive RC network in function as FO capacitors, total mathematical order of this circuit is 17.</p> Full article ">Figure 13
<p>Approximate entropy calculated for memory with two CPEs of different orders <span class="html-italic">α</span> and <span class="html-italic">β</span>, associated impedance norms <span class="html-italic">ξ</span><sub>1</sub>, <span class="html-italic">ξ</span><sub>2</sub> and threshold <span class="html-italic">r</span>: (<b>a</b>) <span class="html-italic">α</span> = <span class="html-italic">β</span> = 9/10, <span class="html-italic">ξ</span><sub>1</sub> = 1, (<b>b</b>) <span class="html-italic">α</span> = <span class="html-italic">β</span> = 9/10, <span class="html-italic">ξ</span><sub>1</sub> = 2, (<b>c</b>) <span class="html-italic">α</span> = <span class="html-italic">β</span> = 9/10, <span class="html-italic">ξ</span><sub>1</sub> = 3, (<b>d</b>) <span class="html-italic">α</span> = <span class="html-italic">β</span> = 9/10, <span class="html-italic">ξ</span><sub>1</sub> = 4, (<b>e</b>) <span class="html-italic">α</span> = <span class="html-italic">β</span> = 9/10, <span class="html-italic">ξ</span><sub>1</sub> = 5, (<b>f</b>) <span class="html-italic">α</span> = <span class="html-italic">β</span> = 9/10, <span class="html-italic">ξ</span><sub>1</sub> = 6, (<b>g</b>) <span class="html-italic">α</span> = 9/10, <span class="html-italic">β</span> = 8/9, <span class="html-italic">ξ</span><sub>1</sub> = 1, (<b>h</b>) <span class="html-italic">α</span> = 9/10, <span class="html-italic">β</span> = 8/9, <span class="html-italic">ξ</span><sub>1</sub> = 2, (<b>i</b>) <span class="html-italic">α</span> = 9/10, <span class="html-italic">β</span> = 8/9, <span class="html-italic">ξ</span><sub>1</sub> = 3, (<b>j</b>) <span class="html-italic">α</span> = 9/10, <span class="html-italic">β</span> = 8/9, <span class="html-italic">ξ</span><sub>1</sub> = 4, (<b>k</b>) <span class="html-italic">α</span> = 9/10, <span class="html-italic">β</span> = 8/9, <span class="html-italic">ξ</span><sub>1</sub> = 5, (<b>l</b>) <span class="html-italic">α</span> = 9/10, <span class="html-italic">β</span> = 8/9, <span class="html-italic">ξ</span><sub>1</sub> = 6, (<b>m</b>) <span class="html-italic">α</span> = 8/9, <span class="html-italic">β</span> = 9/10, <span class="html-italic">ξ</span><sub>1</sub> = 1, (<b>n</b>) <span class="html-italic">α</span> = 8/9, <span class="html-italic">β</span> = 9/10, <span class="html-italic">ξ</span><sub>1</sub> = 2, (<b>o</b>) <span class="html-italic">α</span> = 8/9, <span class="html-italic">β</span> = 9/10, <span class="html-italic">ξ</span><sub>1</sub> = 3, (<b>p</b>) <span class="html-italic">α</span> = 8/9, <span class="html-italic">β</span> = 9/10, <span class="html-italic">ξ</span><sub>1</sub> = 4, (<b>q</b>) <span class="html-italic">α</span> = 8/9, <span class="html-italic">β</span> = 9/10, <span class="html-italic">ξ</span><sub>1</sub> = 5, (<b>r</b>) <span class="html-italic">α</span> = 8/9, <span class="html-italic">β</span> = 9/10, <span class="html-italic">ξ</span><sub>1</sub> = 6.</p> Full article ">Figure 14
<p>Approximate entropy calculated for memory with two CPEs of equivalent orders <span class="html-italic">α</span> = <span class="html-italic">β</span>, associated impedance norms <span class="html-italic">ξ</span><sub>1</sub>, <span class="html-italic">ξ</span><sub>2</sub> and threshold <span class="html-italic">r</span>: (<b>a</b>) <span class="html-italic">α</span> = <span class="html-italic">β</span> = 8/9, <span class="html-italic">ξ</span><sub>1</sub> = 1, (<b>b</b>) <span class="html-italic">α</span> = <span class="html-italic">β</span> = 8/9, <span class="html-italic">ξ</span><sub>1</sub> = 2, (<b>c</b>) <span class="html-italic">α</span> = <span class="html-italic">β</span> = 8/9, <span class="html-italic">ξ</span><sub>1</sub> = 3, (<b>d</b>) <span class="html-italic">α</span> = <span class="html-italic">β</span> = 8/9, <span class="html-italic">ξ</span><sub>1</sub> = 4, (<b>e</b>) <span class="html-italic">α</span> = <span class="html-italic">β</span> = 8/9, <span class="html-italic">ξ</span><sub>1</sub> = 5, (<b>f</b>) <span class="html-italic">α</span> = <span class="html-italic">β</span> = 8/9, <span class="html-italic">ξ</span><sub>1</sub> = 6, (<b>g</b>) <span class="html-italic">α</span> = <span class="html-italic">β</span> = 4/5, <span class="html-italic">ξ</span><sub>1</sub> = 1, (<b>h</b>) <span class="html-italic">α</span> = <span class="html-italic">β</span> = 4/5, <span class="html-italic">ξ</span><sub>1</sub> = 2, (<b>i</b>) <span class="html-italic">α</span> = <span class="html-italic">β</span> = 4/5, <span class="html-italic">ξ</span><sub>1</sub> = 3, (<b>j</b>) <span class="html-italic">α</span> = <span class="html-italic">β</span> = 4/5, <span class="html-italic">ξ</span><sub>1</sub> = 4, (<b>k</b>) <span class="html-italic">α</span> = <span class="html-italic">β</span> = 4/5, <span class="html-italic">ξ</span><sub>1</sub> = 5, (<b>l</b>) <span class="html-italic">α</span> = <span class="html-italic">β</span> = 4/5, <span class="html-italic">ξ</span><sub>1</sub> = 6, (<b>m</b>) <span class="html-italic">α</span> = <span class="html-italic">β</span> = 7/9, <span class="html-italic">ξ</span><sub>1</sub> = 1, (<b>n</b>) <span class="html-italic">α</span> = <span class="html-italic">β</span> = 7/9, <span class="html-italic">ξ</span><sub>1</sub> = 2, (<b>o</b>) <span class="html-italic">α</span> = <span class="html-italic">β</span> = 7/9, <span class="html-italic">ξ</span><sub>1</sub> = 3, (<b>p</b>) <span class="html-italic">α</span> = <span class="html-italic">β</span> = 7/9, <span class="html-italic">ξ</span><sub>1</sub> = 4, (<b>q</b>) <span class="html-italic">α</span> = <span class="html-italic">β</span> = 7/9, <span class="html-italic">ξ</span><sub>1</sub> = 5, (<b>r</b>) <span class="html-italic">α</span> = <span class="html-italic">β</span> = 7/9, <span class="html-italic">ξ</span><sub>1</sub> = 6.</p> Full article ">Figure 15
<p>Topographically scaled surface-contour plot of LLE as a function of slopes of both PWL functions; high resolution is achieved by using the uniform parameter step 0.01.</p> Full article ">Figure 16
<p>Orcad Pspice circuit simulation of IO memory in robust chaotic regime; plane projection: (<b>a</b>) <span class="html-italic">v<sub>C</sub></span><sub>1</sub> vs. <span class="html-italic">i<sub>L</sub></span>, (<b>b</b>) <span class="html-italic">v<sub>C</sub></span><sub>2</sub> vs. <span class="html-italic">i<sub>L</sub></span>, and (<b>c</b>) <span class="html-italic">v<sub>C</sub></span><sub>2</sub> vs. <span class="html-italic">v<sub>C</sub></span><sub>1</sub> together with the experiment; frequency spectrum of generated signal: (<b>d</b>) <span class="html-italic">v<sub>C</sub></span><sub>1</sub>, (<b>e</b>) <span class="html-italic">v<sub>C</sub></span><sub>2</sub>, and (<b>f</b>) <span class="html-italic">i<sub>L</sub></span>.</p> Full article ">Figure 17
<p>Orcad Pspice circuit simulation of FO memory of total mathematical order 2.7 working in robust chaotic regime; plane projection: (<b>a</b>) <span class="html-italic">v<sub>C</sub></span><sub>1</sub> vs. <span class="html-italic">i<sub>L</sub></span>, (<b>b</b>) <span class="html-italic">v<sub>C</sub></span><sub>2</sub> vs. <span class="html-italic">i<sub>L</sub></span>, and (<b>c</b>) <span class="html-italic">v<sub>C</sub></span><sub>2</sub> vs. <span class="html-italic">v<sub>C</sub></span><sub>1</sub> along with oscilloscope screenshot; frequency spectrum of generated signal: (<b>d</b>) <span class="html-italic">v<sub>C</sub></span><sub>1</sub>, (<b>e</b>) <span class="html-italic">v<sub>C</sub></span><sub>2</sub>, and (<b>f</b>) <span class="html-italic">i<sub>L</sub></span>.</p> Full article ">Figure 18
<p>Orcad Pspice circuit simulation of FO memory of total mathematical order 2.4 working in robust chaotic regime; plane projection: (<b>a</b>) <span class="html-italic">v<sub>C</sub></span><sub>1</sub> vs. <span class="html-italic">i<sub>L</sub></span>, (<b>b</b>) <span class="html-italic">v<sub>C</sub></span><sub>2</sub> vs. <span class="html-italic">i<sub>L</sub></span>, and (<b>c</b>) <span class="html-italic">v<sub>C</sub></span><sub>2</sub> vs. <span class="html-italic">v<sub>C</sub></span><sub>1</sub> with experimental verification; frequency spectrum of generated signal: (<b>d</b>) <span class="html-italic">v<sub>C</sub></span><sub>1</sub>, (<b>e</b>) <span class="html-italic">v<sub>C</sub></span><sub>2</sub>, and (<b>f</b>) <span class="html-italic">i<sub>L</sub></span>.</p> Full article ">Figure 19
<p>Orcad Pspice circuit simulation of FO memory of total mathematical order 2.4 working in the operation of a double-spiral generation; plane projection: (<b>a</b>) <span class="html-italic">v<sub>C</sub></span><sub>1</sub> vs. <span class="html-italic">i<sub>L</sub></span>, (<b>b</b>) <span class="html-italic">v<sub>C</sub></span><sub>2</sub> vs. <span class="html-italic">i<sub>L</sub></span> and (<b>c</b>) <span class="html-italic">v<sub>C</sub></span><sub>2</sub> vs. <span class="html-italic">v<sub>C</sub></span><sub>1</sub>; frequency spectrum of generated signal: (<b>d</b>) <span class="html-italic">v<sub>C</sub></span><sub>1</sub>, (<b>e</b>) <span class="html-italic">v<sub>C</sub></span><sub>2</sub>, and (<b>f</b>) <span class="html-italic">i<sub>L</sub></span>.</p> Full article ">
8 pages, 4645 KiB  
Article
Investigation of Early Warning Indexes in a Three-Dimensional Chaotic System with Zero Eigenvalues
by Lianyu Chen, Fahimeh Nazarimehr, Sajad Jafari, Esteban Tlelo-Cuautle and Iqtadar Hussain
Entropy 2020, 22(3), 341; https://doi.org/10.3390/e22030341 - 17 Mar 2020
Cited by 7 | Viewed by 2555
Abstract
A rare three-dimensional chaotic system with all eigenvalues equal to zero is proposed, and its dynamical properties are investigated. The chaotic system has one equilibrium point at the origin. Numerical analysis shows that the equilibrium point is unstable. Bifurcation analysis of the system [...] Read more.
A rare three-dimensional chaotic system with all eigenvalues equal to zero is proposed, and its dynamical properties are investigated. The chaotic system has one equilibrium point at the origin. Numerical analysis shows that the equilibrium point is unstable. Bifurcation analysis of the system shows various dynamics in a period-doubling route to chaos. We highlight that from the evaluation of the entropy, bifurcation points can be predicted by identifying early warning signals. In this manner, bifurcation points of the system are analyzed using Shannon and Kolmogorov-Sinai entropy. The results are compared with Lyapunov exponents. Full article
(This article belongs to the Special Issue Nonlinear Dynamics and Entropy of Complex Systems with Hidden and Self-Excited Attractors II)
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Figure 1

Figure 1
<p>Time-series and phase-space projections of the chaotic attractor System (1), using <math display="inline"><semantics> <mrow> <mi>a</mi> <mo>=</mo> <mn>0.37</mn> </mrow> </semantics></math> and initial conditions <math display="inline"><semantics> <mrow> <mrow> <mo>(</mo> <mrow> <msub> <mi>x</mi> <mn>0</mn> </msub> <mo>,</mo> <msub> <mi>y</mi> <mn>0</mn> </msub> <mo>,</mo> <msub> <mi>z</mi> <mn>0</mn> </msub> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <mo>−</mo> <mn>10.9</mn> <mo>,</mo> <mo>−</mo> <mn>12.04</mn> <mo>,</mo> <mn>33.68</mn> </mrow> <mo>)</mo> </mrow> </mrow> </semantics></math>. (<b>a</b>) time-series; (<b>b</b>) projection in <math display="inline"><semantics> <mrow> <mi>X</mi> <mo>−</mo> <mi>Y</mi> </mrow> </semantics></math> plane; (<b>c</b>) projection in <math display="inline"><semantics> <mrow> <mi>Y</mi> <mo>−</mo> <mi>Z</mi> </mrow> </semantics></math> plane; and (<b>d</b>) projection in <math display="inline"><semantics> <mrow> <mi>t</mi> <mi>h</mi> <mi>e</mi> <mo> </mo> <mi>X</mi> <mo>−</mo> <mi>Z</mi> </mrow> </semantics></math> plane. It can be seen that the attractor is symmetric around the line <math display="inline"><semantics> <mrow> <mi>y</mi> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math>.</p> Full article ">Figure 2
<p>Bifurcation diagram of System (1) with respect to changing parameter <math display="inline"><semantics> <mi>a</mi> </semantics></math> and forward continuation method; (<b>a</b>) bifurcation diagram of peak values of <math display="inline"><semantics> <mi>x</mi> </semantics></math> variable; (<b>b</b>) bifurcation diagram of peak values of <math display="inline"><semantics> <mi>y</mi> </semantics></math> variable; (<b>c</b>) bifurcation diagram of peak values of <math display="inline"><semantics> <mi>z</mi> </semantics></math> variable.</p> Full article ">Figure 2 Cont.
