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Entropy
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Complex and Fractional Dynamics
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Complex and Fractional Dynamics

A special issue of Entropy (ISSN 1099-4300).

Deadline for manuscript submissions: closed (30 November 2015) | Viewed by 172814

Special Issue Editors

Prof. Dr. José A. Tenreiro Machado

grade Website
Guest Editor
Department of Electrical Engineering, Institute of Engineering, Polytechnic Institute of Porto, 4249-015 Porto, Portugal
Interests: nonlinear dynamics; fractional calculus; modeling; control; evolutionary computing; genomics; robotics, complex systems
Special Issues, Collections and Topics in MDPI journals
Dr. António Lopes

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Guest Editor
Faculty of Engineering, University of Porto, Rua Dr. Roberto Frias, 4200-465 Porto, Portugal
Interests: complex systems modelling; automation and robotics; fractional order systems modelling and control; data analysis and visualization; machine learning
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

Complex systems are pervasive in many areas of science and we find them everyday and everywhere. Examples include financial markets, highway transportation networks, telecommunication networks, world and country economies, social networks, immunological systems, living organisms, computational systems, and electrical and mechanical structures. Complex systems are often composed of large numbers of interconnected and interacting entities exhibiting much richer global scale dynamics than can be inferred from the properties and behavior of individual entities. Complex systems are studied in many areas of natural sciences, social sciences, engineering, and mathematical sciences.

This Special Issue focuses on original and new research results concerning systems dynamics in science and engineering. Manuscripts regarding complex dynamical systems, nonlinearity, chaos, and fractional dynamics in the thermodynamics or information processing perspectives are solicited. We welcome submissions addressing novel issues, as well as those on more specific topics that illustrate the broad impact of entropy-based techniques in complexity, nonlinearity, and fractionality.

Papers should fit the scope of the journal Entropy and topics of interest include (but are not limited to):
- Complex dynamics- Nonlinear dynamical systems
- Advanced control systems
- Fractional calculus and its applications
- Evolutionary computing
- Finance and economy dynamics
- Fractals and chaos
- Biological systems and bioinformatics
- Nonlinear waves and acoustics
- Image and signal processing
- Transportation systems
- Geosciences
- Astronomy and cosmology
- Nuclear physics

Prof. Dr. J. A. Tenreiro Machado
Prof. Dr. António M. Lopes
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.

Keywords

  • dynamics
  • complex systems
  • fractional calculus
  • entropy
  • information

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Published Papers (22 papers)

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Editorial

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163 KiB  
Editorial
Complex and Fractional Dynamics
by J. A. Tenreiro Machado and António M. Lopes
Entropy 2017, 19(2), 62; https://doi.org/10.3390/e19020062 - 8 Feb 2017
Cited by 8 | Viewed by 4017
Abstract
Complex systems (CS) are pervasive in many areas, namely financial markets; highway transportation; telecommunication networks; world and country economies; social networks; immunological systems; living organisms; computational systems; and electrical and mechanical structures. CS are often composed of a large number of interconnected and [...] Read more.
Complex systems (CS) are pervasive in many areas, namely financial markets; highway transportation; telecommunication networks; world and country economies; social networks; immunological systems; living organisms; computational systems; and electrical and mechanical structures. CS are often composed of a large number of interconnected and interacting entities exhibiting much richer global scale dynamics than could be inferred from the properties and behavior of individual elements. [...]
Full article
(This article belongs to the Special Issue Complex and Fractional Dynamics)

Research

Jump to: Editorial

5283 KiB  
Article
Chaos on the Vallis Model for El Niño with Fractional Operators
by Badr Saad T. Alkahtani and Abdon Atangana
Entropy 2016, 18(4), 100; https://doi.org/10.3390/e18040100 - 23 Mar 2016
Cited by 25 | Viewed by 5932
Abstract
The Vallis model for El Niño is an important model describing a very interesting physical problem. The aim of this paper is to investigate and compare the models using both integer and non-integer order derivatives. We first studied the model with the local [...] Read more.
The Vallis model for El Niño is an important model describing a very interesting physical problem. The aim of this paper is to investigate and compare the models using both integer and non-integer order derivatives. We first studied the model with the local derivative by presenting for the first time the exact solution for equilibrium points, and then we presented the exact solutions with the numerical simulations. We further examined the model within the scope of fractional order derivatives. The fractional derivatives used here are the Caputo derivative and Caputo–Fabrizio type. Within the scope of fractional derivatives, we presented the existence and unique solutions of the model. We derive special solutions of both models with Caputo and Caputo–Fabrizio derivatives. Some numerical simulations are presented to compare the models. We obtained more chaotic behavior from the model with Caputo–Fabrizio derivative than other one with local and Caputo derivative. When compare the three models, we realized that, the Caputo derivative plays a role of low band filter when the Caputo–Fabrizio presents more information that were not revealed in the model with local derivative. Full article
(This article belongs to the Special Issue Complex and Fractional Dynamics)
►▼ Show Figures

Figure 1

Figure 1
<p>Solution as function of time for <span class="html-italic">p</span> = 0.3.</p> Full article ">Figure 2
<p>Solution as function of time for <span class="html-italic">p</span> = 0.</p> Full article ">Figure 3
<p>Parametric for <span class="html-italic">p</span> = 0.3.</p> Full article ">Figure 4
<p>Parametric plot for <span class="html-italic">p</span> = 0.</p> Full article ">Figure 5
<p>2-Dimensional parametric plot of <span class="html-italic">y(t)</span> and <span class="html-italic">z(t)</span> for <span class="html-italic">p</span> = 0.3.</p> Full article ">Figure 6
<p>2-Dimensional parametric plot of <span class="html-italic">y(t)</span> and <span class="html-italic">z(t)</span> for <span class="html-italic">p</span> = 0.3.</p> Full article ">Figure 7
<p>3-Dimensional representation for <span class="html-italic">α</span> = 0.85.</p> Full article ">Figure 8
<p>3-Dimensional representation for <span class="html-italic">α</span> = 0.55.</p> Full article ">Figure 9
<p>3-Dimensional representation for <span class="html-italic">α</span> = 0.25.</p> Full article ">Figure 10
<p>3-Dimensional representation for <span class="html-italic">α</span> = 0.85.</p> Full article ">Figure 11
<p>3-Dimensional representation for <span class="html-italic">α</span> = 0.55.</p> Full article ">Figure 12
<p>3-Dimensional representation for <span class="html-italic">α</span> = 0.25.</p> Full article ">
240 KiB  
Article
A Novel Weak Fuzzy Solution for Fuzzy Linear System
by Soheil Salahshour, Ali Ahmadian, Fudziah Ismail and Dumitru Baleanu
Entropy 2016, 18(3), 68; https://doi.org/10.3390/e18030068 - 11 Mar 2016
Cited by 8 | Viewed by 4736
Abstract
This article proposes a novel weak fuzzy solution for the fuzzy linear system. As a matter of fact, we define the right-hand side column of the fuzzy linear system as a piecewise fuzzy function to overcome the related shortcoming, which exists in the [...] Read more.
This article proposes a novel weak fuzzy solution for the fuzzy linear system. As a matter of fact, we define the right-hand side column of the fuzzy linear system as a piecewise fuzzy function to overcome the related shortcoming, which exists in the previous findings. The strong point of this proposal is that the weak fuzzy solution is always a fuzzy number vector. Two complex and non-complex linear systems under uncertainty are tested to validate the effectiveness and correctness of the presented method. Full article
(This article belongs to the Special Issue Complex and Fractional Dynamics)
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Figure 1

Figure 1
<p>A comparison between our proposed solution with the solution in [<a href="#B17-entropy-18-00068" class="html-bibr">17</a>]. It can be noticed, and was proved in [<a href="#B19-entropy-18-00068" class="html-bibr">19</a>] that (<b>a</b>) the solution in [<a href="#B17-entropy-18-00068" class="html-bibr">17</a>] is not a fuzzy vector; while using our new definition (Definition 6), (<b>b</b>) the obtained solution is a fuzzy vector. (<span class="html-italic">y</span>-axis represents the <span class="html-italic">α</span>-cuts and <span class="html-italic">x</span>-axis is for the value of fuzzy solution (<math display="inline"> <msub> <mi mathvariant="bold">z</mi> <mn>1</mn> </msub> </math>)).</p> Full article ">
719 KiB  
Article
Stability Analysis and Synchronization for a Class of Fractional-Order Neural Networks
by Guanjun Li and Heng Liu
Entropy 2016, 18(2), 55; https://doi.org/10.3390/e18020055 - 6 Feb 2016
Cited by 25 | Viewed by 6195
Abstract
Stability of a class of fractional-order neural networks (FONNs) is analyzed in this paper. First, two sufficient conditions for convergence of the solution for such systems are obtained by utilizing Gronwall–Bellman lemma and Laplace transform technique. Then, according to the fractional-order Lyapunov second [...] Read more.
Stability of a class of fractional-order neural networks (FONNs) is analyzed in this paper. First, two sufficient conditions for convergence of the solution for such systems are obtained by utilizing Gronwall–Bellman lemma and Laplace transform technique. Then, according to the fractional-order Lyapunov second method and linear feedback control, the synchronization problem between two fractional-order chaotic neural networks is investigated. Finally, several numerical examples are presented to justify the feasibility of the proposed methods. Full article
(This article belongs to the Special Issue Complex and Fractional Dynamics)
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Figure 1