<p>Bifurcation diagram of System (1) with respect to changing parameter <math display="inline"><semantics> <mi>a</mi> </semantics></math> and forward continuation method; (<b>a</b>) bifurcation diagram of peak values of <math display="inline"><semantics> <mi>x</mi> </semantics></math> variable; (<b>b</b>) bifurcation diagram of peak values of <math display="inline"><semantics> <mi>y</mi> </semantics></math> variable; (<b>c</b>) bifurcation diagram of peak values of <math display="inline"><semantics> <mi>z</mi> </semantics></math> variable.</p> Full article ">Figure 3
<p>The entropy of the chaotic system (1), which is calculated using the peak values of the state variable <span class="html-italic">y</span>. (<b>a</b>) Shannon entropy for changing parameter <math display="inline"><semantics> <mi>a</mi> </semantics></math>; (<b>b</b>) Kolmogorov-Sinai entropy for changing parameter <math display="inline"><semantics> <mi>a</mi> </semantics></math>.</p> Full article ">Figure 4
<p>The extracted period of the chaotic system (1) for changing parameter (a).</p> Full article ">Figure 5
<p>The mean value of the entropies in all vectors of cycles by changing parameter <math display="inline"><semantics> <mi>a</mi> </semantics></math>; (<b>a</b>) Shannon entropy; (<b>b</b>) Kolmogorov-Sinai entropy.</p> Full article ">Figure 6
<p>Lyapunov exponents of the chaotic system (1) using the forward continuation method; (<b>a</b>) three Lyapunov exponents; (<b>b</b>) the two largest Lyapunov exponents.</p> Full article ">
15 pages, 4325 KiB  
Article
Synchronization of a Non-Equilibrium Four-Dimensional Chaotic System Using a Disturbance-Observer-Based Adaptive Terminal Sliding Mode Control Method
by Shaojie Wang, Amin Yousefpour, Abdullahi Yusuf, Hadi Jahanshahi, Raúl Alcaraz, Shaobo He and Jesus M. Munoz-Pacheco
Entropy 2020, 22(3), 271; https://doi.org/10.3390/e22030271 - 27 Feb 2020
Cited by 36 | Viewed by 3089
Abstract
In this paper, dynamical behavior and synchronization of a non-equilibrium four-dimensional chaotic system are studied. The system only includes one constant term and has hidden attractors. Some dynamical features of the governing system, such as invariance and symmetry, the existence of attractors and [...] Read more.
In this paper, dynamical behavior and synchronization of a non-equilibrium four-dimensional chaotic system are studied. The system only includes one constant term and has hidden attractors. Some dynamical features of the governing system, such as invariance and symmetry, the existence of attractors and dissipativity, chaotic flow with a plane of equilibria, and offset boosting of the chaotic attractor, are stated and discussed and a new disturbance-observer-based adaptive terminal sliding mode control (ATSMC) method with input saturation is proposed for the control and synchronization of the chaotic system. To deal with unexpected noises, an extended Kalman filter (EKF) is implemented along with the designed controller. Through the concept of Lyapunov stability, the proposed control technique guarantees the finite time convergence of the uncertain system in the presence of disturbances and control input limits. Furthermore, to decrease the chattering phenomena, a genetic algorithm is used to optimize the controller parameters. Finally, numerical simulations are presented to demonstrate the performance of the designed control scheme in the presence of noise, disturbances, and control input saturation. Full article
(This article belongs to the Special Issue Nonlinear Dynamics and Entropy of Complex Systems with Hidden and Self-Excited Attractors II)
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Figure 1

Figure 1
<p>Projections in the stated planes with a suitable choice of parameter values. (<b>a</b>) <math display="inline"><semantics> <mrow> <mi>x</mi> <mi>y</mi> <mo>,</mo> <mo> </mo> <mi>y</mi> <mi>z</mi> <mo>,</mo> <mo> </mo> <mi>w</mi> <mi>z</mi> </mrow> </semantics></math>, and <math display="inline"><semantics> <mrow> <mi>x</mi> <mi>y</mi> <mi>z</mi> </mrow> </semantics></math> planes with initial conditions <math display="inline"><semantics> <mrow> <mrow> <mo>[</mo> <mrow> <mi>x</mi> <mrow> <mo>(</mo> <mn>0</mn> <mo>)</mo> </mrow> <mo>,</mo> <mi>y</mi> <mrow> <mo>(</mo> <mn>0</mn> <mo>)</mo> </mrow> <mo>,</mo> <mi>z</mi> <mrow> <mo>(</mo> <mn>0</mn> <mo>)</mo> </mrow> <mo>,</mo> <mi>w</mi> <mrow> <mo>(</mo> <mn>0</mn> <mo>)</mo> </mrow> </mrow> <mo>]</mo> </mrow> <mo>=</mo> <mrow> <mo>[</mo> <mrow> <mn>1.5</mn> <mo>,</mo> <mn>0</mn> <mo>,</mo> <mo>−</mo> <mn>1</mn> <mo>,</mo> <mo>−</mo> <mn>0.3</mn> </mrow> <mo>]</mo> </mrow> </mrow> </semantics></math>. (<b>b</b>) <math display="inline"><semantics> <mrow> <mi>x</mi> <mi>y</mi> <mo>,</mo> <mo> </mo> <mi>y</mi> <mi>z</mi> <mo>,</mo> <mo> </mo> <mi>w</mi> <mi>z</mi> </mrow> </semantics></math>, and <math display="inline"><semantics> <mrow> <mi>x</mi> <mi>y</mi> <mi>z</mi> </mrow> </semantics></math> planes with initial conditions <math display="inline"><semantics> <mrow> <mrow> <mo>[</mo> <mrow> <mi>x</mi> <mrow> <mo>(</mo> <mn>0</mn> <mo>)</mo> </mrow> <mo>,</mo> <mi>y</mi> <mrow> <mo>(</mo> <mn>0</mn> <mo>)</mo> </mrow> <mo>,</mo> <mi>z</mi> <mrow> <mo>(</mo> <mn>0</mn> <mo>)</mo> </mrow> <mo>,</mo> <mi>w</mi> <mrow> <mo>(</mo> <mn>0</mn> <mo>)</mo> </mrow> </mrow> <mo>]</mo> </mrow> <mo>=</mo> <mrow> <mo>[</mo> <mrow> <mn>3.5</mn> <mo>,</mo> <mo>−</mo> <mn>1</mn> <mo>,</mo> <mn>0</mn> <mo>,</mo> <mn>0.5</mn> </mrow> <mo>]</mo> </mrow> </mrow> </semantics></math>.</p> Full article ">Figure 2
<p>A block diagram describing the proposed disturbance-observer-based adaptive terminal sliding mode control (ATSMC) technique with the extended Kalman filter (EKF) algorithm.</p> Full article ">Figure 3
<p>System states with disturbance-observer-based ATSMC with EKF (<span class="html-italic">Tstart</span> = 10).</p> Full article ">Figure 4
<p>Time history of the control input for the proposed control scheme (<span class="html-italic">Tstart</span> = 10).</p> Full article ">Figure 5
<p>Synchronization results for the chaotic system using disturbance-observer-based ATSMC with the EKF (<span class="html-italic">Tstart</span> = 10).</p> Full article ">Figure 6
<p>Synchronization errors in the chaotic system using disturbance-observer-based ATSMC with the EKF (<span class="html-italic">Tstart</span> = 10).</p> Full article ">Figure 7
<p>Control input for synchronization of the chaotic system using disturbance-observer-based ATSMC with the EKF (<span class="html-italic">Tstart</span> = 10).</p> Full article ">
11 pages, 547 KiB  
Article
Dynamic Effects Arise Due to Consumers’ Preferences Depending on Past Choices
by Sameh S. Askar and A. Al-khedhairi
Entropy 2020, 22(2), 173; https://doi.org/10.3390/e22020173 - 3 Feb 2020
Cited by 4 | Viewed by 2036
Abstract
We analyzed a dynamic duopoly game where players adopt specific preferences. These preferences are derived from Cobb–Douglas utility function with the assumption that they depend on past choices. For this paper, we investigated two possible cases for the suggested game. The first case [...] Read more.
We analyzed a dynamic duopoly game where players adopt specific preferences. These preferences are derived from Cobb–Douglas utility function with the assumption that they depend on past choices. For this paper, we investigated two possible cases for the suggested game. The first case considers only focusing on the action done by one player. This action reduces the game’s map to a one-dimensional map, which is the logistic map. Using analytical and numerical simulation, the stability of fixed points of this map is studied. In the second case, we focus on the actions applied by both players. The fixed points, in this case, are calculated, and their stability is discussed. The conditions of stability are provided in terms of the game’s parameters. Numerical simulation is carried out to give local and global investigations of the chaotic behavior of the game’s map. In addition, we use a statistical measure, such as entropy, to get more evidences on the regularity and predictability of time series associated with this case. Full article
(This article belongs to the Special Issue Nonlinear Dynamics and Entropy of Complex Systems with Hidden and Self-Excited Attractors II)
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Figure 1
<p>(<b>a</b>) The cobweb diagram for the stable fixed point; (<b>b</b>) 2D-Bifurcation diagram for the parameters <span class="html-italic">m</span> and <span class="html-italic">b</span>. (<b>c</b>) The cobweb diagram for the unstable fixed point. (<b>d</b>) Bifurcation diagram when varying the parameter <span class="html-italic">m</span>. (<b>e</b>) Bifurcation diagram when varying the parameter <span class="html-italic">b</span>. (<b>f</b>) Maximum Lyapunov exponents of <span class="html-italic">m</span> and <span class="html-italic">b</span>.</p> Full article ">Figure 2
<p>(<b>a</b>) Bifurcation diagram for <math display="inline"><semantics> <msub> <mi>x</mi> <mn>1</mn> </msub> </semantics></math> when varying <span class="html-italic">m</span>. (<b>b</b>) Bifurcation diagram for <math display="inline"><semantics> <msub> <mi>x</mi> <mn>2</mn> </msub> </semantics></math> when varying <span class="html-italic">m</span>. (<b>c</b>) Bifurcation diagram for <math display="inline"><semantics> <msub> <mi>x</mi> <mn>1</mn> </msub> </semantics></math> when varying <span class="html-italic">b</span>. (<b>d</b>) Bifurcation diagram for <math display="inline"><semantics> <msub> <mi>x</mi> <mn>2</mn> </msub> </semantics></math> when varying <span class="html-italic">b</span>. (<b>e</b>) 2D-Bifurcation diagram in the <math display="inline"><semantics> <mrow> <mo>(</mo> <mi>m</mi> <mo>,</mo> <mi>b</mi> <mo>)</mo> </mrow> </semantics></math>-plane for the map (<a href="#FD5-entropy-22-00173" class="html-disp-formula">5</a>). (<b>f</b>) The phase portrait of the stable fixed point <math display="inline"><semantics> <msub> <mi>e</mi> <mn>2</mn> </msub> </semantics></math>.</p> Full article ">Figure 3
<p>(<b>a</b>) The region of period 2-cycles. (<b>b</b>) The basin of attraction of period 2-cycle. (<b>c</b>) The region of period 3-cycles. (<b>d</b>) The basin of attraction of period 3-cycle. (<b>e</b>) The region of period 4-cycles. (<b>f</b>) The basin of attraction of period 4-cycle. (<b>g</b>) The region of period 5-cycles. (<b>h</b>) The basin of attraction of period 5-cycle. (<b>i</b>) Time series for the map’s variables at <math display="inline"><semantics> <mrow> <mi>m</mi> <mo>=</mo> <mn>0.7372</mn> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mi>b</mi> <mo>=</mo> <mn>0.4312</mn> </mrow> </semantics></math>. (<b>j</b>) The basin of attraction of period 5-cycle. (<b>k</b>) The region of period 6-cycles. (<b>l</b>) The basin of attraction of period 3-cycle.</p> Full article ">Figure 4
<p>(<b>a</b>) Maximum Lyapunov exponent when varying the parameter <span class="html-italic">m</span>. (<b>b</b>) Maximum Lyapunov exponent when varying the parameter <span class="html-italic">b</span>. (<b>c</b>) A two-piece chaotic attractor. (<b>d</b>) Time series for the two-piece chaotic attractor. (<b>e</b>) One piece chaotic attractor. (<b>f</b>) Time series for the one piece chaotic attractor.</p> Full article ">
15 pages, 826 KiB  
Article
Stabilization of Port Hamiltonian Chaotic Systems with Hidden Attractors by Adaptive Terminal Sliding Mode Control
by Ahmad Taher Azar and Fernando E. Serrano
Entropy 2020, 22(1), 122; https://doi.org/10.3390/e22010122 - 19 Jan 2020
Cited by 24 | Viewed by 3412
Abstract
In this study, the design of an adaptive terminal sliding mode controller for the stabilization of port Hamiltonian chaotic systems with hidden attractors is proposed. This study begins with the design methodology of a chaotic oscillator with a hidden attractor implementing the topological [...] Read more.
In this study, the design of an adaptive terminal sliding mode controller for the stabilization of port Hamiltonian chaotic systems with hidden attractors is proposed. This study begins with the design methodology of a chaotic oscillator with a hidden attractor implementing the topological framework for its respective design. With this technique it is possible to design a 2-D chaotic oscillator, which is then converted into port-Hamiltonia to track and analyze these models for the stabilization of the hidden chaotic attractors created by this analysis. Adaptive terminal sliding mode controllers (ATSMC) are built when a Hamiltonian system has a chaotic behavior and a hidden attractor is detected. A Lyapunov approach is used to formulate the adaptive device controller by creating a control law and the adaptive law, which are used online to make the system states stable while at the same time suppressing its chaotic behavior. The empirical tests obtaining the discussion and conclusions of this thesis should verify the theoretical findings. Full article
(This article belongs to the Special Issue Nonlinear Dynamics and Entropy of Complex Systems with Hidden and Self-Excited Attractors II)
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Figure 1

Figure 1
<p>Phase portrait of <math display="inline"><semantics> <msub> <mi>p</mi> <mn>1</mn> </msub> </semantics></math> and <math display="inline"><semantics> <msub> <mi>p</mi> <mn>2</mn> </msub> </semantics></math>.</p> Full article ">Figure 2
<p>Phase portrait of <math display="inline"><semantics> <mrow> <msub> <mi>x</mi> <mn>1</mn> </msub> <mo>=</mo> <msub> <mi>q</mi> <mn>1</mn> </msub> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <msub> <mi>x</mi> <mn>2</mn> </msub> <mo>=</mo> <msub> <mi>q</mi> <mn>2</mn> </msub> </mrow> </semantics></math>.</p> Full article ">Figure 3
<p>Phase portrait of <math display="inline"><semantics> <mrow> <msub> <mi>x</mi> <mn>1</mn> </msub> <mo>=</mo> <msub> <mi>q</mi> <mn>1</mn> </msub> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <msub> <mi>x</mi> <mn>2</mn> </msub> <mo>=</mo> <msub> <mi>q</mi> <mn>2</mn> </msub> </mrow> </semantics></math>.</p> Full article ">Figure 4
<p>Phase portrait of <math display="inline"><semantics> <msub> <mi>p</mi> <mn>1</mn> </msub> </semantics></math> and <math display="inline"><semantics> <msub> <mi>p</mi> <mn>2</mn> </msub> </semantics></math>.</p> Full article ">Figure 5
<p>Sliding surface <math display="inline"><semantics> <msub> <mi>σ</mi> <mn>3</mn> </msub> </semantics></math> fo the experiment 1.</p> Full article ">Figure 6
<p>Evolution in time of the variables <math display="inline"><semantics> <msub> <mi>q</mi> <mn>2</mn> </msub> </semantics></math> and <math display="inline"><semantics> <msub> <mi>p</mi> <mn>1</mn> </msub> </semantics></math>.</p> Full article ">Figure 7
<p>Input variables <math display="inline"><semantics> <msub> <mi>u</mi> <mn>2</mn> </msub> </semantics></math> and <math display="inline"><semantics> <msub> <mi>u</mi> <mn>4</mn> </msub> </semantics></math>.</p> Full article ">Figure 8
<p>Evolution in time of the gain variable <math display="inline"><semantics> <msub> <mi>k</mi> <mn>2</mn> </msub> </semantics></math>.</p> Full article ">Figure 9
<p>Evolution in time of the variables <math display="inline"><semantics> <msub> <mover accent="true"> <mi>q</mi> <mo>˙</mo> </mover> <mn>3</mn> </msub> </semantics></math> and <math display="inline"><semantics> <msub> <mover accent="true"> <mi>p</mi> <mo>˙</mo> </mover> <mn>3</mn> </msub> </semantics></math>.</p> Full article ">Figure 10
<p>Evolution in time of the input variables <math display="inline"><semantics> <msub> <mi>u</mi> <mn>3</mn> </msub> </semantics></math> and <math display="inline"><semantics> <msub> <mi>u</mi> <mn>6</mn> </msub> </semantics></math>.</p> Full article ">Figure 11
<p>Evolution in time of the gain variable <math display="inline"><semantics> <msub> <mi>k</mi> <mn>2</mn> </msub> </semantics></math>.</p> Full article ">Figure 12
<p>Evolution in time of the variable <math display="inline"><semantics> <msub> <mi>σ</mi> <mn>3</mn> </msub> </semantics></math>.</p> Full article ">
16 pages, 2255 KiB  
Article
Image Parallel Encryption Technology Based on Sequence Generator and Chaotic Measurement Matrix
by Jiayin Yu, Shiyu Guo, Xiaomeng Song, Yaqin Xie and Erfu Wang
Entropy 2020, 22(1), 76; https://doi.org/10.3390/e22010076 - 6 Jan 2020
Cited by 16 | Viewed by 3032
Abstract
In this paper, a new image encryption transmission algorithm based on the parallel mode is proposed. This algorithm aims to improve information transmission efficiency and security based on existing hardware conditions. To improve efficiency, this paper adopts the method of parallel compressed sensing [...] Read more.
In this paper, a new image encryption transmission algorithm based on the parallel mode is proposed. This algorithm aims to improve information transmission efficiency and security based on existing hardware conditions. To improve efficiency, this paper adopts the method of parallel compressed sensing to realize image transmission. Compressed sensing can perform data sampling and compression at a rate much lower than the Nyquist sampling rate. To enhance security, this algorithm combines a sequence signal generator with chaotic cryptography. The initial sensitivity of chaos, used in a measurement matrix, makes it possible to improve the security of an encryption algorithm. The cryptographic characteristics of chaotic signals can be fully utilized by the flexible digital logic circuit. Simulation experiments and analyses show that the algorithm achieves the goal of improving transmission efficiency and has the capacity to resist illegal attacks. Full article
(This article belongs to the Special Issue Nonlinear Dynamics and Entropy of Complex Systems with Hidden and Self-Excited Attractors II)
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<p>Block diagram of compressed sensing implementation.</p> Full article ">Figure 2
<p>Parallel compression sensing encryption algorithm based on sequence generator.</p> Full article ">Figure 3
<p>Circuit diagram of sequence signal generator.</p> Full article ">Figure 4
<p>Parallel sampling compression of compressed sensing process.</p> Full article ">Figure 5
<p>Results of gray image parallel compression perception encryption. (<b>a</b>) Original image, (<b>b</b>) compressed sensing encrypted image, (<b>c</b>) diffused cipher text image, (<b>d</b>) difference between (<b>b</b>) and (<b>c</b>).</p> Full article ">Figure 6
<p>Histogram of encryption and decryption image. (<b>a</b>) Original image, (<b>b</b>) cipher image, (<b>c</b>) histogram of plaintext image, (<b>d</b>) Histogram of cipher text image.</p> Full article ">Figure 7
<p>Distribution of adjacent pixels. (<b>a</b>) Plaintext horizontal adjacent pixels, (<b>b</b>) plaintext vertical adjacent pixels, (<b>c</b>) plaintext diagonal adjacent pixels, (<b>d</b>) cipher text horizontal adjacent pixels, (<b>e</b>) cipher text vertical adjacent pixels, (<b>f</b>) cipher text diagonal adjacent.</p> Full article ">Figure 8
<p>Encrypted image reconstruction. (<b>a</b>) Original image, (<b>b</b>) reconstructed image.</p> Full article ">Figure 9
<p>PSNR of reconstructed image.</p> Full article ">Figure 10
<p>Key sensitivity analysis. (<b>a</b>) Initial value change <math display="inline"><semantics> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>14</mn> </mrow> </msup> </semantics></math>, (<b>b</b>) initial value change <math display="inline"><semantics> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>15</mn> </mrow> </msup> </semantics></math>, (<b>c</b>) initial value change <math display="inline"><semantics> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>16</mn> </mrow> </msup> </semantics></math>.</p> Full article ">
19 pages, 5951 KiB  
Article
Low-Element Image Restoration Based on an Out-of-Order Elimination Algorithm
by Yaqin Xie, Jiayin Yu, Xinwu Chen, Qun Ding and Erfu Wang
Entropy 2019, 21(12), 1192; https://doi.org/10.3390/e21121192 - 4 Dec 2019
Cited by 3 | Viewed by 2374
Abstract
To reduce the consumption of receiving devices, a number of devices at the receiving end undergo low-element treatment (the number of devices at the receiving end is less than that at the transmitting ends). The underdetermined blind-source separation system is a classic low-element [...] Read more.
To reduce the consumption of receiving devices, a number of devices at the receiving end undergo low-element treatment (the number of devices at the receiving end is less than that at the transmitting ends). The underdetermined blind-source separation system is a classic low-element model at the receiving end. Blind signal extraction in an underdetermined system remains an ill-posed problem, as it is difficult to extract all the source signals. To realize fewer devices at the receiving end without information loss, this paper proposes an image restoration method for underdetermined blind-source separation based on an out-of-order elimination algorithm. Firstly, a chaotic system is used to perform hidden transmission of source signals, where the source signals can hardly be observed and confidentiality is guaranteed. Secondly, empirical mode decomposition is used to decompose and complement the missing observed signals, and the fast independent component analysis (FastICA) algorithm is used to obtain part of the source signals. Finally, all the source signals are successfully separated using the out-of-order elimination algorithm and the FastICA algorithm. The results show that the performance of the underdetermined blind separation algorithm is related to the configuration of the transceiver antenna. When the signal is 3 × 4 antenna configuration, the algorithm in this paper is superior to the comparison algorithm in signal recovery, and its separation performance is better for a lower degree of missing array elements. The end result is that the algorithms discussed in this paper can effectively and completely extract all the source signals. Full article
(This article belongs to the Special Issue Nonlinear Dynamics and Entropy of Complex Systems with Hidden and Self-Excited Attractors II)
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<p>Underdetermined state blind extraction mathematical model.</p> Full article ">Figure 2
<p>A 3D discrete chaotic schematic: (<b>a</b>) three-dimensional view of the chaotic system and (<b>b</b>) time domain response graph of x variable of the chaotic system.</p> Full article ">Figure 3
<p>A virtual receiving array model with multicomponent complement method.</p> Full article ">Figure 4
<p>Implementation of extraction and reduction.</p> Full article ">Figure 5
<p>Overall flow chart.</p> Full article ">Figure 6
<p>The image information of the source signal: (<b>a</b>) image information of the first source; (<b>b</b>) image information of the second source; (<b>c</b>) image information of the third source; and (<b>d</b>) image information of the fourth source.</p> Full article ">Figure 7
<p>The image information of the observed signal in the multicomponent complement model: (<b>a</b>) image information of the first observation signal; (<b>b</b>) image information of the second observation signal; and (<b>c</b>) image information of the third observation signal.</p> Full article ">Figure 8
<p>Statistical histograms of hiding images. (<b>a</b>)–(<b>c</b>) is the statistical histogram of the hiding image in <a href="#entropy-21-01192-f007" class="html-fig">Figure 7</a>.</p> Full article ">Figure 9
<p>Multi-component complement method first extracts signal image information. (<b>a</b>)–(<b>c</b>) three effective Lena graphs were extracted blind at the first blind extraction; (<b>d)</b> the effective Lake information was extracted blind at the first blind extraction; and (<b>e</b>) information of no interest.</p> Full article ">Figure 10
<p>Image information of the second extraction of signal in multicomponent complement method. (<b>a</b>) The second blind extraction of the remaining information of valid Peppers image information; (<b>b</b>) The second blind extraction of the remaining information of valid Cameraman image information; and (<b>c</b>) Information of no interest.</p> Full article ">Figure 11
<p>Decryption effects of 5 × 3 signals of source and receiver at different signal-to-noise ratio (SNRs).</p> Full article ">Figure 12
<p>Comparisons of (<b>a</b>) mean-square error (MSE) and (<b>b</b>) peak signal-to-noise ratio (PSNR) values when the number of source signals and receiving sensors are different.</p> Full article ">Figure 13
<p>Performance comparison of proposed algorithm and that in [<a href="#B53-entropy-21-01192" class="html-bibr">53</a>].</p> Full article ">
24 pages, 28547 KiB  
Article
A High Spectral Entropy (SE) Memristive Hidden Chaotic System with Multi-Type Quasi-Periodic and its Circuit
by Licai Liu, Chuanhong Du, Lixiu Liang and Xiefu Zhang
Entropy 2019, 21(10), 1026; https://doi.org/10.3390/e21101026 - 22 Oct 2019
Cited by 25 | Viewed by 3707
Abstract
As a new type of nonlinear electronic component, a memristor can be used in a chaotic system to increase the complexity of the system. In this paper, a flux-controlled memristor is applied to an existing chaotic system, and a novel five-dimensional chaotic system [...] Read more.
As a new type of nonlinear electronic component, a memristor can be used in a chaotic system to increase the complexity of the system. In this paper, a flux-controlled memristor is applied to an existing chaotic system, and a novel five-dimensional chaotic system with high complexity and hidden attractors is proposed. Analyzing the nonlinear characteristics of the system, we can find that the system has new chaotic attractors and many novel quasi-periodic limit cycles; the unique attractor structure of the Poincaré map also reflects the complexity and novelty of the hidden attractor for the system; the system has a very high complexity when measured through spectral entropy. In addition, under different initial conditions, the system exhibits the coexistence of chaotic attractors with different topologies, quasi-periodic limit cycles, and chaotic attractors. At the same time, an interesting transient chaos phenomenon, one kind of novel quasi-periodic, and weak chaotic hidden attractors are found. Finally, we realize the memristor model circuit and the proposed chaotic system use off-the-shelf electronic components. The experimental results of the circuit are consistent with the numerical simulation, which shows that the system is physically achievable and provides a new option for the application of memristive chaotic systems. Full article
(This article belongs to the Special Issue Nonlinear Dynamics and Entropy of Complex Systems with Hidden and Self-Excited Attractors II)
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<p>Memristor model: (<b>a</b>) the relationship of magnetic flux and charge; (<b>b</b>) the I-V characteristic curve.</p> Full article ">Figure 2
<p>3-D chaotic attractor of system (2): (<b>a</b>) <math display="inline"><semantics> <mrow> <mi>x</mi> <mo>−</mo> <mi>z</mi> <mo>−</mo> <mi>w</mi> </mrow> </semantics></math> space, (<b>b</b>) <math display="inline"><semantics> <mrow> <mi>x</mi> <mo>−</mo> <mi>y</mi> <mo>−</mo> <mi>u</mi> </mrow> </semantics></math> space, (<b>c</b>) <math display="inline"><semantics> <mrow> <mi>x</mi> <mo>−</mo> <mi>w</mi> <mo>−</mo> <mi>u</mi> </mrow> </semantics></math> space, and (<b>d</b>) <math display="inline"><semantics> <mrow> <mi>z</mi> <mo>−</mo> <mi>w</mi> <mo>−</mo> <mi>u</mi> </mrow> </semantics></math> space.</p> Full article ">Figure 3
<p>2-D chaotic attractor of system: (<b>a</b>) <math display="inline"><semantics> <mrow> <mi>y</mi> <mo>−</mo> <mi>w</mi> </mrow> </semantics></math> plane; (<b>b</b>) <math display="inline"><semantics> <mrow> <mi>z</mi> <mo>−</mo> <mi>w</mi> </mrow> </semantics></math> plane; (<b>c</b>) <math display="inline"><semantics> <mrow> <mi>w</mi> <mo>−</mo> <mi>u</mi> </mrow> </semantics></math> plane; (<b>d</b>) <math display="inline"><semantics> <mrow> <mi>x</mi> <mo>−</mo> <mi mathvariant="normal">u</mi> </mrow> </semantics></math> plane; (<b>e</b>) <math display="inline"><semantics> <mrow> <mi>z</mi> <mo>−</mo> <mi>u</mi> </mrow> </semantics></math> plane; and (<b>f</b>) <math display="inline"><semantics> <mrow> <mi mathvariant="normal">w</mi> <mo>−</mo> <mi>i</mi> </mrow> </semantics></math> plane of memristor model (1).</p> Full article ">Figure 3 Cont.