Figure 1
<p>Boundedness of the solution <math display="inline"> <mrow> <mi>x</mi> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </math> for fractional-order neural network (44) with <math display="inline"> <mrow> <msub> <mi>c</mi> <mn>1</mn> </msub> <mo>=</mo> <mn>0</mn> <mo>.</mo> <mn>3</mn> </mrow> </math> and <math display="inline"> <mrow> <msub> <mi>c</mi> <mn>2</mn> </msub> <mo>=</mo> <mn>0</mn> <mo>.</mo> <mn>35</mn> </mrow> </math>.</p> Full article ">Figure 2
<p>Convergence of the solution <math display="inline"> <mrow> <mi>x</mi> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </math> for fractional-order neural network (44) with <math display="inline"> <mrow> <msub> <mi>c</mi> <mn>1</mn> </msub> <mo>=</mo> <mn>2</mn> </mrow> </math> and <math display="inline"> <mrow> <msub> <mi>c</mi> <mn>2</mn> </msub> <mo>=</mo> <mn>3</mn> </mrow> </math>.</p> Full article ">Figure 3
<p>Time responses of <math display="inline"> <mrow> <mi>x</mi> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </math> of system (16) with initial value <math display="inline"> <mrow> <msup> <mrow> <mo>[</mo> <mo>−</mo> <mn>0</mn> <mo>.</mo> <mn>3</mn> <mo>,</mo> <mn>0</mn> <mo>.</mo> <mn>4</mn> <mo>,</mo> <mn>0</mn> <mo>.</mo> <mn>3</mn> <mo>]</mo> </mrow> <mi>T</mi> </msup> <mo>.</mo> </mrow> </math></p> Full article ">Figure 4
<p>Chaotic trajectories of system (16) with initial value <math display="inline"> <mrow> <msup> <mrow> <mo>[</mo> <mo>−</mo> <mn>0</mn> <mo>.</mo> <mn>3</mn> <mo>,</mo> <mn>0</mn> <mo>.</mo> <mn>4</mn> <mo>,</mo> <mn>0</mn> <mo>.</mo> <mn>3</mn> <mo>]</mo> </mrow> <mi>T</mi> </msup> <mo>.</mo> </mrow> </math></p> Full article ">Figure 5
<p>Synchronization trajectories of (16) and (37).</p> Full article ">Figure 6
<p>Synchronization errors.</p> Full article ">
204 KiB  
Article
Particular Solutions of the Confluent Hypergeometric Differential Equation by Using the Nabla Fractional Calculus Operator
by Resat Yilmazer, Mustafa Inc, Fairouz Tchier and Dumitru Baleanu
Entropy 2016, 18(2), 49; https://doi.org/10.3390/e18020049 - 5 Feb 2016
Cited by 19 | Viewed by 5077
Abstract
In this work; we present a method for solving the second-order linear ordinary differential equation of hypergeometric type. The solutions of this equation are given by the confluent hypergeometric functions (CHFs). Unlike previous studies, we obtain some different new solutions of the equation [...] Read more.
In this work; we present a method for solving the second-order linear ordinary differential equation of hypergeometric type. The solutions of this equation are given by the confluent hypergeometric functions (CHFs). Unlike previous studies, we obtain some different new solutions of the equation without using the CHFs. Therefore, we obtain new discrete fractional solutions of the homogeneous and non-homogeneous confluent hypergeometric differential equation (CHE) by using a discrete fractional Nabla calculus operator. Thus, we obtain four different new discrete complex fractional solutions for these equations. Full article
(This article belongs to the Special Issue Complex and Fractional Dynamics)
258 KiB  
Article
New Derivatives on the Fractal Subset of Real-Line
by Alireza Khalili Golmankhaneh and Dumitru Baleanu
Entropy 2016, 18(2), 1; https://doi.org/10.3390/e18020001 - 29 Jan 2016
Cited by 48 | Viewed by 6974
Abstract
In this manuscript we introduced the generalized fractional Riemann-Liouville and Caputo like derivative for functions defined on fractal sets. The Gamma, Mittag-Leffler and Beta functions were defined on the fractal sets. The non-local Laplace transformation is given and applied for solving linear and [...] Read more.
In this manuscript we introduced the generalized fractional Riemann-Liouville and Caputo like derivative for functions defined on fractal sets. The Gamma, Mittag-Leffler and Beta functions were defined on the fractal sets. The non-local Laplace transformation is given and applied for solving linear and non-linear fractal equations. The advantage of using these new nonlocal derivatives on the fractals subset of real-line lies in the fact that they are better at modeling processes with memory effect. Full article
(This article belongs to the Special Issue Complex and Fractional Dynamics)
►▼ Show Figures

Figure 1

Figure 1
<p>The finite iteration for constructing the triadic Cantor set.</p> Full article ">Figure 2
<p>We plot the integral staircase function for triadic Cantor.</p> Full article ">Figure 3
<p>We sketch the fractal Gamma function which is compared with the standard case.</p> Full article ">Figure 4
<p>We plot <math display="inline"> <mrow> <mi>y</mi> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> <mo>=</mo> <msup> <mi>x</mi> <mn>2</mn> </msup> </mrow> </math> and <math display="inline"> <mrow> <mi>f</mi> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> <mo>=</mo> <msubsup> <mi>S</mi> <mrow> <mi>F</mi> </mrow> <mi>α</mi> </msubsup> <msup> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> <mn>2</mn> </msup> </mrow> </math> and their non-local derivative <math display="inline"> <mrow> <msub> <mrow/> <mn>0</mn> </msub> <msubsup> <mi>D</mi> <mrow> <mi>x</mi> </mrow> <mrow> <mn>0</mn> <mo>.</mo> <mn>5</mn> </mrow> </msubsup> <mi>y</mi> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> </mrow> </math> and <math display="inline"> <mrow> <msub> <mrow/> <mn>0</mn> </msub> <msubsup> <mi mathvariant="script">D</mi> <mrow> <mi>x</mi> </mrow> <mrow> <mn>0</mn> <mo>.</mo> <mn>5</mn> </mrow> </msubsup> <mi>f</mi> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> </mrow> </math>, respectively.</p> Full article ">Figure 5
<p>We show the graph of <math display="inline"> <mrow> <mi>g</mi> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> <mo>=</mo> <msup> <mi>x</mi> <mn>2</mn> </msup> </mrow> </math> and <math display="inline"> <mrow> <mi>f</mi> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> <mo>=</mo> <msubsup> <mi>S</mi> <mrow> <mi>F</mi> </mrow> <mi>α</mi> </msubsup> <msup> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> <mn>2</mn> </msup> </mrow> </math> and their non-local integral <math display="inline"> <mrow> <msub> <mrow/> <mn>0</mn> </msub> <msubsup> <mi>I</mi> <mrow> <mi>x</mi> </mrow> <mrow> <mn>0</mn> <mo>.</mo> <mn>5</mn> </mrow> </msubsup> <mi>g</mi> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> </mrow> </math> and <math display="inline"> <mrow> <msub> <mrow/> <mn>0</mn> </msub> <msubsup> <mi mathvariant="script">I</mi> <mrow> <mi>x</mi> </mrow> <mrow> <mn>0</mn> <mo>.</mo> <mn>5</mn> </mrow> </msubsup> <mi>f</mi> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> </mrow> </math>, respectively.</p> Full article ">Figure 6
<p>We present the solution of Equation (51) on the real-line and Cantor set.</p> Full article ">Figure 7
<p>We give the graph of the solution of Equation (54) on the real-line and Cantor set.</p> Full article ">Figure 8
<p>We plot the solution of Equation (57) on the real-line and Cantor set.</p> Full article ">
1039 KiB  
Article
Predicting Traffic Flow in Local Area Networks by the Largest Lyapunov Exponent
by Yan Liu and Jiazhong Zhang
Entropy 2016, 18(1), 32; https://doi.org/10.3390/e18010032 - 19 Jan 2016
Cited by 14 | Viewed by 5559
Abstract
The dynamics of network traffic are complex and nonlinear, and chaotic behaviors and their prediction, which play an important role in local area networks (LANs), are studied in detail, using the largest Lyapunov exponent. With the introduction of phase space reconstruction based on [...] Read more.
The dynamics of network traffic are complex and nonlinear, and chaotic behaviors and their prediction, which play an important role in local area networks (LANs), are studied in detail, using the largest Lyapunov exponent. With the introduction of phase space reconstruction based on the time sequence, the high-dimensional traffic is projected onto the low dimension reconstructed phase space, and a reduced dynamic system is obtained from the dynamic system viewpoint. Then, a numerical method for computing the largest Lyapunov exponent of the low-dimensional dynamic system is presented. Further, the longest predictable time, which is related to chaotic behaviors in the system, is studied using the largest Lyapunov exponent, and the Wolf method is used to predict the evolution of the traffic in a local area network by both Dot and Interval predictions, and a reliable result is obtained by the presented method. As the conclusion, the results show that the largest Lyapunov exponent can be used to describe the sensitivity of the trajectory in the reconstructed phase space to the initial values. Moreover, Dot Prediction can effectively predict the flow burst. The numerical simulation also shows that the presented method is feasible and efficient for predicting the complex dynamic behaviors in LAN traffic, especially for congestion and attack in networks, which are the main two complex phenomena behaving as chaos in networks. Full article
(This article belongs to the Special Issue Complex and Fractional Dynamics)
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Figure 1