<p>2-D chaotic attractor of system: (<b>a</b>) <math display="inline"><semantics> <mrow> <mi>y</mi> <mo>−</mo> <mi>w</mi> </mrow> </semantics></math> plane; (<b>b</b>) <math display="inline"><semantics> <mrow> <mi>z</mi> <mo>−</mo> <mi>w</mi> </mrow> </semantics></math> plane; (<b>c</b>) <math display="inline"><semantics> <mrow> <mi>w</mi> <mo>−</mo> <mi>u</mi> </mrow> </semantics></math> plane; (<b>d</b>) <math display="inline"><semantics> <mrow> <mi>x</mi> <mo>−</mo> <mi mathvariant="normal">u</mi> </mrow> </semantics></math> plane; (<b>e</b>) <math display="inline"><semantics> <mrow> <mi>z</mi> <mo>−</mo> <mi>u</mi> </mrow> </semantics></math> plane; and (<b>f</b>) <math display="inline"><semantics> <mrow> <mi mathvariant="normal">w</mi> <mo>−</mo> <mi>i</mi> </mrow> </semantics></math> plane of memristor model (1).</p> Full article ">Figure 4
<p>Frequency spectrum and time series of <math display="inline"><semantics> <mi>x</mi> </semantics></math> variable for system (2): (<b>a</b>) frequency spectrum; (<b>b</b>) time series.</p> Full article ">Figure 5
<p>Poincaré map of system (2): (<b>a</b>) <math display="inline"><semantics> <mrow> <mi>x</mi> <mo>−</mo> <mi>y</mi> </mrow> </semantics></math> plane; (<b>b</b>) <math display="inline"><semantics> <mrow> <mi>y</mi> <mo>−</mo> <mi>u</mi> </mrow> </semantics></math> plane; (<b>c</b>) <math display="inline"><semantics> <mrow> <mi>y</mi> <mo>−</mo> <mi>w</mi> </mrow> </semantics></math> plane; (<b>d</b>) <math display="inline"><semantics> <mrow> <mi>x</mi> <mo>−</mo> <mi>w</mi> </mrow> </semantics></math> plane.</p> Full article ">Figure 6
<p>Bifurcation diagram of <math display="inline"><semantics> <mi>x</mi> </semantics></math> versus <math display="inline"><semantics> <mi>a</mi> </semantics></math> for system (2) when <math display="inline"><semantics> <mrow> <mi>b</mi> <mo>=</mo> <mn>0.05</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>d</mi> <mo>=</mo> <mn>0.1</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>c</mi> <mo>=</mo> <mi>e</mi> <mo>=</mo> <mi>g</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mn>0</mn> </msub> <mo>,</mo> <msub> <mi>y</mi> <mn>0</mn> </msub> <mo>,</mo> <msub> <mi>z</mi> <mn>0</mn> </msub> <mo>,</mo> <msub> <mi>w</mi> <mn>0</mn> </msub> <mo>,</mo> <msub> <mi>u</mi> <mn>0</mn> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <mo>,</mo> <mo>−</mo> <mn>1</mn> <mo>,</mo> <mn>0</mn> <mo>,</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </semantics></math>, and <math display="inline"><semantics> <mrow> <mi>a</mi> <mo>∈</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>4</mn> <mo stretchy="false">)</mo> </mrow> </semantics></math>.</p> Full article ">Figure 7
<p>Les of system (2) versus <math display="inline"><semantics> <mi>a</mi> </semantics></math>, when <math display="inline"><semantics> <mrow> <mi>b</mi> <mo>=</mo> <mn>0.05</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>d</mi> <mo>=</mo> <mn>0.1</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>c</mi> <mo>=</mo> <mi>e</mi> <mo>=</mo> <mi>g</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mn>0</mn> </msub> <mo>,</mo> <msub> <mi>y</mi> <mn>0</mn> </msub> <mo>,</mo> <msub> <mi>z</mi> <mn>0</mn> </msub> <mo>,</mo> <msub> <mi>w</mi> <mn>0</mn> </msub> <mo>,</mo> <msub> <mi>u</mi> <mn>0</mn> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <mo>,</mo> <mo>−</mo> <mn>1</mn> <mo>,</mo> <mn>0</mn> <mo>,</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </semantics></math>, and <math display="inline"><semantics> <mrow> <mi>a</mi> <mo>∈</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>4</mn> <mo stretchy="false">)</mo> </mrow> </semantics></math>.</p> Full article ">Figure 8
<p>Projections of 2-D phase diagram with parameter <math display="inline"><semantics> <mrow> <mi>a</mi> <mo>=</mo> <mn>0.3</mn> </mrow> </semantics></math>: (<b>a</b>) attractor on the <math display="inline"><semantics> <mrow> <mi>x</mi> <mo>−</mo> <mi>z</mi> </mrow> </semantics></math> plane; (<b>b</b>) attractor on the <math display="inline"><semantics> <mrow> <mi>y</mi> <mo>−</mo> <mi>z</mi> </mrow> </semantics></math> plane; (<b>c</b>) attractor on the <math display="inline"><semantics> <mrow> <mi>y</mi> <mo>−</mo> <mi>w</mi> </mrow> </semantics></math> plane.</p> Full article ">Figure 9
<p>Projections of a 2-D phase diagram with parameter <math display="inline"><semantics> <mrow> <mi>a</mi> <mo>=</mo> <mn>0.4</mn> </mrow> </semantics></math>: (<b>a</b>) attractor on the <math display="inline"><semantics> <mrow> <mi>x</mi> <mo>−</mo> <mi>z</mi> </mrow> </semantics></math> plane; (<b>b</b>) attractor on the <math display="inline"><semantics> <mrow> <mi>y</mi> <mo>−</mo> <mi>z</mi> </mrow> </semantics></math> plane; (<b>c</b>) attractor on the <math display="inline"><semantics> <mrow> <mi>y</mi> <mo>−</mo> <mi>w</mi> </mrow> </semantics></math> plane.</p> Full article ">Figure 10
<p>Projections of 2-D phase diagram with parameter <math display="inline"><semantics> <mrow> <mi>a</mi> <mo>=</mo> <mn>0.70</mn> </mrow> </semantics></math>: (<b>a</b>) attractor on the <math display="inline"><semantics> <mrow> <mi>x</mi> <mo>−</mo> <mi>z</mi> </mrow> </semantics></math> plane; (<b>b</b>) attractor on the <math display="inline"><semantics> <mrow> <mi>y</mi> <mo>−</mo> <mi>z</mi> </mrow> </semantics></math> plane; (<b>c</b>) attractor on the <math display="inline"><semantics> <mrow> <mi>y</mi> <mo>−</mo> <mi>w</mi> </mrow> </semantics></math> plane.</p> Full article ">Figure 11
<p>Projections of 2-D phase diagram with parameter <math display="inline"><semantics> <mrow> <mi>a</mi> <mo>=</mo> <mn>0.75</mn> </mrow> </semantics></math>: (<b>a</b>) attractor on the <math display="inline"><semantics> <mrow> <mi>x</mi> <mo>−</mo> <mi>z</mi> </mrow> </semantics></math> plane; (<b>b</b>) attractor on the <math display="inline"><semantics> <mrow> <mi>y</mi> <mo>−</mo> <mi>z</mi> </mrow> </semantics></math> plane; (<b>c</b>) attractor on the <math display="inline"><semantics> <mrow> <mi>y</mi> <mo>−</mo> <mi>w</mi> </mrow> </semantics></math> plane.</p> Full article ">Figure 12
<p>Projections of 2-D phase diagram with parameter <math display="inline"><semantics> <mrow> <mi>a</mi> <mo>=</mo> <mn>1.17</mn> </mrow> </semantics></math>: (<b>a</b>) attractor on the <math display="inline"><semantics> <mrow> <mi>x</mi> <mo>−</mo> <mi>z</mi> </mrow> </semantics></math> plane; (<b>b</b>) attractor on the <math display="inline"><semantics> <mrow> <mi>y</mi> <mo>−</mo> <mi>z</mi> </mrow> </semantics></math> plane; (<b>c</b>) attractor on the <math display="inline"><semantics> <mrow> <mi>y</mi> <mo>−</mo> <mi>w</mi> </mrow> </semantics></math> plane.</p> Full article ">Figure 13
<p>Projections of 2-D phase diagram with parameter <math display="inline"><semantics> <mrow> <mi>a</mi> <mo>=</mo> <mn>1.28</mn> </mrow> </semantics></math>: (<b>a</b>) attractor on the <math display="inline"><semantics> <mrow> <mi>x</mi> <mo>−</mo> <mi>z</mi> </mrow> </semantics></math> plane; (<b>b</b>) attractor on the <math display="inline"><semantics> <mrow> <mi>y</mi> <mo>−</mo> <mi>z</mi> </mrow> </semantics></math> plane; (<b>c</b>) attractor on the <math display="inline"><semantics> <mrow> <mi>y</mi> <mo>−</mo> <mi>w</mi> </mrow> </semantics></math> plane.</p> Full article ">Figure 14
<p>Projections of 2-D phase diagram with parameter <math display="inline"><semantics> <mrow> <mi>a</mi> <mo>=</mo> <mn>1.891</mn> </mrow> </semantics></math>: (<b>a</b>) attractor on the <math display="inline"><semantics> <mrow> <mi>x</mi> <mo>−</mo> <mi>z</mi> </mrow> </semantics></math> plane; (<b>b</b>) attractor on the <math display="inline"><semantics> <mrow> <mi>y</mi> <mo>−</mo> <mi>z</mi> </mrow> </semantics></math> plane; (<b>c</b>) attractor on the <math display="inline"><semantics> <mrow> <mi>y</mi> <mo>−</mo> <mi>w</mi> </mrow> </semantics></math> plane.</p> Full article ">Figure 15
<p>Projections of 2-D phase diagram with parameter <math display="inline"><semantics> <mrow> <mi>a</mi> <mo>=</mo> <mn>1.89678</mn> </mrow> </semantics></math>: (<b>a</b>) attractor on the <math display="inline"><semantics> <mrow> <mi>x</mi> <mo>−</mo> <mi>z</mi> </mrow> </semantics></math> plane; (<b>b</b>) attractor on the <math display="inline"><semantics> <mrow> <mi>y</mi> <mo>−</mo> <mi>z</mi> </mrow> </semantics></math> plane; (<b>c</b>) attractor on the <math display="inline"><semantics> <mrow> <mi>y</mi> <mo>−</mo> <mi>w</mi> </mrow> </semantics></math> plane.</p> Full article ">Figure 16
<p>Projections of 2-D phase diagram with parameter <math display="inline"><semantics> <mrow> <mi>a</mi> <mo>=</mo> <mn>2.56</mn> </mrow> </semantics></math>: (<b>a</b>) attractor on the <math display="inline"><semantics> <mrow> <mi>x</mi> <mo>−</mo> <mi>z</mi> </mrow> </semantics></math> plane; (<b>b</b>) attractor on the <math display="inline"><semantics> <mrow> <mi>y</mi> <mo>−</mo> <mi>z</mi> </mrow> </semantics></math> plane; (<b>c</b>) attractor on the <math display="inline"><semantics> <mrow> <mi>y</mi> <mo>−</mo> <mi>w</mi> </mrow> </semantics></math> plane.</p> Full article ">Figure 17
<p>Projections of 2-D phase diagram with parameter <math display="inline"><semantics> <mrow> <mi>a</mi> <mo>=</mo> <mn>2.65</mn> </mrow> </semantics></math>: (<b>a</b>) attractor on the <math display="inline"><semantics> <mrow> <mi>x</mi> <mo>−</mo> <mi>z</mi> </mrow> </semantics></math> plane; (<b>b</b>) attractor on the <math display="inline"><semantics> <mrow> <mi>y</mi> <mo>−</mo> <mi>z</mi> </mrow> </semantics></math> plane; (<b>c</b>) attractor on the <math display="inline"><semantics> <mrow> <mi>y</mi> <mo>−</mo> <mi>w</mi> </mrow> </semantics></math> plane.</p> Full article ">Figure 18
<p>Bifurcation diagram of <math display="inline"><semantics> <mi>y</mi> </semantics></math> and Les versus <math display="inline"><semantics> <mi>u</mi> </semantics></math> for system (2) with initial value <math display="inline"><semantics> <mrow> <msub> <mi>O</mi> <mn>0</mn> </msub> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mn>0</mn> </msub> <mo>,</mo> <msub> <mi>y</mi> <mn>0</mn> </msub> <mo>,</mo> <msub> <mi>z</mi> <mn>0</mn> </msub> <mo>,</mo> <msub> <mi>w</mi> <mn>0</mn> </msub> <mo>,</mo> <msub> <mi>u</mi> <mn>0</mn> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mi>u</mi> <mo>,</mo> <mn>0</mn> <mo>,</mo> <mn>0</mn> <mo>,</mo> <mn>0</mn> <mo>,</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mrow> </semantics></math>, and <math display="inline"><semantics> <mrow> <mi>u</mi> <mo>∈</mo> <mo stretchy="false">[</mo> <mn>0</mn> <mi>,</mi> <mn>4</mn> <mo stretchy="false">]</mo> </mrow> </semantics></math>: (<b>a</b>) bifurcation diagram; (<b>b</b>) Les graph.</p> Full article ">Figure 19
<p>Bifurcation diagram of <math display="inline"><semantics> <mi>y</mi> </semantics></math> and Les versus <math display="inline"><semantics> <mi>u</mi> </semantics></math> for system (2) with initial value <math display="inline"><semantics> <mrow> <msub> <mi>O</mi> <mn>1</mn> </msub> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mn>0</mn> </msub> <mo>,</mo> <msub> <mi>y</mi> <mn>0</mn> </msub> <mo>,</mo> <msub> <mi>z</mi> <mn>0</mn> </msub> <mo>,</mo> <msub> <mi>w</mi> <mn>0</mn> </msub> <mo>,</mo> <msub> <mi>u</mi> <mn>0</mn> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mi>u</mi> <mo>,</mo> <mi>u</mi> <mo>,</mo> <mn>0</mn> <mo>,</mo> <mn>0</mn> <mo>,</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mrow> </semantics></math>, and <math display="inline"><semantics> <mrow> <mi>u</mi> <mo>∈</mo> <mo stretchy="false">[</mo> <mn>0</mn> <mi>,</mi> <mn>4</mn> <mo stretchy="false">]</mo> </mrow> </semantics></math>: (<b>a</b>) bifurcation diagram; (<b>b</b>) Les graph.</p> Full article ">Figure 20
<p>Phase diagram on the <math display="inline"><semantics> <mrow> <mi>y</mi> <mo>−</mo> <mi>z</mi> </mrow> </semantics></math> plane for system (2) with different initial values: (<b>a</b>) <math display="inline"><semantics> <mrow> <mrow> <mo>(</mo> <mrow> <mn>0.51</mn> <mo>,</mo> <mn>0.51</mn> <mo>,</mo> <mn>0</mn> <mo>,</mo> <mn>0</mn> <mo>,</mo> <mn>0</mn> </mrow> <mo>)</mo> </mrow> </mrow> </semantics></math> (blue), <math display="inline"><semantics> <mrow> <mrow> <mo>(</mo> <mrow> <mn>3.34</mn> <mo>,</mo> <mn>3.34</mn> <mo>,</mo> <mn>0</mn> <mo>,</mo> <mn>0</mn> <mo>,</mo> <mn>0</mn> </mrow> <mo>)</mo> </mrow> </mrow> </semantics></math> (red), <math display="inline"><semantics> <mrow> <mrow> <mo>(</mo> <mrow> <mn>0.01</mn> <mo>,</mo> <mn>0</mn> <mo>,</mo> <mn>0</mn> <mo>,</mo> <mn>0</mn> <mo>,</mo> <mn>0</mn> </mrow> <mo>)</mo> </mrow> </mrow> </semantics></math> (green); (<b>b</b>) <math display="inline"><semantics> <mrow> <mrow> <mo>(</mo> <mrow> <mn>3</mn> <mo>,</mo> <mn>3</mn> <mo>,</mo> <mn>0</mn> <mo>,</mo> <mn>0</mn> <mo>,</mo> <mn>0</mn> </mrow> <mo>)</mo> </mrow> </mrow> </semantics></math> (blue), <math display="inline"><semantics> <mrow> <mrow> <mo>(</mo> <mrow> <mn>0.9711</mn> <mo>,</mo> <mn>0.9711</mn> <mo>,</mo> <mn>0</mn> <mo>,</mo> <mn>0</mn> <mo>,</mo> <mn>0</mn> </mrow> <mo>)</mo> </mrow> </mrow> </semantics></math> (red), <math display="inline"><semantics> <mrow> <mrow> <mo>(</mo> <mrow> <mn>1.922</mn> <mo>,</mo> <mn>0</mn> <mo>,</mo> <mn>0</mn> <mo>,</mo> <mn>0</mn> <mo>,</mo> <mn>0</mn> </mrow> <mo>)</mo> </mrow> </mrow> </semantics></math> (green).</p> Full article ">Figure 21
<p>Multi-stable nonlinear dynamic behavior distribution of chaotic for memristive system with different initial values.</p> Full article ">Figure 22
<p>Time series and phase diagram of attractors for transient chaotic: (<b>a</b>) time series of <math display="inline"><semantics> <mi>z</mi> </semantics></math> when <math display="inline"><semantics> <mrow> <mi>t</mi> <mo>∈</mo> <mrow> <mo>(</mo> <mrow> <mn>0</mn> <mo>,</mo> <mn>8000</mn> </mrow> <mo>)</mo> </mrow> </mrow> </semantics></math>; (<b>b</b>) phase diagrams in the <math display="inline"><semantics> <mrow> <mi>x</mi> <mo>−</mo> <mi>y</mi> <mo>−</mo> <mi>u</mi> </mrow> </semantics></math> space when <math display="inline"><semantics> <mrow> <mi>t</mi> <mo>∈</mo> <mrow> <mo>(</mo> <mrow> <mn>0</mn> <mo>,</mo> <mn>2000</mn> </mrow> <mo>)</mo> </mrow> </mrow> </semantics></math>; (<b>c</b>) phase diagrams on the <math display="inline"><semantics> <mrow> <mi>x</mi> <mo>−</mo> <mi>u</mi> </mrow> </semantics></math> plane when <math display="inline"><semantics> <mrow> <mi>t</mi> <mo>∈</mo> <mrow> <mo>(</mo> <mrow> <mn>0</mn> <mo>,</mo> <mn>2000</mn> </mrow> <mo>)</mo> </mrow> </mrow> </semantics></math>; (<b>d</b>) phase diagrams in the <math display="inline"><semantics> <mrow> <mi>x</mi> <mo>−</mo> <mi>y</mi> <mo>−</mo> <mi>u</mi> </mrow> </semantics></math> space when <math display="inline"><semantics> <mrow> <mi>t</mi> <mo>∈</mo> <mrow> <mo>(</mo> <mrow> <mn>2000</mn> <mo>,</mo> <mn>8000</mn> </mrow> <mo>)</mo> </mrow> </mrow> </semantics></math>; (<b>e</b>) phase diagrams on the <math display="inline"><semantics> <mrow> <mi>x</mi> <mo>−</mo> <mi>u</mi> </mrow> </semantics></math> plane when <math display="inline"><semantics> <mrow> <mi>t</mi> <mo>∈</mo> <mrow> <mo>(</mo> <mrow> <mn>2000</mn> <mo>,</mo> <mn>8000</mn> </mrow> <mo>)</mo> </mrow> </mrow> </semantics></math>.</p> Full article ">Figure 22 Cont.