Figure 1
<p>Real LAN traffic data in the week. (<b>a</b>) Real network traffic; (<b>b</b>) Average daily traffic sequence.</p> Full article ">Figure 1 Cont.
<p>Real LAN traffic data in the week. (<b>a</b>) Real network traffic; (<b>b</b>) Average daily traffic sequence.</p> Full article ">Figure 2
<p>Real LAN traffic data on Monday. (<b>a</b>) Real network traffic data; (<b>b</b>) Average traffic sequence per hour.</p> Full article ">Figure 3
<p><span class="html-italic">y</span>(<span class="html-italic">i</span>) in the working periods. (<b>a</b>) Morning; (<b>b</b>) Afternoon.</p> Full article ">Figure 4
<p>Approximate linear part of <span class="html-italic">y</span>(<span class="html-italic">i</span>) in the working periods. (<b>a</b>) Morning; (<b>b</b>) Afternoon.</p> Full article ">Figure 5
<p>Predicted data and the real data of the network traffic. (<b>a</b>) Saturday; (<b>b</b>) Sunday.</p> Full article ">
2373 KiB  
Article
Increment Entropy as a Measure of Complexity for Time Series
by Xiaofeng Liu, Aimin Jiang, Ning Xu and Jianru Xue
Entropy 2016, 18(1), 22; https://doi.org/10.3390/e18010022 - 8 Jan 2016
Cited by 55 | Viewed by 9930 | Correction
Abstract
Entropy has been a common index to quantify the complexity of time series in a variety of fields. Here, we introduce an increment entropy to measure the complexity of time series in which each increment is mapped onto a word of two letters, [...] Read more.
Entropy has been a common index to quantify the complexity of time series in a variety of fields. Here, we introduce an increment entropy to measure the complexity of time series in which each increment is mapped onto a word of two letters, one corresponding to the sign and the other corresponding to the magnitude. Increment entropy (IncrEn) is defined as the Shannon entropy of the words. Simulations on synthetic data and tests on epileptic electroencephalogram (EEG) signals demonstrate its ability of detecting abrupt changes, regardless of the energetic (e.g., spikes or bursts) or structural changes. The computation of IncrEn does not make any assumption on time series, and it can be applicable to arbitrary real-world data. Full article
(This article belongs to the Special Issue Complex and Fractional Dynamics)
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Figure 1

Figure 1
<p>Schematic description of the steps of the increment entropy.</p> Full article ">Figure 2
<p>Logistic equation for varying control parameter <span class="html-italic">r</span> and corresponding IncrEn with varying scale. (<b>a</b>) bifurcation diagram; (<b>b</b>) increment Entropy, <math display="inline"> <msub> <mi>h</mi> <mn>6</mn> </msub> </math>; (<b>c</b>) <math display="inline"> <mrow> <msub> <mi>h</mi> <mn>8</mn> </msub> <mo>;</mo> </mrow> </math> (<b>d</b>) <math display="inline"> <msub> <mi>h</mi> <mn>8</mn> </msub> </math> with Gaussian observational noise at standard deviations.</p> Full article ">Figure 3
<p>Order <span class="html-italic">m</span> choice and data length effect. (<b>a</b>) mean <math display="inline"> <mrow> <mo>&lt;</mo> <msub> <mi>h</mi> <mi>m</mi> </msub> <mo>&gt;</mo> </mrow> </math> of logistic map (<math display="inline"> <mrow> <mi>r</mi> <mo>=</mo> <mn>4</mn> </mrow> </math>) with <math display="inline"> <msup> <mn>10</mn> <mi>k</mi> </msup> </math> data points (<math display="inline"> <mrow> <mi>k</mi> <mo>=</mo> <mn>2</mn> <mo>,</mo> <mn>3</mn> <mo>,</mo> <mn>4</mn> <mo>,</mo> <mn>5</mn> <mo>,</mo> <mn>6</mn> </mrow> </math>). Horizontal: embeding dimension <span class="html-italic">m</span>; (<b>b</b>) corresponding standard deviation <span class="html-italic">σ</span> of <math display="inline"> <msub> <mi>h</mi> <mi>m</mi> </msub> </math>.</p> Full article ">Figure 4
<p>Analogous motifs (<b>a–g</b>) and their corresponding patterns in IncrEn and PE.</p> Full article ">Figure 5
<p>Detection of energetic and structual change. (<b>a</b>) Regular time series consists of 300 identical atomic epochs that contain four random numbers; (<b>b</b>) Time series interspersed with three energetic mutation epochs (attenuation); (<b>c</b>) Time series interspersed with three energetic mutation epochs (enhancement); (<b>d</b>) Time series interspersed with three structural mutation epochs.</p> Full article ">Figure 6
<p>Invariance of IncrEn, PE and SampEn on random noise.</p> Full article ">Figure 7
<p>Average of IncrEn (<b>a</b>); PE (<b>b</b>); and SampEn (<b>c</b>) over 14 epileptic EEG signals at preictal, crossover, and ictal stages.</p> Full article ">Figure 8
<p>Detecting the seizure onset in a seizure record (<b>a</b>); using IncrEn (<b>b</b>); PE (<b>c</b>); and SampEn (<b>d</b>). The left vertical dashed line denotes the seizure onset, and the right vertical dashed line denotes the end of seizure.</p> Full article ">Figure 9
<p>IncrEn, PE and SampEn of vibration acceleration signals recorded on a bearing with a fault on rolling element. A sliding time window of 1000 samples with 500 overlapped samples is adopted.</p> Full article ">
647 KiB  
Article
Cloud Entropy Management System Involving a Fractional Power
by Rabha W. Ibrahim, Hamid A. Jalab and Abdullah Gani
Entropy 2016, 18(1), 14; https://doi.org/10.3390/e18010014 - 29 Dec 2015
Cited by 12 | Viewed by 5180
Abstract
Cloud computing (CC) capacities deliver high quality, connected with demand services and service-oriented construction. Nevertheless, a cloud service (CS) is normally derived from numerous stages of facilities and concert features, which determine the value of the cloud service. Therefore, it is problematic for [...] Read more.
Cloud computing (CC) capacities deliver high quality, connected with demand services and service-oriented construction. Nevertheless, a cloud service (CS) is normally derived from numerous stages of facilities and concert features, which determine the value of the cloud service. Therefore, it is problematic for the users to estimate these cloud services and select them to appropriate their requirements. In this study, a new algorithm is carried out for a multi-agent system (MAS) based on fractional power. The study depends on a fractional difference equation of type two point boundary value problem (BVP) based on the fractional entropy. We discuss the existence of solutions for the system as well as the stability, utilizing the Hadamard well-posed problem. Experimental results show that the proposed method demonstrates stability and performance. Full article
(This article belongs to the Special Issue Complex and Fractional Dynamics)
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<p>The proposed algorithm.</p> Full article ">Figure 2
<p>Convergence of the algorithm: four-agent system: demand with respect to time.</p> Full article ">
6403 KiB  
Article
Complexity Analysis and DSP Implementation of the Fractional-Order Lorenz Hyperchaotic System
by Shaobo He, Kehui Sun and Huihai Wang
Entropy 2015, 17(12), 8299-8311; https://doi.org/10.3390/e17127882 - 18 Dec 2015
Cited by 185 | Viewed by 8661
Abstract
The fractional-order hyperchaotic Lorenz system is solved as a discrete map by applying the Adomian decomposition method (ADM). Lyapunov Characteristic Exponents (LCEs) of this system are calculated according to this deduced discrete map. Complexity of this system versus parameters are analyzed by LCEs, [...] Read more.
The fractional-order hyperchaotic Lorenz system is solved as a discrete map by applying the Adomian decomposition method (ADM). Lyapunov Characteristic Exponents (LCEs) of this system are calculated according to this deduced discrete map. Complexity of this system versus parameters are analyzed by LCEs, bifurcation diagrams, phase portraits, complexity algorithms. Results show that this system has rich dynamical behaviors. Chaos and hyperchaos can be generated by decreasing fractional order q in this system. It also shows that the system is more complex when q takes smaller values. SE and C 0 complexity algorithms provide a parameter choice criteria for practice applications of fractional-order chaotic systems. The fractional-order system is implemented by digital signal processor (DSP), and a pseudo-random bit generator is designed based on the implemented system, which passes the NIST test successfully. Full article
(This article belongs to the Special Issue Complex and Fractional Dynamics)
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Figure 1

Figure 1
<p>Bifurcation and Lyapunov Characteristic Exponents (LCEs) with different <span class="html-italic">q</span> (<math display="inline"> <mrow> <mi>k</mi> <mo>=</mo> <mn>0</mn> <mo>.</mo> <mn>26</mn> </mrow> </math>) (<b>a</b>) Bifurcation diagram; (<b>b</b>) Lyapunov characteristic exponents.</p> Full article ">Figure 2
<p>Bifurcation and LCEs with different <span class="html-italic">k</span> (<math display="inline"> <mrow> <mi>q</mi> <mo>=</mo> <mn>0</mn> <mo>.</mo> <mn>96</mn> </mrow> </math>) (<b>a</b>) Bifurcation diagram; (<b>b</b>) Lyapunov characteristic exponents.</p> Full article ">Figure 3
<p>Phase diagrams of fractional-order hyperchaotic Lorenz system (<math display="inline"> <mrow> <mi>k</mi> <mo>=</mo> <mn>0</mn> <mo>.</mo> <mn>26</mn> </mrow> </math>) (<b>a</b>) periodic orbits (<math display="inline"> <mrow> <mi>q</mi> <mo>=</mo> <mn>0</mn> <mo>.</mo> <mn>65</mn> </mrow> </math>); (<b>b</b>) hyperchaos (<math display="inline"> <mrow> <mi>q</mi> <mo>=</mo> <mn>0</mn> <mo>.</mo> <mn>72</mn> </mrow> </math>); (<b>c</b>) chaos (<math display="inline"> <mrow> <mi>q</mi> <mo>=</mo> <mn>0</mn> <mo>.</mo> <mn>89</mn> </mrow> </math>); (<b>d</b>) periodic orbits (<math display="inline"> <mrow> <mi>q</mi> <mo>=</mo> <mn>1</mn> <mo>.</mo> <mn>00</mn> </mrow> </math>).</p> Full article ">Figure 4
<p>Phase diagrams of fractional-order hyperchaotic Lorenz system <math display="inline"> <mrow> <mo>(</mo> <mi>q</mi> <mo>=</mo> <mn>0</mn> <mo>.</mo> <mn>96</mn> <mo>)</mo> </mrow> </math> (<b>a</b>) hyperchaos (<math display="inline"> <mrow> <mi>k</mi> <mo>=</mo> <mn>0</mn> <mo>.</mo> <mn>05</mn> </mrow> </math>); (<b>b</b>) chaos (<math display="inline"> <mrow> <mi>k</mi> <mo>=</mo> <mn>0</mn> <mo>.</mo> <mn>20</mn> </mrow> </math>); (<b>c</b>) periodic orbits (<math display="inline"> <mrow> <mi>k</mi> <mo>=</mo> <mn>0</mn> <mo>.</mo> <mn>25</mn> </mrow> </math>); (<b>d</b>) periodic orbits (<math display="inline"> <mrow> <mi>k</mi> <mo>=</mo> <mn>0</mn> <mo>.</mo> <mn>30</mn> </mrow> </math>); (<b>e</b>) chaos (<math display="inline"> <mrow> <mi>k</mi> <mo>=</mo> <mn>0</mn> <mo>.</mo> <mn>40</mn> </mrow> </math>); (<b>f</b>) quasi-periodic orbits (<math display="inline"> <mrow> <mi>k</mi> <mo>=</mo> <mn>0</mn> <mo>.</mo> <mn>50</mn> </mrow> </math>); (<b>g</b>) periodic orbits (<math display="inline"> <mrow> <mi>k</mi> <mo>=</mo> <mn>0</mn> <mo>.</mo> <mn>80</mn> </mrow> </math>); (<b>h</b>) periodic orbits (<math display="inline"> <mrow> <mi>k</mi> <mo>=</mo> <mn>1</mn> <mo>.</mo> <mn>00</mn> </mrow> </math>).</p> Full article ">Figure 5
<p>Spectral entropy (SE) complexity results (<b>a</b>) SE complexity <span class="html-italic">versus</span> fractional order <span class="html-italic">q</span> (<span class="html-italic">k</span> = 0.26); (<b>b</b>) SE complexity <span class="html-italic">versus</span> fractional order <span class="html-italic">k</span> (<span class="html-italic">q</span> = 0.96); (<b>c</b>) SE complexity in the <math display="inline"> <mrow> <mi>q</mi> <mo>−</mo> <mi>k</mi> </mrow> </math> parameter plane.</p> Full article ">Figure 6
<p>C<math display="inline"> <msub> <mrow/> <mn>0</mn> </msub> </math> complexity results (<b>a</b>) C<math display="inline"> <msub> <mrow/> <mn>0</mn> </msub> </math> complexity <span class="html-italic">versus</span> fractional order <span class="html-italic">q</span> (<span class="html-italic">k</span> = 0.26); (<b>b</b>) C<math display="inline"> <msub> <mrow/> <mn>0</mn> </msub> </math> complexity <span class="html-italic">versus</span> fractional order <span class="html-italic">k</span> (<span class="html-italic">q</span> = 0.96); (<b>c</b>) C<math display="inline"> <msub> <mrow/> <mn>0</mn> </msub> </math> complexity in the <math display="inline"> <mrow> <mi>q</mi> <mo>−</mo> <mi>k</mi> </mrow> </math> parameter plane.</p> Full article ">Figure 7
<p>Hardware block diagram.</p> Full article ">Figure 8
<p>Digital signal processor (DSP) board used to perform digital implementation.</p> Full article ">Figure 9
<p>Flow diagram for DSP implementation of the fractional-order system.</p> Full article ">Figure 10
<p>Phase diagrams of the fractional-order Lorenz hyperchaotic system by DSP <math display="inline"> <mrow> <mo>(</mo> <mi>k</mi> <mo>=</mo> <mn>0</mn> <mo>.</mo> <mn>26</mn> <mo>)</mo> </mrow> </math> (<b>a</b>) periodic orbits (<math display="inline"> <mrow> <mi>q</mi> <mo>=</mo> <mn>0</mn> <mo>.</mo> <mn>65</mn> </mrow> </math>); (<b>b</b>) hyperchaos (<math display="inline"> <mrow> <mi>q</mi> <mo>=</mo> <mn>0</mn> <mo>.</mo> <mn>72</mn> </mrow> </math>); (<b>c</b>) chaos (<math display="inline"> <mrow> <mi>q</mi> <mo>=</mo> <mn>0</mn> <mo>.</mo> <mn>89</mn> </mrow> </math>); (<b>d</b>) periodic orbits (<math display="inline"> <mrow> <mi>q</mi> <mo>=</mo> <mn>1</mn> <mo>.</mo> <mn>00</mn> </mrow> </math>).</p> Full article ">Figure 11
<p>Phase diagrams of the fractional-order Lorenz hyperchaotic system by DSP <math display="inline"> <mrow> <mo>(</mo> <mi>q</mi> <mo>=</mo> <mn>0</mn> <mo>.</mo> <mn>96</mn> <mo>)</mo> </mrow> </math> (<b>a</b>) hyperchaos (<math display="inline"> <mrow> <mi>k</mi> <mo>=</mo> <mn>0</mn> <mo>.</mo> <mn>05</mn> </mrow> </math>); (<b>b</b>) chaos (<math display="inline"> <mrow> <mi>k</mi> <mo>=</mo> <mn>0</mn> <mo>.</mo> <mn>20</mn> </mrow> </math>); (<b>c</b>) quasi-periodic orbits (<math display="inline"> <mrow> <mi>k</mi> <mo>=</mo> <mn>0</mn> <mo>.</mo> <mn>50</mn> </mrow> </math>); (<b>d</b>) periodic orbits (<math display="inline"> <mrow> <mi>k</mi> <mo>=</mo> <mn>1</mn> <mo>.</mo> <mn>00</mn> </mrow> </math>).</p> Full article ">
4030 KiB  
Article
Identify the Rotating Stall in Centrifugal Compressors by Fractal Dimension in Reconstructed Phase Space
by Le Wang, Jiazhong Zhang and Wenfan Zhang
Entropy 2015, 17(12), 7888-7899; https://doi.org/10.3390/e17127848 - 30 Nov 2015
Cited by 13 | Viewed by 6588
Abstract
Based on phase space reconstruction and fractal dynamics in nonlinear dynamics, a method is proposed to extract and analyze the dynamics of the rotating stall in the impeller of centrifugal compressor, and some numerical examples are given to verify the results as well. [...] Read more.