<p>Time series and phase diagram of attractors for transient chaotic: (<b>a</b>) time series of <math display="inline"><semantics> <mi>z</mi> </semantics></math> when <math display="inline"><semantics> <mrow> <mi>t</mi> <mo>∈</mo> <mrow> <mo>(</mo> <mrow> <mn>0</mn> <mo>,</mo> <mn>8000</mn> </mrow> <mo>)</mo> </mrow> </mrow> </semantics></math>; (<b>b</b>) phase diagrams in the <math display="inline"><semantics> <mrow> <mi>x</mi> <mo>−</mo> <mi>y</mi> <mo>−</mo> <mi>u</mi> </mrow> </semantics></math> space when <math display="inline"><semantics> <mrow> <mi>t</mi> <mo>∈</mo> <mrow> <mo>(</mo> <mrow> <mn>0</mn> <mo>,</mo> <mn>2000</mn> </mrow> <mo>)</mo> </mrow> </mrow> </semantics></math>; (<b>c</b>) phase diagrams on the <math display="inline"><semantics> <mrow> <mi>x</mi> <mo>−</mo> <mi>u</mi> </mrow> </semantics></math> plane when <math display="inline"><semantics> <mrow> <mi>t</mi> <mo>∈</mo> <mrow> <mo>(</mo> <mrow> <mn>0</mn> <mo>,</mo> <mn>2000</mn> </mrow> <mo>)</mo> </mrow> </mrow> </semantics></math>; (<b>d</b>) phase diagrams in the <math display="inline"><semantics> <mrow> <mi>x</mi> <mo>−</mo> <mi>y</mi> <mo>−</mo> <mi>u</mi> </mrow> </semantics></math> space when <math display="inline"><semantics> <mrow> <mi>t</mi> <mo>∈</mo> <mrow> <mo>(</mo> <mrow> <mn>2000</mn> <mo>,</mo> <mn>8000</mn> </mrow> <mo>)</mo> </mrow> </mrow> </semantics></math>; (<b>e</b>) phase diagrams on the <math display="inline"><semantics> <mrow> <mi>x</mi> <mo>−</mo> <mi>u</mi> </mrow> </semantics></math> plane when <math display="inline"><semantics> <mrow> <mi>t</mi> <mo>∈</mo> <mrow> <mo>(</mo> <mrow> <mn>2000</mn> <mo>,</mo> <mn>8000</mn> </mrow> <mo>)</mo> </mrow> </mrow> </semantics></math>.</p> Full article ">Figure 23
<p>SE with different <math display="inline"><semantics> <mi>a</mi> </semantics></math> and <math display="inline"><semantics> <mi>b</mi> </semantics></math>: (<b>a</b>) SE vs. <math display="inline"><semantics> <mi>a</mi> </semantics></math> (<math display="inline"><semantics> <mrow> <mi>b</mi> <mo>=</mo> <mn>0.05</mn> </mrow> </semantics></math>); (<b>b</b>) SE vs. <math display="inline"><semantics> <mi>b</mi> </semantics></math> (<math display="inline"><semantics> <mrow> <mi>a</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math>) .</p> Full article ">Figure 24
<p>SE distribution of the system under different conditions: (<b>a</b>) the interaction of parameters <math display="inline"><semantics> <mi>a</mi> </semantics></math> and <math display="inline"><semantics> <mi>b</mi> </semantics></math>; (<b>b</b>) the interaction of initial values <math display="inline"><semantics> <mrow> <msub> <mi>x</mi> <mn>0</mn> </msub> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <msub> <mi>y</mi> <mn>0</mn> </msub> </mrow> </semantics></math>.</p> Full article ">Figure 25
<p>The memristor model circuit: (<b>a</b>) memristor unit; (<b>b</b>) absolute circuit.</p> Full article ">Figure 26
<p>Schematic of the memristor-based chaotic system.</p> Full article ">Figure 27
<p>Multisim circuit simulation of system (2).</p> Full article ">Figure 28
<p>Phase diagrams observed by oscilloscope: (<b>a</b>) <math display="inline"><semantics> <mrow> <msub> <mi>u</mi> <mi>y</mi> </msub> <mo>−</mo> <msub> <mi>u</mi> <mi>w</mi> </msub> </mrow> </semantics></math> plane; (<b>b</b>) <math display="inline"><semantics> <mrow> <msub> <mi>u</mi> <mi>z</mi> </msub> <mo>−</mo> <msub> <mi>u</mi> <mi>w</mi> </msub> </mrow> </semantics></math> plane; (<b>c</b>) <math display="inline"><semantics> <mrow> <msub> <mi>u</mi> <mi>w</mi> </msub> <mo>−</mo> <msub> <mi>u</mi> <mi>u</mi> </msub> </mrow> </semantics></math> plane; (<b>d</b>) <math display="inline"><semantics> <mrow> <msub> <mi>u</mi> <mi>x</mi> </msub> <mo>−</mo> <msub> <mi>u</mi> <mi>u</mi> </msub> </mrow> </semantics></math> plane; (<b>e</b>) <math display="inline"><semantics> <mrow> <msub> <mi>u</mi> <mi>z</mi> </msub> <mo>−</mo> <msub> <mi>u</mi> <mi>u</mi> </msub> </mrow> </semantics></math> plane; (<b>f</b>) <math display="inline"><semantics> <mrow> <msub> <mi>u</mi> <mi>x</mi> </msub> <mo>−</mo> <msub> <mi>u</mi> <mi>z</mi> </msub> </mrow> </semantics></math> plane.</p> Full article ">
14 pages, 6336 KiB  
Article
Entropy Analysis and Image Encryption Application Based on a New Chaotic System Crossing a Cylinder
by Alaa Kadhim Farhan, Nadia M.G. Al-Saidi, Abeer Tariq Maolood, Fahimeh Nazarimehr and Iqtadar Hussain
Entropy 2019, 21(10), 958; https://doi.org/10.3390/e21100958 - 30 Sep 2019
Cited by 44 | Viewed by 4196
Abstract
Designing chaotic systems with specific features is a hot topic in nonlinear dynamics. In this study, a novel chaotic system is presented with a unique feature of crossing inside and outside of a cylinder repeatedly. This new system is thoroughly analyzed by the [...] Read more.
Designing chaotic systems with specific features is a hot topic in nonlinear dynamics. In this study, a novel chaotic system is presented with a unique feature of crossing inside and outside of a cylinder repeatedly. This new system is thoroughly analyzed by the help of the bifurcation diagram, Lyapunov exponents’ spectrum, and entropy measurement. Bifurcation analysis of the proposed system with two initiation methods reveals its multistability. As an engineering application, the system’s efficiency is tested in image encryption. The complexity of the chaotic attractor of the proposed system makes it a proper choice for encryption. States of the chaotic attractor are used to shuffle the rows and columns of the image, and then the shuffled image is XORed with the states of chaotic attractor. The unpredictability of the chaotic attractor makes the encryption method very safe. The performance of the encryption method is analyzed using the histogram, correlation coefficient, Shannon entropy, and encryption quality. The results show that the encryption method using the proposed chaotic system has reliable performance. Full article
(This article belongs to the Special Issue Nonlinear Dynamics and Entropy of Complex Systems with Hidden and Self-Excited Attractors II)
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Figure 1

Figure 1
<p>Chaotic attractor of System (1) in three different planes, (<b>a</b>) <math display="inline"><semantics> <mrow> <mi>x</mi> <mo>−</mo> <mi>y</mi> </mrow> </semantics></math> plane, (<b>b</b>) <math display="inline"><semantics> <mrow> <mi>x</mi> <mo>−</mo> <mi>z</mi> </mrow> </semantics></math> plane, (<b>c</b>) <math display="inline"><semantics> <mrow> <mi>y</mi> <mo>−</mo> <mi>z</mi> </mrow> </semantics></math> plane.</p> Full article ">Figure 2
<p>Real and imaginary parts of equilibrium points in a∈[1.7,2.4]. (<b>a</b>) Real part of Eigenvalues of <math display="inline"><semantics> <mrow> <mi>E</mi> <msub> <mi>q</mi> <mn>1</mn> </msub> </mrow> </semantics></math>, (<b>b</b>) imaginary part of Eigenvalues of <math display="inline"><semantics> <mrow> <mi>E</mi> <msub> <mi>q</mi> <mn>1</mn> </msub> </mrow> </semantics></math>, (<b>c</b>) real part of Eigenvalues of <math display="inline"><semantics> <mrow> <mi>E</mi> <msub> <mi>q</mi> <mn>2</mn> </msub> </mrow> </semantics></math>, (<b>d</b>) imaginary part of Eigenvalues of <math display="inline"><semantics> <mrow> <mi>E</mi> <msub> <mi>q</mi> <mn>2</mn> </msub> </mrow> </semantics></math>, (<b>e</b>) real part of Eigenvalues of <math display="inline"><semantics> <mrow> <mi>E</mi> <msub> <mi>q</mi> <mn>3</mn> </msub> </mrow> </semantics></math>, (<b>f</b>) imaginary part of Eigenvalues of <math display="inline"><semantics> <mrow> <mi>E</mi> <msub> <mi>q</mi> <mn>3</mn> </msub> </mrow> </semantics></math>, (<b>g</b>) real part of Eigenvalues of <math display="inline"><semantics> <mrow> <mi>E</mi> <msub> <mi>q</mi> <mn>4</mn> </msub> </mrow> </semantics></math>, (<b>h</b>) imaginary part of Eigenvalues of <math display="inline"><semantics> <mrow> <mi>E</mi> <msub> <mi>q</mi> <mn>4</mn> </msub> </mrow> </semantics></math>.</p> Full article ">Figure 3
<p>The chaotic attractor in <math display="inline"><semantics> <mrow> <mi>a</mi> <mo> </mo> <mo>=</mo> <mtext> </mtext> <mn>1.7</mn> </mrow> </semantics></math> and (<b>a</b>) the condition <math display="inline"><semantics> <mrow> <mrow> <mo>〈</mo> <mrow> <msup> <mi>x</mi> <mn>2</mn> </msup> <mo>+</mo> <msup> <mi>y</mi> <mn>2</mn> </msup> </mrow> <mo>〉</mo> </mrow> <mo> </mo> <mo>=</mo> <msup> <mi>a</mi> <mn>2</mn> </msup> </mrow> </semantics></math>, (<b>b</b>) the condition <math display="inline"><semantics> <mrow> <mrow> <mo>〈</mo> <mrow> <msup> <mi>x</mi> <mn>2</mn> </msup> <mo>+</mo> <msup> <mi>y</mi> <mn>2</mn> </msup> </mrow> <mo>〉</mo> </mrow> <mo> </mo> <mo>=</mo> <msup> <mi>a</mi> <mn>2</mn> </msup> </mrow> </semantics></math> from another viewpoint, (<b>c</b>) the condition <math display="inline"><semantics> <mrow> <mrow> <mo>〈</mo> <mrow> <mn>0.4</mn> <mi>z</mi> <mo>+</mo> <mn>3</mn> <mi>x</mi> <mi>y</mi> </mrow> <mo>〉</mo> </mrow> <mo> </mo> <mo>=</mo> <mo> </mo> <mn>0</mn> </mrow> </semantics></math>, (<b>d</b>) the condition <math display="inline"><semantics> <mrow> <mrow> <mo>〈</mo> <mi>z</mi> <mo>〉</mo> </mrow> <mo> </mo> <mo>=</mo> <mo> </mo> <mn>0</mn> </mrow> </semantics></math>.</p> Full article ">Figure 4
<p>Bifurcation diagram of System (1) with backward continuation and the first initial conditions <math display="inline"><semantics> <mrow> <mrow> <mo>(</mo> <mrow> <msub> <mi>x</mi> <mn>0</mn> </msub> <mo>,</mo> <msub> <mi>y</mi> <mn>0</mn> </msub> <mo>,</mo> <msub> <mi>z</mi> <mn>0</mn> </msub> </mrow> <mo>)</mo> </mrow> <mo> </mo> <mo>=</mo> <mo> </mo> <mrow> <mo>(</mo> <mrow> <mn>0.29</mn> <mo>,</mo> <mo>−</mo> <mn>1.81</mn> <mo>,</mo> <mn>0.17</mn> </mrow> <mo>)</mo> </mrow> </mrow> </semantics></math>. (<b>a</b>) maximum values of <math display="inline"><semantics> <mi>x</mi> </semantics></math> variable by changing parameter <math display="inline"><semantics> <mi>a</mi> </semantics></math>; (<b>b</b>) maximum values of <math display="inline"><semantics> <mi>y</mi> </semantics></math> variable by changing parameter <math display="inline"><semantics> <mi>a</mi> </semantics></math>; (<b>c</b>) maximum values of <math display="inline"><semantics> <mi>z</mi> </semantics></math> variable by changing parameter <math display="inline"><semantics> <mi>a</mi> </semantics></math>.</p> Full article ">Figure 5
<p>Bifurcation diagram of System (1) with constant initial conditions <math display="inline"><semantics> <mrow> <mrow> <mo>(</mo> <mrow> <msub> <mi>x</mi> <mn>0</mn> </msub> <mo>,</mo> <msub> <mi>y</mi> <mn>0</mn> </msub> <mo>,</mo> <msub> <mi>z</mi> <mn>0</mn> </msub> </mrow> <mo>)</mo> </mrow> <mo> </mo> <mo>=</mo> <mo> </mo> <mrow> <mo>(</mo> <mrow> <mn>0.29</mn> <mo>,</mo> <mo>−</mo> <mn>1.81</mn> <mo>,</mo> <mn>0.17</mn> </mrow> <mo>)</mo> </mrow> </mrow> </semantics></math>. (<b>a</b>) maximum values of <math display="inline"><semantics> <mi>x</mi> </semantics></math> variable by changing parameter <math display="inline"><semantics> <mi>a</mi> </semantics></math>; (<b>b</b>) maximum values of <math display="inline"><semantics> <mi>y</mi> </semantics></math> variable by changing parameter <math display="inline"><semantics> <mi>a</mi> </semantics></math>; (<b>c</b>) maximum values of <math display="inline"><semantics> <mi>z</mi> </semantics></math> variable by changing parameter <math display="inline"><semantics> <mi>a</mi> </semantics></math>.</p> Full article ">Figure 6
<p>Lyapunov exponents of System (1) (<b>a</b>) with backward continuation, (<b>b</b>) with constant initial conditions.</p> Full article ">Figure 7
<p>(<b>a</b>) Shannon entropy and (<b>b</b>) Kolmogorov<tt>–</tt>Sinai entropy of the proposed system with respect to changing parameter <math display="inline"><semantics> <mi>a</mi> </semantics></math> and backward continuation method.</p> Full article ">Figure 8
<p>The attractor of System (1) in (<b>a</b>) <math display="inline"><semantics> <mrow> <mi>a</mi> <mo>=</mo> <mn>1.74</mn> </mrow> </semantics></math>, (<b>b</b>) <math display="inline"><semantics> <mrow> <mi>a</mi> <mo>=</mo> <mn>1.8</mn> </mrow> </semantics></math>, (<b>c</b>) <math display="inline"><semantics> <mrow> <mi>a</mi> <mo>=</mo> <mn>2</mn> </mrow> </semantics></math>.</p> Full article ">Figure 9
<p>(<b>a</b>) The original Lena image, (<b>b</b>) the shuffled rows and columns of the Lena image, (<b>c</b>) the encrypted image, (<b>d</b>) the decrypted image.</p> Full article ">Figure 10
<p>(<b>a</b>) The original baby image, (<b>b</b>) the shuffled rows and columns of the baby image, (<b>c</b>) the encrypted image, (<b>d</b>) the decrypted image.</p> Full article ">Figure 11
<p>(<b>a</b>) The histogram of Lena image, (<b>b</b>) the histogram of the encrypted Lena image, (<b>c</b>) the histogram of the baby image, (<b>d</b>) the histogram of the encrypted baby image.</p> Full article ">Figure 12
<p>(<b>a</b>) The relation between pixels of Lena, (<b>b</b>) the correlation of the encrypted images of Lena, (<b>c</b>) the relationship between pixels of the baby image, (<b>d</b>) the correlation of the encrypted images of the baby.</p> Full article ">
37 pages, 39282 KiB  
Article
New Nonlinear Active Element Dedicated to Modeling Chaotic Dynamics with Complex Polynomial Vector Fields
by Jiri Petrzela and Roman Sotner
Entropy 2019, 21(9), 871; https://doi.org/10.3390/e21090871 - 6 Sep 2019
Cited by 6 | Viewed by 3453
Abstract
This paper describes evolution of new active element that is able to significantly simplify the design process of lumped chaotic oscillator, especially if the concept of analog computer or state space description is adopted. The major advantage of the proposed active device lies [...] Read more.
This paper describes evolution of new active element that is able to significantly simplify the design process of lumped chaotic oscillator, especially if the concept of analog computer or state space description is adopted. The major advantage of the proposed active device lies in the incorporation of two fundamental mathematical operations into a single five-port voltage-input current-output element: namely, differentiation and multiplication. The developed active device is verified inside three different synthesis scenarios: circuitry realization of a third-order cyclically symmetrical vector field, hyperchaotic system based on the Lorenz equations and fourth- and fifth-order hyperjerk function. Mentioned cases represent complicated vector fields that cannot be implemented without the necessity of utilizing many active elements. The captured oscilloscope screenshots are compared with numerically integrated trajectories to demonstrate good agreement between theory and measurement. Full article
(This article belongs to the Special Issue Nonlinear Dynamics and Entropy of Complex Systems with Hidden and Self-Excited Attractors II)
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Figure 1

Figure 1
<p>Symbols of active cells employed in designed chaotic oscillators: multiplier with internal voltage-to-current conversion (<b>left</b>) and summation/subtraction (<b>right</b>) unit.</p> Full article ">Figure 2
<p>Internal topologies of the active cells fabricated in a single integrated circuit (IC) package.</p> Full article ">Figure 3
<p>Comparison between measured and simulated responses of bipolar core-based multiplier: DC transfer responses X<sub>1</sub>→Z for <span class="html-italic">V</span><sub>Y1</sub> controlled by DC voltage (<b>upper plot</b>), DC transfer responses Y<sub>1</sub>→Z for <span class="html-italic">V</span><sub>X1</sub> controlled by DC voltage (<b>middle plot</b>), and dependence of <span class="html-italic">g</span><sub>m</sub> on <span class="html-italic">V</span><sub>Y1</sub>/<span class="html-italic">V</span><sub>X1</sub> (<b>lower plot</b>).</p> Full article ">Figure 4
<p>Magnitude of AC transfer responses X<sub>1</sub>→Z for <span class="html-italic">V</span><sub>Y1</sub> controlled by DC voltage (<b>upper left plot</b>), magnitude of AC transfer responses Y<sub>1</sub>→Z for <span class="html-italic">V</span><sub>X1</sub> controlled by DC voltage (<b>upper right graph</b>), magnitude vs. frequency plot of input impedance at X<sub>1</sub> terminal (<b>middle left plot</b>), magnitude vs. frequency plot of input impedance at Y<sub>1</sub> terminal (<b>middle right plot</b>), and magnitude vs. frequency plot of output impedance at Z terminal, example for <span class="html-italic">V</span><sub>X,Y</sub> = −0.5 V (<b>lower plot</b>).</p> Full article ">Figure 5
<p>Comparison between measured and simulated responses of CMOS core-based multiplier: DC transfer responses X<sub>1</sub>→Z for <span class="html-italic">V</span><sub>Y1</sub> controlled by DC voltage (<b>upper image</b>), DC transfer responses Y<sub>1</sub>→Z for <span class="html-italic">V</span><sub>X1</sub> controlled by DC voltage (<b>middle plot</b>), and dependence of <span class="html-italic">g</span><sub>m</sub> on <span class="html-italic">V</span><sub>Y1,X1</sub> (<b>lower image</b>).</p> Full article ">Figure 5 Cont.
<p>Comparison between measured and simulated responses of CMOS core-based multiplier: DC transfer responses X<sub>1</sub>→Z for <span class="html-italic">V</span><sub>Y1</sub> controlled by DC voltage (<b>upper image</b>), DC transfer responses Y<sub>1</sub>→Z for <span class="html-italic">V</span><sub>X1</sub> controlled by DC voltage (<b>middle plot</b>), and dependence of <span class="html-italic">g</span><sub>m</sub> on <span class="html-italic">V</span><sub>Y1,X1</sub> (<b>lower image</b>).</p> Full article ">Figure 6
<p>Measured magnitude of AC transfer X<sub>1</sub>→Z responses for <span class="html-italic">V</span><sub>Y1</sub> controlled by DC voltage (<b>upper left plot</b>), magnitude of AC transfer responses Y<sub>1</sub>→Z for <span class="html-italic">V</span><sub>X1</sub> controlled by DC voltage (<b>upper right plot</b>), magnitude vs. frequency plot of input impedance at X<sub>1</sub> node (<b>middle left image</b>), magnitude vs. frequency plot of input impedance at Y<sub>1</sub> node (<b>middle right plot</b>), and magnitude vs. frequency plot of output impedance at Z terminal, example for <span class="html-italic">V</span><sub>X1</sub> = −0.5 V (<b>lower graph</b>).</p> Full article ">Figure 7
<p>Comparison between measured and simulated responses of a summation/subtraction unit (see text for evaluation): DC transfer responses between terminals Y<sub>1,2,3</sub>→W.</p> Full article ">Figure 8
<p>Summation/subtraction unit and magnitude AC transfer responses between terminals Y<sub>1,2,3</sub>→W (<b>upper left plot</b>), magnitude of the input impedances Y<sub>1,2,3</sub> vs. frequency (<b>upper right graph</b>), and magnitude vs. frequency plot of output impedance measured at W terminal (<b>lower image</b>).</p> Full article ">Figure 9
<p>Three-dimensional perspective view on the typical strange attractor associated with the dynamical system (3) (<b>black color</b>) and corresponding plane projections: <span class="html-italic">x</span> vs. <span class="html-italic">y</span> (<b>green</b>), <span class="html-italic">x</span> vs. <span class="html-italic">z</span> (<b>blue</b>) and <span class="html-italic">y</span> vs. <span class="html-italic">z</span> (<b>orange</b>) for initial conditions <b>x</b><sub>0</sub> = (0.1, 0, 0)<sup>T</sup>. Sensitivity of the dynamical system (3) to initial conditions for different final time of integration; see text for clarification.</p> Full article ">Figure 10
<p>Energy distribution over strange attractor (see text): static (<b>blue</b>) and dynamic (<b>orange</b>).</p> Full article ">Figure 11
<p>2D and 3D view of the rainbow color-scaled surface contour plot of the largest Lyapunov exponent (LLE) as a function of the internal system parameters <span class="html-italic">a</span> ∈ (1, 2) with step size 0.01 and <span class="html-italic">b</span> ∈ (0.3, 0.5) with step size 0.001.</p> Full article ">Figure 12
<p>Perspective view on a typical strange attractor associated with system (4) is demonstrated in the left plot and circuit-oriented equivalent flow (10) is provided on the middle plot. The next pictures give generated chaotic waveforms in time and frequency domain for normalized and real circuit.</p> Full article ">Figure 13
<p>Three-dimensional perspective view (black) of the typical strange attractor associated with system (5) is provided in the left plot: <span class="html-italic">x</span> vs. <span class="html-italic">dx</span>/<span class="html-italic">dt</span> (green), <span class="html-italic">x</span> vs. <span class="html-italic">dx</span><sup>2</sup>/<span class="html-italic">dt</span><sup>2</sup> (blue) and <span class="html-italic">x</span> vs. <span class="html-italic">dx</span><sup>3</sup>/<span class="html-italic">dt</span><sup>3</sup> (orange). Remaining images represent the individual plane projections of a fifth-dimensional strange attractor: (<b>a</b>) <span class="html-italic">x</span> vs. <span class="html-italic">dx</span>/<span class="html-italic">dt</span>, (<b>b</b>) <span class="html-italic">x</span> vs. <span class="html-italic">d</span><sup>2</sup><span class="html-italic">x</span>/<span class="html-italic">dt</span><sup>2</sup>, (<b>c</b>) <span class="html-italic">x</span> vs. <span class="html-italic">dx</span><sup>3</sup>/<span class="html-italic">dt</span><sup>3</sup>, (<b>d</b>) <span class="html-italic">x</span> vs. <span class="html-italic">dx</span><sup>4</sup>/<span class="html-italic">dt</span><sup>4</sup>, (<b>e</b>) <span class="html-italic">dx</span>/<span class="html-italic">dt</span> vs. <span class="html-italic">dx</span><sup>2</sup>/<span class="html-italic">dt</span><sup>2</sup>, (<b>f</b>) <span class="html-italic">dx</span>/<span class="html-italic">dt</span> vs. <span class="html-italic">dx</span><sup>3</sup>/<span class="html-italic">dt</span><sup>3</sup>, (<b>g</b>) <span class="html-italic">dx</span>/<span class="html-italic">dt</span> vs. <span class="html-italic">dx</span><sup>4</sup>/<span class="html-italic">dt</span><sup>4</sup>, (<b>h</b>) <span class="html-italic">dx</span><sup>2</sup>/<span class="html-italic">dt</span><sup>2</sup> vs. <span class="html-italic">dx</span><sup>3</sup>/<span class="html-italic">dt</span><sup>3</sup>, (<b>i</b>) <span class="html-italic">dx</span><sup>2</sup>/<span class="html-italic">dt</span><sup>2</sup> vs. <span class="html-italic">dx</span><sup>4</sup>/<span class="html-italic">dt</span><sup>4</sup>, (<b>j</b>) <span class="html-italic">dx</span><sup>3</sup>/<span class="html-italic">dt</span><sup>3</sup> vs. <span class="html-italic">dx</span><sup>4</sup>/<span class="html-italic">dt</span><sup>4</sup>.</p> Full article ">Figure 14
<p>Basin of attraction calculated for dynamical system (4), state space degraded into cubes defined by <span class="html-italic">x</span> = 0 and plotted as planes <span class="html-italic">d</span><sup>4</sup><span class="html-italic">x</span>/<span class="html-italic">dt</span><sup>4</sup> (from left to right), upper row is set {−2, −1, 0, 1, 2, 3, 4}, lower row is continuation {5, 6, 7, 8, 9, 10, 11}.</p> Full article ">Figure 15
<p>Kaplan–Yorke dimension calculated in close neighborhood of nominal values of internal system parameters, low-resolution one-dimensional bifurcation diagram calculated with respect to the external voltage <span class="html-italic">V<sub>a</sub></span> ∈ (0.4, 0.5)V with a step of 1 mV (red) and <span class="html-italic">V<sub>b</sub></span> ∈ (−0.45, −0.2)V with a step of 1 mV (green).</p> Full article ">Figure 16
<p>3D perspective plots of typical strange attractor generated by 4D hyperchaotic system (6). Upper left plot: <span class="html-italic">y</span> vs. <span class="html-italic">z</span> plane (<b>orange</b>), <span class="html-italic">x</span> vs. <span class="html-italic">z</span> plane (<b>blue</b>) and <span class="html-italic">x</span> vs. <span class="html-italic">y</span> plane (<b>green</b>). Upper middle plot: <span class="html-italic">z</span> vs. <span class="html-italic">w</span> plane (<b>orange</b>), <span class="html-italic">y</span> vs. <span class="html-italic">w</span> plane (<b>blue</b>) and <span class="html-italic">y</span> vs. <span class="html-italic">z</span> plane (<b>green</b>). Upper right plot: <span class="html-italic">x</span> vs. <span class="html-italic">w</span> plane (purple). Lower row represents 3D cross sections defined by planes <span class="html-italic">x</span> = {−0.25, −0.2, −0.1, 0, 0.1, 0.25}.</p> Full article ">Figure 17
<p>Three-dimensional views of a typical strange attractor associated with 4D hyperchaotic system (6) for increased zoom: original dynamical system (<b>blue</b>) and transformed system (<b>red orbit</b>). Size of cube side of individual plane projection, from left to right: 50, 10, 4, 0.4.</p> Full article ">Figure 18
<p>Rainbow color-scaled plots of two LLE as function of internal parameters of hyperchaotic system (6). Colored contours represent maximal LLE while line contours second largest LLE; see text for clarification. Overlapped maxima mark dynamical behavior expanding in two directions.</p> Full article ">Figure 19
<p>Four examples of differential voltage trans-conductance multiplier (DV-TC-M) subcircuits that are interesting in terms of synthesis of the linear part of the vector field: (<b>a</b>) negative grounded resistor, (<b>b</b>) unilateral resistor, (<b>c</b>) negative trans-admittance two-port, and (<b>d</b>) voltage controlled current source.</p> Full article ">Figure 20
<p>DV-TC-M based sine-type resistor that considers power series approximation with three terms, cosine- type resistor can be implemented by a cascade of three squarers, negative resistor, and constant current source (each can be implemented by single proposed summation/subtraction block).</p> Full article ">Figure 21
<p>Circuit realization of fully analog chaotic oscillator based on mathematical model having cyclically symmetrical vector field and three pieces of DV-TC-M.</p> Full article ">Figure 22
<p>Circuitry implementation of fourth-order (<b>left schematic</b>) and fifth-order (<b>right picture</b>) jerky dynamical system, both with only a single DV-TC-M active element.</p> Full article ">Figure 23
<p>Circuitry realization of a hyperchaotic system with only three DV-TC-M active elements.</p> Full article ">Figure 24
<p>Realization of the hyperchaotic system based on the integrator block schematic ready for OrCAD PSpice circuit simulation, five active devices needed, nodes with the same label are connected, individual capacitors can be pre-charged using pseudo-component IC1 (not mandatory).</p> Full article ">Figure 25
<p>Circuit simulation of hyperchaotic system based on the integrator block schematic. Upper plot: <span class="html-italic">v<sub>x</sub></span> vs. <span class="html-italic">v<sub>y</sub></span> plane (<b>blue</b>), <span class="html-italic">v<sub>x</sub></span> vs. <span class="html-italic">v<sub>z</sub></span> plane (<b>red</b>). Lower plot: <span class="html-italic">v<sub>w</sub></span> vs. <span class="html-italic">v<sub>y</sub></span> plane (<b>blue</b>), <span class="html-italic">v<sub>w</sub></span> vs. <span class="html-italic">v<sub>z</sub></span> plane (<b>red</b>).</p> Full article ">Figure 25 Cont.