Based on phase space reconstruction and fractal dynamics in nonlinear dynamics, a method is proposed to extract and analyze the dynamics of the rotating stall in the impeller of centrifugal compressor, and some numerical examples are given to verify the results as well. First, the rotating stall of an existing low speed centrifugal compressor (LSCC) is numerically simulated, and the time series of pressure in the rotating stall is obtained at various locations near the impeller outlet. Then, the phase space reconstruction is applied to these pressure time series, and a low-dimensional dynamical system, which the dynamics properties are included in, is reconstructed. In phase space reconstruction, C–C method is used to obtain the key parameters, such as time delay and the embedding dimension of the reconstructed phase space. Further, the fractal characteristics of the rotating stall are analyzed in detail, and the fractal dimensions are given for some examples to measure the complexity of the flow in the post-rotating stall. The results show that the fractal structures could reveal the intrinsic dynamics of the rotating stall flow and could be considered as a characteristic to identify the rotating stall. Full article
(This article belongs to the Special Issue Complex and Fractional Dynamics)
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<p>Performance comparisons between the numerical and the experimental results.</p> Full article ">Figure 2
<p>(<b>a</b>) Pressure distribution as the flow rates is 30 kg/s at 42.8% blade height; (<b>b</b>) pressure distribution as the flow rates is 23.6 kg/s at 42.8% blade height; (<b>c</b>) pressure distribution as the flow rates is 12.6 kg/s at 42.8% blade height.</p> Full article ">Figure 3
<p>Sampling points of the pressure time series.</p> Full article ">Figure 4
<p>Pressure time series at P1 under the stable state.</p> Full article ">Figure 5
<p>Pressure time series at P1 under the rotating stall.</p> Full article ">Figure 6
<p>The relationship of correlation coefficients and time delay applying the C–C method to time series at P1.</p> Full article ">Figure 7
<p>Phase space reconstruction for the pressure time series at P1 under the stable state.</p> Full article ">Figure 8
<p>Phase space reconstruction for the pressure time series at P1 under the rotating stall. (k is the time delay.)</p> Full article ">Figure 9
<p>Fractal dimension in the reconstructed phase space for pressure time series at P1 under the rotating stall.</p> Full article ">Figure 10
<p>(<b>a</b>) Reconstructed phase space for pressure time series at P2 under the rotating stall; (<b>b</b>) Reconstructed phase space for pressure time series at P3 under the rotating stall; (<b>c</b>) Reconstructed phase space for the pressure time series at P4 under the rotating stall; (<b>d</b>) Reconstructed phase space for the pressure time series at P5 under the rotating stall.</p> Full article ">Figure 11
<p>(<b>a</b>) Fractal dimension in reconstructed phase space for pressure time series at P2 under the rotating stall; (<b>b</b>) Fractal dimension in reconstructed phase space for pressure time series at P3 under the rotating stall.</p> Full article ">Figure 12
<p>(<b>a</b>) Fractal dimension in reconstructed phase space for pressure time series at P4 under the rotating stall; (<b>b</b>) Fractal dimension in reconstructed phase space for pressure time series at P5 under the rotating stall.</p> Full article ">
1800 KiB  
Article
A Memristor-Based Complex Lorenz System and Its Modified Projective Synchronization
by Shibing Wang, Xingyuan Wang and Yufei Zhou
Entropy 2015, 17(11), 7628-7644; https://doi.org/10.3390/e17117628 - 5 Nov 2015
Cited by 29 | Viewed by 8519
Abstract
The aim of this paper is to introduce and investigate a novel complex Lorenz system with a flux-controlled memristor, and to realize its synchronization. The system has an infinite number of stable and unstable equilibrium points, and can generate abundant dynamical behaviors with [...] Read more.
The aim of this paper is to introduce and investigate a novel complex Lorenz system with a flux-controlled memristor, and to realize its synchronization. The system has an infinite number of stable and unstable equilibrium points, and can generate abundant dynamical behaviors with different parameters and initial conditions, such as limit cycle, torus, chaos, transient phenomena, etc., which are explored by means of time-domain waveforms, phase portraits, bifurcation diagrams, and Lyapunov exponents. Furthermore, an active controller is designed to achieve modified projective synchronization (MPS) of this system based on Lyapunov stability theory. The corresponding numerical simulations agree well with the theoretical analysis, and demonstrate that the response system is asymptotically synchronized with the drive system within a short time. Full article
(This article belongs to the Special Issue Complex and Fractional Dynamics)
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<p>Dynamical behaviors with computational time interval 0–3000 s. (<b>a</b>) Lyapunov exponents; (<b>b</b>) bifurcation diagram.</p> Full article ">Figure 2
<p>Dynamical behaviors with computational time interval 0–30000 s. (<b>a</b>) Lyapunov exponents; (<b>b</b>) bifurcation diagram.</p> Full article ">Figure 3
<p>Fixed point (<math display="inline"> <semantics> <mrow> <msub> <mi>a</mi> <mn>3</mn> </msub> <mo>=</mo> <mn>5</mn> </mrow> </semantics> </math>). (<b>a1</b>) Waveforms of <math display="inline"> <semantics> <mrow> <msub> <mi>u</mi> <mn>1</mn> </msub> </mrow> </semantics> </math>–<math display="inline"> <semantics> <mrow> <msub> <mi>u</mi> <mn>5</mn> </msub> </mrow> </semantics> </math>; (<b>a2</b>) waveform of <math display="inline"> <semantics> <mrow> <msub> <mi>u</mi> <mn>6</mn> </msub> </mrow> </semantics> </math>; (<b>b1</b>) phase portrait of transient state; (<b>b2</b>) phase portrait of steady state; (<b>c1</b>) Lyapunov exponents of <math display="inline"> <semantics> <mrow> <msub> <mi>L</mi> <mn>1</mn> </msub> </mrow> </semantics> </math>–<math display="inline"> <semantics> <mrow> <msub> <mi>L</mi> <mn>4</mn> </msub> </mrow> </semantics> </math>; (<b>c2</b>) Lyapunov exponents of <math display="inline"> <semantics> <mrow> <msub> <mi>L</mi> <mn>5</mn> </msub> </mrow> </semantics> </math>– <math display="inline"> <semantics> <mrow> <msub> <mi>L</mi> <mn>6</mn> </msub> </mrow> </semantics> </math>.</p> Full article ">Figure 4
<p>Transient chaos to fixed point (<math display="inline"> <semantics> <mrow> <msub> <mi>a</mi> <mn>3</mn> </msub> <mo>=</mo> <mn>25</mn> </mrow> </semantics> </math>). (<b>a1</b>) Waveforms of <math display="inline"> <semantics> <mrow> <msub> <mi>u</mi> <mn>1</mn> </msub> </mrow> </semantics> </math>–<math display="inline"> <semantics> <mrow> <msub> <mi>u</mi> <mn>5</mn> </msub> </mrow> </semantics> </math>; (<b>a2</b>) waveform of <math display="inline"> <semantics> <mrow> <msub> <mi>u</mi> <mn>6</mn> </msub> </mrow> </semantics> </math>; (<b>b1</b>) phase portrait of transient state; (<b>b2</b>) phase portrait of steady state; (<b>c1</b>) Lyapunov exponents of <math display="inline"> <semantics> <mrow> <msub> <mi>L</mi> <mn>1</mn> </msub> </mrow> </semantics> </math>–<math display="inline"> <semantics> <mrow> <msub> <mi>L</mi> <mn>4</mn> </msub> </mrow> </semantics> </math>; (<b>c2</b>) Lyapunov exponents of <math display="inline"> <semantics> <mrow> <msub> <mi>L</mi> <mn>5</mn> </msub> </mrow> </semantics> </math>–<math display="inline"> <semantics> <mrow> <msub> <mi>L</mi> <mn>6</mn> </msub> </mrow> </semantics> </math>.</p> Full article ">Figure 5
<p>Chaos (<math display="inline"> <semantics> <mrow> <msub> <mi>a</mi> <mn>3</mn> </msub> <mo>=</mo> <mn>50</mn> </mrow> </semantics> </math>). (<b>a1</b>) Waveform of<math display="inline"> <semantics> <mrow> <msub> <mi>u</mi> <mn>1</mn> </msub> </mrow> </semantics> </math>; (<b>a2</b>) subinterval waveform of <math display="inline"> <semantics> <mrow> <msub> <mi>u</mi> <mn>1</mn> </msub> </mrow> </semantics> </math>; (<b>b1</b>) phase portrait of transient state; (<b>b2</b>) phase portrait of steady state; (<b>c1</b>) Lyapunov exponents of <math display="inline"> <semantics> <mrow> <msub> <mi>L</mi> <mn>1</mn> </msub> </mrow> </semantics> </math>–<math display="inline"> <semantics> <mrow> <msub> <mi>L</mi> <mn>4</mn> </msub> </mrow> </semantics> </math>; (<b>c2</b>) Lyapunov exponents of <math display="inline"> <semantics> <mrow> <msub> <mi>L</mi> <mn>5</mn> </msub> </mrow> </semantics> </math>–<math display="inline"> <semantics> <mrow> <msub> <mi>L</mi> <mn>6</mn> </msub> </mrow> </semantics> </math>.</p> Full article ">Figure 6
<p>Transient chaos to Period-5 (<math display="inline"> <semantics> <mrow> <msub> <mi>a</mi> <mn>3</mn> </msub> <mo>=</mo> <mn>50</mn> </mrow> </semantics> </math>). (<b>a1</b>) Waveform of <math display="inline"> <semantics> <mrow> <msub> <mi>u</mi> <mn>1</mn> </msub> </mrow> </semantics> </math>; (<b>a2</b>) subinterval waveform of <math display="inline"> <semantics> <mrow> <msub> <mi>u</mi> <mn>1</mn> </msub> </mrow> </semantics> </math>; (<b>b1</b>) phase portrait of transient state; (<b>b2</b>) phase portrait of steady state; (<b>c1</b>) Lyapunov exponents of <math display="inline"> <semantics> <mrow> <msub> <mi>L</mi> <mn>1</mn> </msub> </mrow> </semantics> </math>–<math display="inline"> <semantics> <mrow> <msub> <mi>L</mi> <mn>4</mn> </msub> </mrow> </semantics> </math>; (<b>c2</b>) Lyapunov exponents of <math display="inline"> <semantics> <mrow> <msub> <mi>L</mi> <mn>5</mn> </msub> </mrow> </semantics> </math>–<math display="inline"> <semantics> <mrow> <msub> <mi>L</mi> <mn>6</mn> </msub> </mrow> </semantics> </math>.</p> Full article ">Figure 7
<p>Transient Period-3 to tours (<math display="inline"> <semantics> <mrow> <msub> <mi>a</mi> <mn>3</mn> </msub> <mo>=</mo> <mn>90</mn> </mrow> </semantics> </math>). (<b>a1</b>) Waveform of <math display="inline"> <semantics> <mrow> <msub> <mi>u</mi> <mn>1</mn> </msub> </mrow> </semantics> </math>; (<b>a2</b>) subinterval waveform of <math display="inline"> <semantics> <mrow> <msub> <mi>u</mi> <mn>1</mn> </msub> </mrow> </semantics> </math>; (<b>b1</b>) phase portrait of transient state; (<b>b2</b>) phase portrait of steady state; (<b>c1</b>) Lyapunov exponents of <math display="inline"> <semantics> <mrow> <msub> <mi>L</mi> <mn>1</mn> </msub> </mrow> </semantics> </math>–<math display="inline"> <semantics> <mrow> <msub> <mi>L</mi> <mn>4</mn> </msub> </mrow> </semantics> </math>; (<b>c2</b>) Lyapunov exponents of <math display="inline"> <semantics> <mrow> <msub> <mi>L</mi> <mn>5</mn> </msub> </mrow> </semantics> </math>–<math display="inline"> <semantics> <mrow> <msub> <mi>L</mi> <mn>6</mn> </msub> </mrow> </semantics> </math>.</p> Full article ">Figure 8
<p>Transient Period-1 to chaos (<math display="inline"> <semantics> <mrow> <msub> <mi>a</mi> <mn>3</mn> </msub> <mo>=</mo> <mn>165</mn> </mrow> </semantics> </math>). (<b>a1</b>) Waveform of <math display="inline"> <semantics> <mrow> <msub> <mi>u</mi> <mn>1</mn> </msub> </mrow> </semantics> </math>; (<b>a2</b>) subinterval waveform of <math display="inline"> <semantics> <mrow> <msub> <mi>u</mi> <mn>1</mn> </msub> </mrow> </semantics> </math>; (<b>b1</b>) phase portrait of transient state; (<b>b2</b>) phase portrait of steady state; (<b>c1</b>) Lyapunov exponents of <math display="inline"> <semantics> <mrow> <msub> <mi>L</mi> <mn>1</mn> </msub> </mrow> </semantics> </math>–<math display="inline"> <semantics> <mrow> <msub> <mi>L</mi> <mn>4</mn> </msub> </mrow> </semantics> </math>; (<b>c2</b>) Lyapunov exponents of <math display="inline"> <semantics> <mrow> <msub> <mi>L</mi> <mn>5</mn> </msub> </mrow> </semantics> </math>–<math display="inline"> <semantics> <mrow> <msub> <mi>L</mi> <mn>6</mn> </msub> </mrow> </semantics> </math>.</p> Full article ">Figure 9
<p>Dynamical behaviors with different initial values of <math display="inline"> <semantics> <mrow> <msub> <mi>u</mi> <mn>6</mn> </msub> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mrow> </semantics> </math>. (<b>a</b>) Period-1 (<math display="inline"> <semantics> <mrow> <msub> <mi>u</mi> <mn>6</mn> </msub> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> </mrow> </semantics> </math>); (<b>b</b>) Transient Period-1 to tours (<math display="inline"> <semantics> <mrow> <msub> <mi>u</mi> <mn>6</mn> </msub> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mn>3</mn> </mrow> </semantics> </math>); (<b>c</b>) Chaos (<math display="inline"> <semantics> <mrow> <msub> <mi>u</mi> <mn>6</mn> </msub> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mn>300</mn> </mrow> </semantics> </math>).</p> Full article ">Figure 10
<p>Time-domain waveforms of the drive system (12) and response system (13). (<b>a</b>) <math display="inline"> <semantics> <mrow> <msub> <mi>u</mi> <mrow> <mn>1</mn> <mi>d</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>u</mi> <mrow> <mn>1</mn> <mi>r</mi> </mrow> </msub> </mrow> </semantics> </math>; (<b>b</b>) <math display="inline"> <semantics> <mrow> <msub> <mi>u</mi> <mrow> <mn>2</mn> <mi>d</mi> </mrow> </msub> <mo>=</mo> <mo>−</mo> <msub> <mi>u</mi> <mrow> <mn>2</mn> <mi>r</mi> </mrow> </msub> </mrow> </semantics> </math>; (<b>c</b>) <math display="inline"> <semantics> <mrow> <msub> <mi>u</mi> <mrow> <mn>3</mn> <mi>d</mi> </mrow> </msub> <mo>=</mo> <mn>2</mn> <msub> <mi>u</mi> <mrow> <mn>3</mn> <mi>r</mi> </mrow> </msub> </mrow> </semantics> </math>; (<b>d</b>) <math display="inline"> <semantics> <mrow> <msub> <mi>u</mi> <mrow> <mn>4</mn> <mi>d</mi> </mrow> </msub> <mo>=</mo> <mo>−</mo> <mn>2</mn> <msub> <mi>u</mi> <mrow> <mn>4</mn> <mi>r</mi> </mrow> </msub> </mrow> </semantics> </math>; (<b>e</b>) <math display="inline"> <semantics> <mrow> <msub> <mi>u</mi> <mrow> <mn>5</mn> <mi>d</mi> </mrow> </msub> <mo>=</mo> <mn>0.5</mn> <msub> <mi>u</mi> <mrow> <mn>5</mn> <mi>r</mi> </mrow> </msub> </mrow> </semantics> </math>; (<b>f</b>) <math display="inline"> <semantics> <mrow> <msub> <mi>u</mi> <mrow> <mn>6</mn> <mi>d</mi> </mrow> </msub> <mo>=</mo> <mo>−</mo> <mn>0.5</mn> <msub> <mi>u</mi> <mrow> <mn>6</mn> <mi>r</mi> </mrow> </msub> </mrow> </semantics> </math>.</p> Full article ">Figure 11