<p>Circuit simulation of hyperchaotic system based on the integrator block schematic. Upper plot: <span class="html-italic">v<sub>x</sub></span> vs. <span class="html-italic">v<sub>y</sub></span> plane (<b>blue</b>), <span class="html-italic">v<sub>x</sub></span> vs. <span class="html-italic">v<sub>z</sub></span> plane (<b>red</b>). Lower plot: <span class="html-italic">v<sub>w</sub></span> vs. <span class="html-italic">v<sub>y</sub></span> plane (<b>blue</b>), <span class="html-italic">v<sub>w</sub></span> vs. <span class="html-italic">v<sub>z</sub></span> plane (<b>red</b>).</p> Full article ">Figure 26
<p>Fabricated printed circuit board (PCB) dedicated to experimental verification of DV-TC-M-based applications: fourth-order jerky function as a robust generator of the chaotic waveforms (<b>left photo</b>) and chaotic oscillator with cyclically symmetrical vector field (<b>right photo</b>).</p> Full article ">Figure 27
<p>Experimental verification of chaotic circuit given in <a href="#entropy-21-00871-f021" class="html-fig">Figure 21</a>: selected plane projections.</p> Full article ">Figure 28
<p>Experimental verification of chaotic circuit given in <a href="#entropy-21-00871-f021" class="html-fig">Figure 21</a>: generated chaotic signals.</p> Full article ">Figure 29
<p>Outputs coming from the numerical integration process (<b>upper row of plots</b>) compared to true experimental results captured as the oscilloscope screenshots: <span class="html-italic">v</span><sub>1</sub> vs. <span class="html-italic">v</span><sub>2</sub> plane projections of the generated state attractors (including chaotic orbits), different values of the external voltage <span class="html-italic">V<sub>a</sub></span> and fixed voltage <span class="html-italic">V<sub>b</sub></span> = 350 mV.</p> Full article ">Figure 30
<p>Experimental results captured as the oscilloscope screenshots: <span class="html-italic">v</span><sub>1</sub> vs. <span class="html-italic">v</span><sub>3</sub> plane projections of generated state attractors (including chaotic orbits), different values of external voltage <span class="html-italic">V<sub>a</sub></span> and fixed voltage <span class="html-italic">V<sub>b</sub></span> = 350 mV.</p> Full article ">Figure 31
<p>Oscilloscope screenshots captured during measurement of fourth-order jerky dynamics, individual rows represent different plane projection of the limit cycles and typical chaotic attractors.</p> Full article ">Figure 32
<p>Oscilloscope screenshots captured during verification of a fifth-order jerky chaotic circuit, plane projections of different limit cycles (left column) and chaotic motion (the rest of photos).</p> Full article ">Figure 33
<p>Oscilloscope screenshots associated with measurement of a hyperchaotic circuit, different plane projections of the compressed strange attractor.</p> Full article ">Figure A1
<p>Layout design of multiplier (Cadence Virtuoso IC6) in bipolar technology.</p> Full article ">Figure A2
<p>Layout design of multiplier (Cadence Virtuoso IC6) in CMOS technology.</p> Full article ">Figure A3
<p>Layout of summation/subtraction cell (Cadence Virtuoso IC6) in CMOS technology.</p> Full article ">
17 pages, 6736 KiB  
Article
Image Encryption Scheme with Compressed Sensing Based on New Three-Dimensional Chaotic System
by Yaqin Xie, Jiayin Yu, Shiyu Guo, Qun Ding and Erfu Wang
Entropy 2019, 21(9), 819; https://doi.org/10.3390/e21090819 - 22 Aug 2019
Cited by 53 | Viewed by 4204
Abstract
In this paper, a new three-dimensional chaotic system is proposed for image encryption. The core of the encryption algorithm is the combination of chaotic system and compressed sensing, which can complete image encryption and compression at the same time. The Lyapunov exponent, bifurcation [...] Read more.
In this paper, a new three-dimensional chaotic system is proposed for image encryption. The core of the encryption algorithm is the combination of chaotic system and compressed sensing, which can complete image encryption and compression at the same time. The Lyapunov exponent, bifurcation diagram and complexity of the new three-dimensional chaotic system are analyzed. The performance analysis shows that the chaotic system has two positive Lyapunov exponents and high complexity. In the encryption scheme, a new chaotic system is used as the measurement matrix for compressed sensing, and Arnold is used to scrambling the image further. The proposed method has better reconfiguration ability in the compressible range of the algorithm compared with other methods. The experimental results show that the proposed encryption scheme has good encryption effect and image compression capability. Full article
(This article belongs to the Special Issue Nonlinear Dynamics and Entropy of Complex Systems with Hidden and Self-Excited Attractors II)
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Figure 1
<p>Chaotic attractor; (<b>a</b>) x-y plane; (<b>b</b>) x-z plane; (<b>c</b>) y-z plane; (<b>d</b>) perspective view.</p> Full article ">Figure 2
<p>Dynamics diagram of Lyapunov exponents of the new three-dimensional chaotic system.</p> Full article ">Figure 3
<p>Bifurcation diagram of the new 3D discrete chaotic map.</p> Full article ">Figure 4
<p>Image encryption and decryption processes.</p> Full article ">Figure 5
<p>Simulation experiment results. (<b>a1–a4</b>) Plaintext images; (<b>b1–b4</b>) Images with chaotic encryption; (<b>c1–c4</b>) Images with scrambling encryption (ciphertext images); (<b>d1–d4</b>) Decrypted images.</p> Full article ">Figure 6
<p>Compression ratios and image reconstruction effects.</p> Full article ">Figure 7
<p>Encryption and decryption histogram analysis: (<b>a</b>) plaintext histogram of Lena; (<b>b</b>) ciphertext histogram of Lena; (<b>c</b>) plaintext histogram of Lake; (<b>d</b>) ciphertext histogram of Lake; (<b>e</b>) plaintext histogram of Cameraman; (<b>f</b>) ciphertext histogram of Cameraman; (<b>g</b>) plaintext histogram of Rice; (<b>h</b>) ciphertext histogram of Rice.</p> Full article ">Figure 8
<p>Correlation analysis of plaintext image and ciphertext image; (<b>a</b>) Lena-horizontal; (<b>b</b>) Encrypted image-horizontal; (<b>c</b>) Lena-vertical; (<b>d</b>) Encrypted image-vertical; (<b>e</b>) Lena-diagonal; (<b>f</b>) Encrypted image-diagonal.</p> Full article ">
19 pages, 12122 KiB  
Article
Coexisting Attractors and Multistability in a Simple Memristive Wien-Bridge Chaotic Circuit
by Yixuan Song, Fang Yuan and Yuxia Li
Entropy 2019, 21(7), 678; https://doi.org/10.3390/e21070678 - 11 Jul 2019
Cited by 33 | Viewed by 3851
Abstract
In this paper, a new voltage-controlled memristor is presented. The mathematical expression of this memristor has an absolute value term, so it is called an absolute voltage-controlled memristor. The proposed memristor is locally active, which is proved by its DC V–I [...] Read more.
In this paper, a new voltage-controlled memristor is presented. The mathematical expression of this memristor has an absolute value term, so it is called an absolute voltage-controlled memristor. The proposed memristor is locally active, which is proved by its DC V–I (Voltage–Current) plot. A simple three-order Wien-bridge chaotic circuit without inductor is constructed on the basis of the presented memristor. The dynamical behaviors of the simple chaotic system are analyzed in this paper. The main properties of this system are coexisting attractors and multistability. Furthermore, an analog circuit of this chaotic system is realized by the Multisim software. The multistability of the proposed system can enlarge the key space in encryption, which makes the encryption effect better. Therefore, the proposed chaotic system can be used as a pseudo-random sequence generator to provide key sequences for digital encryption systems. Thus, the chaotic system is discretized and implemented by Digital Signal Processing (DSP) technology. The National Institute of Standards and Technology (NIST) test and Approximate Entropy analysis of the proposed chaotic system are conducted in this paper. Full article
(This article belongs to the Special Issue Nonlinear Dynamics and Entropy of Complex Systems with Hidden and Self-Excited Attractors II)
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Figure 1
<p>The <span class="html-italic">v</span>–<span class="html-italic">i</span> pinched hysteresis loops of the proposed memristor with different frequencies <span class="html-italic">f</span>.</p> Full article ">Figure 2
<p>DC <span class="html-italic">V</span>–<span class="html-italic">I</span> plot of the proposed memristor.</p> Full article ">Figure 3
<p>The simple memristive Wien-bridge circuit.</p> Full article ">Figure 4
<p>Phase portraits of the proposed chaotic system on (<b>a</b>) <span class="html-italic">x-y-z</span> plane, (<b>b</b>) <span class="html-italic">x-y</span> plane, (<b>c</b>) <span class="html-italic">y-z</span> plane, and (<b>d</b>) <span class="html-italic">x-z</span> plane.</p> Full article ">Figure 4 Cont.
<p>Phase portraits of the proposed chaotic system on (<b>a</b>) <span class="html-italic">x-y-z</span> plane, (<b>b</b>) <span class="html-italic">x-y</span> plane, (<b>c</b>) <span class="html-italic">y-z</span> plane, and (<b>d</b>) <span class="html-italic">x-z</span> plane.</p> Full article ">Figure 5
<p>(<b>a</b>) The time domain waveform of the state variable <span class="html-italic">x</span>, (<b>b</b>) the corresponding Poincare mapping on <span class="html-italic">z</span> = −1.3 section.</p> Full article ">Figure 6
<p>The stable region colored in blue and the unstable region colored in yellow in the region of <span class="html-italic">a</span> ∈ [−20, 10] and <span class="html-italic">b</span> ∈ [−20, 10].</p> Full article ">Figure 7
<p>Bifurcation diagram and Lyapunov exponent spectra varying with <span class="html-italic">a</span>. (<b>a</b>) Bifurcation diagram, (<b>b</b>) Lyapunov exponent spectra.</p> Full article ">Figure 8
<p>Various phase portraits with different <span class="html-italic">a</span> on the <span class="html-italic">x</span>-<span class="html-italic">y</span> plane. (<b>a</b>) <span class="html-italic">a</span> = 1.5 in red, <span class="html-italic">a</span> = 5.5 in blue, (<b>b</b>) <span class="html-italic">a</span> = 2.0, (<b>c</b>) <span class="html-italic">a</span> = 4.0, (<b>d</b>) <span class="html-italic">a</span> = 5.1.</p> Full article ">Figure 9
<p>Coexisting attractors, indicated in red at initial conditions of (0, 0.1, 0) and indicated in blue at initial conditions of (0, −0.1, 0). (<b>a</b>) coexisting attractors on the <span class="html-italic">y</span>-<span class="html-italic">z</span> plane (<b>b</b>) coexisting attractors on the <span class="html-italic">x</span>-<span class="html-italic">z</span> plane.</p> Full article ">Figure 10
<p>Various coexisting attractors with different values of <span class="html-italic">b</span> under initial conditions of (0, 0.1, 0) in red and (0, −0.1, 0) in blue. (<b>a</b>) <span class="html-italic">b</span> = 4.83, (<b>b</b>) <span class="html-italic">b</span> = 5.1, (<b>c</b>) <span class="html-italic">b</span> = 5.5, (<b>d</b>) <span class="html-italic">b</span> = 5.9, (<b>e</b>) <span class="html-italic">b</span> = 6.6, (<b>f</b>) <span class="html-italic">b</span> = 7.0.</p> Full article ">Figure 10 Cont.
<p>Various coexisting attractors with different values of <span class="html-italic">b</span> under initial conditions of (0, 0.1, 0) in red and (0, −0.1, 0) in blue. (<b>a</b>) <span class="html-italic">b</span> = 4.83, (<b>b</b>) <span class="html-italic">b</span> = 5.1, (<b>c</b>) <span class="html-italic">b</span> = 5.5, (<b>d</b>) <span class="html-italic">b</span> = 5.9, (<b>e</b>) <span class="html-italic">b</span> = 6.6, (<b>f</b>) <span class="html-italic">b</span> = 7.0.</p> Full article ">Figure 11
<p>Coexisting bifurcation diagram and the corresponding Lyapunov exponent spectra varying with <span class="html-italic">b</span>. (<b>a</b>) Coexisting bifurcation diagram of the variable <span class="html-italic">x</span> at the initial conditions of (0, 0.1, 0) in red and at the initial conditions of (0, −0.1, 0) in blue, (<b>b</b>) corresponding Lyapunov exponent spectra.</p> Full article ">Figure 12
<p>Bifurcation diagram of the state variable <span class="html-italic">x</span> varying with different initial values. (<b>a</b>) Variation with initial value <span class="html-italic">x</span>(0), (<b>b</b>) variation with initial value <span class="html-italic">y</span>(0), (<b>c</b>) variation with initial value <span class="html-italic">z</span>(0).</p> Full article ">Figure 13
<p>Corresponding Lyapunov exponent spectra varying with different initial values. (<b>a</b>) Variation with initial value <span class="html-italic">x</span>(0), (<b>b</b>) variation with initial value <span class="html-italic">y</span>(0), (<b>c</b>) variation with initial value <span class="html-italic">z</span>(0).</p> Full article ">Figure 14
<p>The analog circuit of the memristive Wien-bridge chaotic oscillator.</p> Full article ">Figure 15
<p>Simulation results obtained from the Multisim software. (<b>a</b>) Attractor on the <span class="html-italic">x</span>-<span class="html-italic">y</span> plane, (<b>b</b>) attractor on the <span class="html-italic">y</span>-<span class="html-italic">z</span> plane, (<b>c</b>) attractor on the <span class="html-italic">x</span>-<span class="html-italic">z</span> plane, (<b>d</b>) coexisting attractors the on <span class="html-italic">x</span>-<span class="html-italic">y</span> plane, (<b>e</b>) coexisting attractors on the <span class="html-italic">y</span>-<span class="html-italic">z</span> plane, (<b>f</b>) coexisting attractors on the <span class="html-italic">x</span>-<span class="html-italic">z</span> plane.</p> Full article ">Figure 15 Cont.
<p>Simulation results obtained from the Multisim software. (<b>a</b>) Attractor on the <span class="html-italic">x</span>-<span class="html-italic">y</span> plane, (<b>b</b>) attractor on the <span class="html-italic">y</span>-<span class="html-italic">z</span> plane, (<b>c</b>) attractor on the <span class="html-italic">x</span>-<span class="html-italic">z</span> plane, (<b>d</b>) coexisting attractors the on <span class="html-italic">x</span>-<span class="html-italic">y</span> plane, (<b>e</b>) coexisting attractors on the <span class="html-italic">y</span>-<span class="html-italic">z</span> plane, (<b>f</b>) coexisting attractors on the <span class="html-italic">x</span>-<span class="html-italic">z</span> plane.</p> Full article ">Figure 16
<p>Experimental results obtained by Digital Signal Processing (DSP) technology. (<b>a</b>) Chaotic pseudo-noise (PN) sequence extracted from <b>the</b> variable <span class="html-italic">x</span>, (<b>b</b>) chaotic attractor on the <span class="html-italic">x</span>-<span class="html-italic">y</span> plane, (<b>c</b>) chaotic attractor on the <span class="html-italic">y</span>-<span class="html-italic">z</span> plane, (<b>d</b>) chaotic attractors on the <span class="html-italic">x</span>-<span class="html-italic">z</span> plane, (<b>e</b>) coexisting attractors on the <span class="html-italic">y</span>-<span class="html-italic">z</span> plane, (<b>f</b>) coexisting attractors on the <span class="html-italic">x</span>-<span class="html-italic">z</span> plane.</p> Full article ">Figure 17
<p>Experimental equipment.</p> Full article ">
13 pages, 1407 KiB  
Article
A Class of Quadratic Polynomial Chaotic Maps and Their Fixed Points Analysis
by Chuanfu Wang and Qun Ding
Entropy 2019, 21(7), 658; https://doi.org/10.3390/e21070658 - 4 Jul 2019
Cited by 13 | Viewed by 3345
Abstract
When chaotic systems are used in different practical applications, such as chaotic secure communication and chaotic pseudorandom sequence generators, a large number of chaotic systems are strongly required. However, for a lack of a systematic construction theory, the construction of chaotic systems mainly [...] Read more.