<p>Synchronization errors between the drive system (12) and response system (13). (<b>a</b>) <math display="inline"> <semantics> <mrow> <msub> <mi>e</mi> <mn>1</mn> </msub> <mo>=</mo> <msub> <mi>e</mi> <mrow> <mi>u</mi> <mn>1</mn> </mrow> </msub> <mo>+</mo> <mi>i</mi> <msub> <mi>e</mi> <mrow> <mi>u</mi> <mn>2</mn> </mrow> </msub> </mrow> </semantics> </math>; (<b>b</b>) <math display="inline"> <semantics> <mrow> <msub> <mi>e</mi> <mn>2</mn> </msub> <mo>=</mo> <msub> <mi>e</mi> <mrow> <mi>u</mi> <mn>3</mn> </mrow> </msub> <mo>+</mo> <mi>i</mi> <msub> <mi>e</mi> <mrow> <mi>u</mi> <mn>4</mn> </mrow> </msub> </mrow> </semantics> </math>; (<b>c</b>) <math display="inline"> <semantics> <mrow> <msub> <mi>e</mi> <mn>3</mn> </msub> <mo>=</mo> <msub> <mi>e</mi> <mrow> <mi>u</mi> <mn>5</mn> </mrow> </msub> </mrow> </semantics> </math>; (<b>d</b>) <math display="inline"> <semantics> <mrow> <msub> <mi>e</mi> <mn>4</mn> </msub> <mo>=</mo> <msub> <mi>e</mi> <mrow> <mi>u</mi> <mn>6</mn> </mrow> </msub> </mrow> </semantics> </math>.</p> Full article ">Figure 12
<p>Synchronization with different projective factors. (<b>a</b>) <math display="inline"> <semantics> <mrow> <msub> <mi>k</mi> <mn>1</mn> </msub> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mtext> </mtext> <msub> <mi>k</mi> <mn>2</mn> </msub> <mo>=</mo> <mo>−</mo> <mn>1</mn> <mo>,</mo> <mtext> </mtext> <msub> <mi>k</mi> <mn>3</mn> </msub> <mo>=</mo> <mn>2</mn> <mo>,</mo> </mrow> </semantics> </math><math display="inline"> <semantics> <mrow> <msub> <mi>k</mi> <mn>4</mn> </msub> <mo>=</mo> <mo>−</mo> <mn>2</mn> <mo>,</mo> <mtext> </mtext> <msub> <mi>k</mi> <mn>5</mn> </msub> <mo>=</mo> <mn>0.5</mn> <mo>,</mo> <mtext> </mtext> <msub> <mi>k</mi> <mn>6</mn> </msub> <mo>=</mo> <mo>−</mo> <mn>0.5</mn> </mrow> </semantics> </math>; (<b>b</b>) <math display="inline"> <semantics> <mrow> <msub> <mi>k</mi> <mn>1</mn> </msub> <mo>=</mo> <msub> <mi>k</mi> <mn>2</mn> </msub> <mo>=</mo> <mo>−</mo> <mn>1</mn> <mo>,</mo> <mtext> </mtext> <msub> <mi>k</mi> <mn>3</mn> </msub> <mo>=</mo> <msub> <mi>k</mi> <mn>4</mn> </msub> <mo>=</mo> <mn>2</mn> <mo>,</mo> <mtext> </mtext> <msub> <mi>k</mi> <mn>5</mn> </msub> <mo>=</mo> <msub> <mi>k</mi> <mn>6</mn> </msub> <mo>=</mo> <mn>0.5</mn> </mrow> </semantics> </math>; (<b>c</b>) <math display="inline"> <semantics> <mrow> <msub> <mi>k</mi> <mi>j</mi> </msub> <mo>=</mo> <mo>−</mo> <mn>1.5</mn> </mrow> </semantics> </math><math display="inline"> <semantics> <mrow> <mo stretchy="false">(</mo> <mi>j</mi> <mo>=</mo> <mn>1</mn> <mo>−</mo> <mn>6</mn> <mo stretchy="false">)</mo> </mrow> </semantics> </math>; (<b>d</b>) <math display="inline"> <semantics> <mrow> <msub> <mi>k</mi> <mi>j</mi> </msub> <mo>=</mo> <mn>0.5</mn> <mo stretchy="false">(</mo> <mi>j</mi> <mo>=</mo> <mn>1</mn> <mo>−</mo> <mn>6</mn> <mo stretchy="false">)</mo> </mrow> </semantics> </math>; (<b>e</b>) <math display="inline"> <semantics> <mrow> <msub> <mi>k</mi> <mi>j</mi> </msub> <mo>=</mo> <mo>−</mo> <mn>1</mn> <mo stretchy="false">(</mo> <mi>j</mi> <mo>=</mo> <mn>1</mn> <mo>−</mo> <mn>6</mn> <mo stretchy="false">)</mo> </mrow> </semantics> </math>;(<b>f</b>) <math display="inline"> <semantics> <mrow> <msub> <mi>k</mi> <mi>j</mi> </msub> <mo>=</mo> <mn>1</mn> <mo stretchy="false">(</mo> <mi>j</mi> <mo>=</mo> <mn>1</mn> <mo>−</mo> <mn>6</mn> <mo stretchy="false">)</mo> </mrow> </semantics> </math>.</p> Full article ">
831 KiB  
Article
Characterization of Complex Fractionated Atrial Electrograms by Sample Entropy: An International Multi-Center Study
by Eva Cirugeda–Roldán, Daniel Novak, Vaclav Kremen, David Cuesta–Frau, Matthias Keller, Armin Luik and Martina Srutova
Entropy 2015, 17(11), 7493-7509; https://doi.org/10.3390/e17117493 - 28 Oct 2015
Cited by 16 | Viewed by 7366
Abstract
Atrial fibrillation (AF) is the most commonly clinically-encountered arrhythmia. Catheter ablation of AF is mainly based on trigger elimination and modification of the AF substrate. Substrate mapping ablation of complex fractionated atrial electrograms (CFAEs) has emerged to be a promising technique. To improve [...] Read more.
Atrial fibrillation (AF) is the most commonly clinically-encountered arrhythmia. Catheter ablation of AF is mainly based on trigger elimination and modification of the AF substrate. Substrate mapping ablation of complex fractionated atrial electrograms (CFAEs) has emerged to be a promising technique. To improve substrate mapping based on CFAE analysis, automatic detection algorithms need to be developed in order to simplify and accelerate the ablation procedures. According to the latest studies, the level of fractionation has been shown to be promisingly well estimated from CFAE measured during radio frequency (RF) ablation of AF. The nature of CFAE is generally nonlinear and nonstationary, so the use of complexity measures is considered to be the appropriate technique for the analysis of AF records. This work proposes the use of sample entropy (SampEn), not only as a way to discern between non-fractionated and fractionated atrial electrograms (A-EGM), Entropy 2015, 17 7494 but also as a tool for characterizing the degree of A-EGM regularity, which is linked to changes in the AF substrate and to heart tissue damage. The use of SampEn combined with a blind parameter estimation optimization process enables the classification between CFAE and non-CFAE with statistical significance (p < 0:001), 0.89 area under the ROC, 86% specificity and 77% sensitivity over a mixed database of A-EGM combined from two independent CFAE signal databases, recorded during RF ablation of AF in two EU countries (542 signals in total). On the basis of the results obtained in this study, it can be suggested that the use of SampEn is suitable for real-time support during navigation of RF ablation of AF, as only 1.5 seconds of signal segments need to be analyzed. Full article
(This article belongs to the Special Issue Complex and Fractional Dynamics)
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Graphical abstract