When chaotic systems are used in different practical applications, such as chaotic secure communication and chaotic pseudorandom sequence generators, a large number of chaotic systems are strongly required. However, for a lack of a systematic construction theory, the construction of chaotic systems mainly depends on the exhaustive search of systematic parameters or initial values, especially for a class of dynamical systems with hidden chaotic attractors. In this paper, a class of quadratic polynomial chaotic maps is studied, and a general method for constructing quadratic polynomial chaotic maps is proposed. The proposed polynomial chaotic maps satisfy the Li–Yorke definition of chaos. This method can accurately control the amplitude of chaotic time series. Through the existence and stability analysis of fixed points, we proved that such class quadratic polynomial maps cannot have hidden chaotic attractors. Full article
(This article belongs to the Special Issue Nonlinear Dynamics and Entropy of Complex Systems with Hidden and Self-Excited Attractors II)
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Figure 1
<p>The output chaotic time series under <math display="inline"><semantics> <mrow> <mi>a</mi> <mo>=</mo> <mo>−</mo> <mn>1.5</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>b</mi> <mo>=</mo> <mn>5.5</mn> </mrow> </semantics></math>, and <math display="inline"><semantics> <mrow> <mi>d</mi> <mo>=</mo> <mo>−</mo> <mn>2</mn> </mrow> </semantics></math>, and iterative route diagram (<b>a</b>) initial value <math display="inline"><semantics> <mrow> <mi>x</mi> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mn>1.3</mn> </mrow> </semantics></math>, (<b>b</b>) initial value <math display="inline"><semantics> <mrow> <mi>x</mi> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0.3</mn> </mrow> </semantics></math>, (<b>c</b>) initial value <math display="inline"><semantics> <mrow> <mi>x</mi> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math>, (<b>d</b>) iterative route diagram.</p> Full article ">Figure 1 Cont.
<p>The output chaotic time series under <math display="inline"><semantics> <mrow> <mi>a</mi> <mo>=</mo> <mo>−</mo> <mn>1.5</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>b</mi> <mo>=</mo> <mn>5.5</mn> </mrow> </semantics></math>, and <math display="inline"><semantics> <mrow> <mi>d</mi> <mo>=</mo> <mo>−</mo> <mn>2</mn> </mrow> </semantics></math>, and iterative route diagram (<b>a</b>) initial value <math display="inline"><semantics> <mrow> <mi>x</mi> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mn>1.3</mn> </mrow> </semantics></math>, (<b>b</b>) initial value <math display="inline"><semantics> <mrow> <mi>x</mi> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0.3</mn> </mrow> </semantics></math>, (<b>c</b>) initial value <math display="inline"><semantics> <mrow> <mi>x</mi> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math>, (<b>d</b>) iterative route diagram.</p> Full article ">Figure 2
<p>The output chaotic time series of chaotic maps (15) (<b>a</b>) <math display="inline"><semantics> <mrow> <mi>t</mi> <mo>=</mo> <mn>0.16</mn> <mo>,</mo> <mi>w</mi> <mo>=</mo> <mn>0.5</mn> <mo>,</mo> <mi>k</mi> <mo>=</mo> <mn>2</mn> </mrow> </semantics></math>, (<b>b</b>) <math display="inline"><semantics> <mrow> <mi>t</mi> <mo>=</mo> <mn>0.64</mn> <mo>,</mo> <mi>w</mi> <mo>=</mo> <mn>0.5</mn> <mo>,</mo> <mi>k</mi> <mo>=</mo> <mn>2</mn> </mrow> </semantics></math>, (<b>c</b>) <math display="inline"><semantics> <mrow> <mi>t</mi> <mo>=</mo> <mn>0.16</mn> <mo>,</mo> <mi>w</mi> <mo>=</mo> <mn>5</mn> <mo>,</mo> <mi>k</mi> <mo>=</mo> <mn>2</mn> </mrow> </semantics></math>, (<b>d</b>) <math display="inline"><semantics> <mrow> <mi>t</mi> <mo>=</mo> <mn>0.64</mn> <mo>,</mo> <mi>w</mi> <mo>=</mo> <mn>0.01</mn> <mo>,</mo> <mi>k</mi> <mo>=</mo> <mn>2</mn> </mrow> </semantics></math>.</p> Full article ">Figure 3
<p>The output chaotic time series of chaotic maps (17) (<b>a</b>) <math display="inline"><semantics> <mrow> <mi>M</mi> <mo>=</mo> <mn>4</mn> </mrow> </semantics></math>, (<b>b</b>) <math display="inline"><semantics> <mrow> <mi>M</mi> <mo>=</mo> <mn>2</mn> </mrow> </semantics></math>, (<b>c</b>) <math display="inline"><semantics> <mrow> <mi>M</mi> <mo>=</mo> <mn>0.5</mn> </mrow> </semantics></math>, (<b>d</b>) <math display="inline"><semantics> <mrow> <mi>M</mi> <mo>=</mo> <mn>0.25</mn> </mrow> </semantics></math>.</p> Full article ">Figure 4
<p>The output chaotic time series of the chaotic map (31).</p> Full article ">
14 pages, 6818 KiB  
Article
A Giga-Stable Oscillator with Hidden and Self-Excited Attractors: A Megastable Oscillator Forced by His Twin
by Thoai Phu Vo, Yeganeh Shaverdi, Abdul Jalil M. Khalaf, Fawaz E. Alsaadi, Tasawar Hayat and Viet-Thanh Pham
Entropy 2019, 21(5), 535; https://doi.org/10.3390/e21050535 - 25 May 2019
Cited by 10 | Viewed by 3948
Abstract
In this paper, inspired by a newly proposed two-dimensional nonlinear oscillator with an infinite number of coexisting attractors, a modified nonlinear oscillator is proposed. The original system has an exciting feature of having layer–layer coexisting attractors. One of these attractors is self-excited while [...] Read more.
In this paper, inspired by a newly proposed two-dimensional nonlinear oscillator with an infinite number of coexisting attractors, a modified nonlinear oscillator is proposed. The original system has an exciting feature of having layer–layer coexisting attractors. One of these attractors is self-excited while the rest are hidden. By forcing this system with its twin, a new four-dimensional nonlinear system is obtained which has an infinite number of coexisting torus attractors, strange attractors, and limit cycle attractors. The entropy, energy, and homogeneity of attractors’ images and their basin of attractions are calculated and reported, which showed an increase in the complexity of attractors when changing the bifurcation parameters. Full article
(This article belongs to the Special Issue Nonlinear Dynamics and Entropy of Complex Systems with Hidden and Self-Excited Attractors II)
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Figure 1

Figure 1
<p>Hidden attractors of System (1) with seven different initial conditions. Except for the inner limit cycle, the rest of the attractors are hidden attractors.</p> Full article ">Figure 2
<p>The basins of attraction of System (1) with values of parameters <math display="inline"><semantics> <mrow> <mi>k</mi> <mo>=</mo> <mn>0.33</mn> </mrow> </semantics></math>.</p> Full article ">Figure 3
<p>(<b>a</b>) The limit cycle of <math display="inline"><semantics> <mi>z</mi> </semantics></math> and <math display="inline"><semantics> <mi>u</mi> </semantics></math> variables in System (3) with <math display="inline"><semantics> <mrow> <mi>ω</mi> <mo>=</mo> <mn>0.5</mn> </mrow> </semantics></math>, (<b>b</b>) <math display="inline"><semantics> <mrow> <mi>ω</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math>, and (<b>c</b>) <math display="inline"><semantics> <mrow> <mi>ω</mi> <mo>=</mo> <mn>2</mn> </mrow> </semantics></math> that are the same attractors and changing the parameter <math display="inline"><semantics> <mi>ω</mi> </semantics></math> has no effect on the attractor’s topology, (<b>d</b>) The time series of variable <math display="inline"><semantics> <mi>z</mi> </semantics></math> in <math display="inline"><semantics> <mrow> <mi>ω</mi> <mo>=</mo> <mn>0.5</mn> </mrow> </semantics></math>, (<b>e</b>) <math display="inline"><semantics> <mrow> <mi>ω</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math>, and (<b>f</b>) <math display="inline"><semantics> <mrow> <mi>ω</mi> <mo>=</mo> <mn>2</mn> </mrow> </semantics></math> that differ with each other in proportion to the value of <math display="inline"><semantics> <mi>ω</mi> </semantics></math> as the frequency tuner.</p> Full article ">Figure 4
<p>(<b>a</b>) The bifurcation diagram of the System (3), and (<b>b</b>) the Lyapunov exponents of the System (3).</p> Full article ">Figure 5
<p>The self-excited attractor (inner limit cycle) and six hidden torus attractors of System (2) with different initial conditions and value of parameter <math display="inline"><semantics> <mrow> <mi>A</mi> <mo>=</mo> <mn>0.1</mn> </mrow> </semantics></math>.</p> Full article ">Figure 6
<p>The basin of attraction of the attractors shown in <a href="#entropy-21-00535-f005" class="html-fig">Figure 5</a> with <math display="inline"><semantics> <mrow> <mi>A</mi> <mo>=</mo> <mn>0.1</mn> </mrow> </semantics></math>.</p> Full article ">Figure 7
<p>The self-excited attractor (inner strange attractor) and other hidden attractors of System (3) with different initial conditions for <math display="inline"><semantics> <mrow> <mi>A</mi> <mo>=</mo> <mn>0.25</mn> </mrow> </semantics></math>.</p> Full article ">Figure 8
<p>Basin of attraction of attractors shown in <a href="#entropy-21-00535-f007" class="html-fig">Figure 7</a> with <math display="inline"><semantics> <mrow> <mi>A</mi> <mo>=</mo> <mn>0.25</mn> </mrow> </semantics></math> with a chaotic solution.</p> Full article ">Figure 9
<p>The GLCM matrix with different directions and angles.</p> Full article ">Figure 10
<p>The Entropy measure of <a href="#entropy-21-00535-f001" class="html-fig">Figure 1</a>, <a href="#entropy-21-00535-f005" class="html-fig">Figure 5</a>, and <a href="#entropy-21-00535-f007" class="html-fig">Figure 7</a> in four directions <math display="inline"><semantics> <mrow> <mn>0</mn> <mo>°</mo> <mo>,</mo> <mo> </mo> <mn>45</mn> <mo>°</mo> <mo>,</mo> <mo> </mo> <mn>90</mn> <mo>°</mo> </mrow> </semantics></math>, and <math display="inline"><semantics> <mrow> <mn>135</mn> <mo>°</mo> </mrow> </semantics></math>. The Entropy of chaotic attractors is larger than that of torus and limit cycle attractors.</p> Full article ">Figure 11
<p>The Energy measure of <a href="#entropy-21-00535-f001" class="html-fig">Figure 1</a>, <a href="#entropy-21-00535-f005" class="html-fig">Figure 5</a>, and <a href="#entropy-21-00535-f007" class="html-fig">Figure 7</a> in four directions <math display="inline"><semantics> <mrow> <mn>0</mn> <mo>°</mo> <mo>,</mo> <mo> </mo> <mn>45</mn> <mo>°</mo> <mo>,</mo> <mo> </mo> <mn>90</mn> <mo>°</mo> </mrow> </semantics></math>, and <math display="inline"><semantics> <mrow> <mn>135</mn> <mo>°</mo> </mrow> </semantics></math>. The Energy of the limit cycle attractors is larger than the Energy of the torus and chaotic attractors contrary to Entropy.</p> Full article ">Figure 12
<p>The Homogeneity measure of <a href="#entropy-21-00535-f001" class="html-fig">Figure 1</a>, <a href="#entropy-21-00535-f005" class="html-fig">Figure 5</a>, and <a href="#entropy-21-00535-f007" class="html-fig">Figure 7</a> in four directions <math display="inline"><semantics> <mrow> <mn>0</mn> <mo>°</mo> <mo>,</mo> <mo> </mo> <mn>45</mn> <mo>°</mo> <mo>,</mo> <mo> </mo> <mn>90</mn> <mo>°</mo> </mrow> </semantics></math>, and <math display="inline"><semantics> <mrow> <mn>135</mn> <mo>°</mo> </mrow> </semantics></math>. The Homogeneity of the limit cycle attractors is larger than the Homogeneity of the torus and chaotic attractors that is the same result for Energy.</p> Full article ">Figure 13
<p>The Entropy measure of <a href="#entropy-21-00535-f006" class="html-fig">Figure 6</a> and <a href="#entropy-21-00535-f008" class="html-fig">Figure 8</a> in four directions <math display="inline"><semantics> <mrow> <mn>0</mn> <mo>°</mo> <mo>,</mo> <mo> </mo> <mn>45</mn> <mo>°</mo> <mo>,</mo> <mo> </mo> <mn>90</mn> <mo>°</mo> </mrow> </semantics></math>, and <math display="inline"><semantics> <mrow> <mn>135</mn> <mo>°</mo> </mrow> </semantics></math>. The Entropy of the limit cycle basins is lower than the Entropy of chaotic basins of attraction.</p> Full article ">Figure 14
<p>The Energy measure of <a href="#entropy-21-00535-f006" class="html-fig">Figure 6</a> and <a href="#entropy-21-00535-f008" class="html-fig">Figure 8</a> in four directions <math display="inline"><semantics> <mrow> <mn>0</mn> <mo>°</mo> <mo>,</mo> <mo> </mo> <mn>45</mn> <mo>°</mo> <mo>,</mo> <mo> </mo> <mn>90</mn> <mo>°</mo> </mrow> </semantics></math>, and <math display="inline"><semantics> <mrow> <mn>135</mn> <mo>°</mo> </mrow> </semantics></math>. The Energy of the chaotic basins of attraction is lower than the Energy torus basins of attraction that is the opposite result for Entropy.</p> Full article ">Figure 15
<p>The Homogeneity measure of <a href="#entropy-21-00535-f006" class="html-fig">Figure 6</a> and <a href="#entropy-21-00535-f008" class="html-fig">Figure 8</a> in four directions <math display="inline"><semantics> <mrow> <mn>0</mn> <mo>°</mo> <mo>,</mo> <mo> </mo> <mn>45</mn> <mo>°</mo> <mo>,</mo> <mo> </mo> <mn>90</mn> <mo>°</mo> </mrow> </semantics></math>, and <math display="inline"><semantics> <mrow> <mn>135</mn> <mo>°</mo> </mrow> </semantics></math>. The Homogeneity of the chaotic basin of attraction is larger than the Homogeneity of torus basin of attraction that is the likewise result for Energy.</p> Full article ">
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