Graphical abstract
Full article ">Figure 1
<p>Block diagram of the new methodology proposed, from the raw atrial electrograms (A-EGM) input signal preprocessing stage, up to the classification results.</p> Full article ">Figure 2
<p>One signal from each fractionation level, ranging from C0 (top) to C3 (bottom). Original raw signals after baseline wander removal. (<b>A</b>) German (GE) database and (<b>B</b>) Czech (CZ) database.</p> Full article ">Figure 3
<p>Boxplot distribution of the SampEn values computed to the initially-optimized parameters for individual levels of fractionation, <math display="inline"> <mrow> <mo>(</mo> <mi>m</mi> <mo>,</mo> <mi>r</mi> <mo>)</mo> <mo>=</mo> </mrow> </math>(4,0.65). (<b>A</b>) The BT database. (<b>B</b>) The GE database. (<b>C</b>) The CZ database.</p> Full article ">Figure 4
<p>Boxplot distribution of the SampEn values computed with the individual optimized parameters for each of the levels of fractionation present in each database. (<b>A</b>) The GE database (4,0.15). (<b>B</b>) The CZ database (5,0.15).</p> Full article ">Figure 5
<p>Comparison between both algorithms used for a specific patient undergoing radio frequency ablation (RFA) of atrial fibrillation (AF). SampEn with optimized parameters (<math display="inline"> <mrow> <mi>m</mi> <mo>=</mo> <mn>4</mn> <mo>,</mo> <mi>r</mi> <mo>=</mo> <mn>0</mn> <mo>.</mo> <mn>65</mn> </mrow> </math>) (bottom) and CFEMean (St. Jude Medical) measurements mapped on a 3D model of heart tissue. The blue color in both measures reveals areas with a higher level of complexity of complex fractionated atrial electrograms (CFAEs). (<b>A</b>) 3D atrial topographical map, whole mapped area, frontal view. (<b>B</b>) A detail of the area around the pulmonary vein.</p> Full article ">
388 KiB  
Article
Adaptive Synchronization for a Class of Uncertain Fractional-Order Neural Networks
by Heng Liu, Shenggang Li, Hongxing Wang, Yuhong Huo and Junhai Luo
Entropy 2015, 17(10), 7185-7200; https://doi.org/10.3390/e17107185 - 22 Oct 2015
Cited by 59 | Viewed by 5965
Abstract
In this paper, synchronization for a class of uncertain fractional-order neural networks subject to external disturbances and disturbed system parameters is studied. Based on the fractional-order extension of the Lyapunov stability criterion, an adaptive synchronization controller is designed, and fractional-order adaptation law is [...] Read more.
In this paper, synchronization for a class of uncertain fractional-order neural networks subject to external disturbances and disturbed system parameters is studied. Based on the fractional-order extension of the Lyapunov stability criterion, an adaptive synchronization controller is designed, and fractional-order adaptation law is proposed to update the controller parameter online. The proposed controller can guarantee that the synchronization errors between two uncertain fractional-order neural networks converge to zero asymptotically. By using some proposed lemmas, the quadratic Lyapunov functions are employed in the stability analysis. Finally, numerical simulations are presented to confirm the effectiveness of the proposed method. Full article
(This article belongs to the Special Issue Complex and Fractional Dynamics)
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Figure 1
<p>Dynamical behavior of system (<a href="#FD15-entropy-17-07185" class="html-disp-formula">15</a>) with initial value <math display="inline"> <msup> <mrow> <mo>[</mo> <mn>2</mn> <mo>,</mo> <mo>−</mo> <mn>2</mn> <mo>,</mo> <mn>3</mn> <mo>]</mo> </mrow> <mi>T</mi> </msup> </math>.</p> Full article ">Figure 2
<p>Synchronization between <math display="inline"> <mrow> <msub> <mi>x</mi> <mn>1</mn> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </mrow> </math> (dashed line) and <math display="inline"> <mrow> <msub> <mi>y</mi> <mn>1</mn> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </mrow> </math> (solid line).</p> Full article ">Figure 3
<p>Synchronization between <math display="inline"> <mrow> <msub> <mi>x</mi> <mn>2</mn> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </mrow> </math> (dashed line) and <math display="inline"> <mrow> <msub> <mi>y</mi> <mn>2</mn> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </mrow> </math> (solid line).</p> Full article ">Figure 4
<p>Synchronization between <math display="inline"> <mrow> <msub> <mi>x</mi> <mn>3</mn> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </mrow> </math> (dashed line) and <math display="inline"> <mrow> <msub> <mi>y</mi> <mn>3</mn> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </mrow> </math> (solid line).</p> Full article ">Figure 5
<p>Synchronization errors.</p> Full article ">Figure 6
<p>Control inputs.</p> Full article ">Figure 7
<p>The time response of control parameters: (a) <math display="inline"> <mrow> <mover accent="true"> <mi>a</mi> <mo>^</mo> </mover> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </mrow> </math>, (b) <math display="inline"> <mrow> <msub> <mover accent="true"> <mi>ρ</mi> <mo>^</mo> </mover> <mn>1</mn> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </mrow> </math>, (c) <math display="inline"> <mrow> <msub> <mover accent="true"> <mi>ρ</mi> <mo>^</mo> </mover> <mn>2</mn> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </mrow> </math>, (d) <math display="inline"> <mrow> <msub> <mover accent="true"> <mi>ρ</mi> <mo>^</mo> </mover> <mn>3</mn> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </mrow> </math>.</p> Full article ">Figure 8
<p>Synchronization errors without chattering phenomenon.</p> Full article ">Figure 9
<p>Control inputs without chattering phenomenon.</p> Full article ">
2927 KiB  
Article
A Novel Image Encryption Algorithm Based on DNA Encoding and Spatiotemporal Chaos
by Chunyan Song and Yulong Qiao
Entropy 2015, 17(10), 6954-6968; https://doi.org/10.3390/e17106954 - 16 Oct 2015
Cited by 88 | Viewed by 8919
Abstract
DNA computing based image encryption is a new, promising field. In this paper, we propose a novel image encryption scheme based on DNA encoding and spatiotemporal chaos. In particular, after the plain image is primarily diffused with the bitwise Exclusive-OR operation, the DNA [...] Read more.
DNA computing based image encryption is a new, promising field. In this paper, we propose a novel image encryption scheme based on DNA encoding and spatiotemporal chaos. In particular, after the plain image is primarily diffused with the bitwise Exclusive-OR operation, the DNA mapping rule is introduced to encode the diffused image. In order to enhance the encryption, the spatiotemporal chaotic system is used to confuse the rows and columns of the DNA encoded image. The experiments demonstrate that the proposed encryption algorithm is of high key sensitivity and large key space, and it can resist brute-force attack, entropy attack, differential attack, chosen-plaintext attack, known-plaintext attack and statistical attack. Full article
(This article belongs to the Special Issue Complex and Fractional Dynamics)
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<p>Spatiotemporal chaos.</p> Full article ">Figure 2
<p>Kolmogorov-Sinai entropy densities. (<b>a</b>) Spatiotemporal chaos based on Logistic chaos; (<b>b</b>) Spatiotemporal chaos based on an NCA map (α =1.51 to 1.57, β = 3.2); (<b>c</b>) Spatiotemporal chaos based on NCA map (α =1.51 to 1.57, β = 3.5).</p> Full article ">Figure 3
<p>Image encryption scheme.</p> Full article ">Figure 4
<p>Experimental results. (<b>a</b>) Plain image Lenna; (<b>b</b>) Cipher image of (a); (<b>c</b>) Decrypted image of (b); (<b>d</b>) Plain image Peppers; (<b>e</b>) Cipher image of (d); (<b>f</b>) Decrypted image of (e); (<b>g</b>) Plain image Boats; (<b>h</b>) Cipher image of (g); (<b>i</b>) Decrypted image of (h).</p> Full article ">Figure 5
<p>Histogram analysis. (<b>a</b>) Histogram of the plain image; (<b>b</b>) Histogram of the cipher image.</p> Full article ">Figure 6
<p>Correlation analysis. (<b>a</b>) Correlation distribution of the plain image along the horizontal direction; (<b>b</b>) Correlation distribution of the encrypted image along the horizontal direction; (<b>c</b>) Correlation distribution of the plain image along the vertical direction; (<b>d</b>) Correlation distribution of the encrypted image along the vertical direction; (<b>e</b>) Correlation distribution of the plain image along the diagonal direction; (<b>f</b>) Correlation distribution of the encrypted image along the diagonal direction.</p> Full article ">Figure 6 Cont.
<p>Correlation analysis. (<b>a</b>) Correlation distribution of the plain image along the horizontal direction; (<b>b</b>) Correlation distribution of the encrypted image along the horizontal direction; (<b>c</b>) Correlation distribution of the plain image along the vertical direction; (<b>d</b>) Correlation distribution of the encrypted image along the vertical direction; (<b>e</b>) Correlation distribution of the plain image along the diagonal direction; (<b>f</b>) Correlation distribution of the encrypted image along the diagonal direction.</p> Full article ">Figure 7
<p>Differential analysis on “Lenna” image. (<b>a</b>) Cipher image <span class="html-italic">C′</span>; (<b>b</b>) Differential image.</p> Full article ">Figure 8
<p>Key sensitivity analysis. (<b>a</b>) Cipher image (α = 1.570000000000001); (<b>b</b>) Differential image; (<b>c</b>) Decrypted image with the wrong key (ε = 0.300000000000001).</p> Full article ">
907 KiB  
Article
Approximate Analytical Solutions of Time Fractional Whitham–Broer–Kaup Equations by a Residual Power Series Method
by Linjun Wang and Xumei Chen
Entropy 2015, 17(9), 6519-6533; https://doi.org/10.3390/e17096519 - 23 Sep 2015
Cited by 65 | Viewed by 5779
Abstract
In this paper, a new analytic iterative technique, called the residual power series method (RPSM), is applied to time fractional Whitham–Broer–Kaup equations. The explicit approximate traveling solutions are obtained by using this method. The efficiency and accuracy of the present method is demonstrated [...] Read more.
In this paper, a new analytic iterative technique, called the residual power series method (RPSM), is applied to time fractional Whitham–Broer–Kaup equations. The explicit approximate traveling solutions are obtained by using this method. The efficiency and accuracy of the present method is demonstrated by two aspects. One is analyzing the approximate solutions graphically. The other is comparing the results with those of the Adomian decomposition method (ADM), the variational iteration method (VIM) and the optimal homotopy asymptotic method (OHAM). Illustrative examples reveal that the present technique outperforms the aforementioned methods and can be used as an alternative for solving fractional equations. Full article
(This article belongs to the Special Issue Complex and Fractional Dynamics)
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Figure 1
<p>The figures of the numerical solutions for Application 1. RPS, residual power series. (<b>a</b>) The fourth RPS solution of <math display="inline"> <mrow> <mi>P</mi> <mo>(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo>)</mo> </mrow> </math> for <math display="inline"> <mrow> <mi>α</mi> <mo>=</mo> <mn>0</mn> <mo>.</mo> <mn>5</mn> </mrow> </math>; (<b>b</b>) The fourth RPS solution of <math display="inline"> <mrow> <mi>Q</mi> <mo>(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo>)</mo> </mrow> </math> for <math display="inline"> <mrow> <mi>α</mi> <mo>=</mo> <mn>0</mn> <mo>.</mo> <mn>5</mn> </mrow> </math>.</p> Full article ">Figure 2
<p>The figures of the numerical solutions for Application 2. (<b>a</b>) The fourth RPS solution of <math display="inline"> <mrow> <mi>P</mi> <mo>(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo>)</mo> </mrow> </math> for <math display="inline"> <mrow> <mi>α</mi> <mo>=</mo> <mn>0</mn> <mo>.</mo> <mn>5</mn> </mrow> </math>; (<b>b</b>) The fourth RPS solution of <math display="inline"> <mrow> <mi>Q</mi> <mo>(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo>)</mo> </mrow> </math> for <math display="inline"> <mrow> <mi>α</mi> <mo>=</mo> <mn>0</mn> <mo>.</mo> <mn>5</mn> </mrow> </math>.</p> Full article ">Figure 3
<p>The figures of the numerical solutions for Application 3. (<b>a</b>) The fourth RPS solution of <math display="inline"> <mrow> <mi>P</mi> <mo>(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo>)</mo> </mrow> </math> for <math display="inline"> <mrow> <mi>α</mi> <mo>=</mo> <mn>0</mn> <mo>.</mo> <mn>5</mn> </mrow> </math>; (<b>b</b>) The fourth RPS solution of <math display="inline"> <mrow> <mi>Q</mi> <mo>(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo>)</mo> </mrow> </math> for <math display="inline"> <mrow> <mi>α</mi> <mo>=</mo> <mn>0</mn> <mo>.</mo> <mn>5</mn> </mrow> </math>.</p> Full article ">
300 KiB  
Article
Modeling of a Mass-Spring-Damper System by Fractional Derivatives with and without a Singular Kernel
by José Francisco Gómez-Aguilar, Huitzilin Yépez-Martínez, Celia Calderón-Ramón, Ines Cruz-Orduña, Ricardo Fabricio Escobar-Jiménez and Victor Hugo Olivares-Peregrino
Entropy 2015, 17(9), 6289-6303; https://doi.org/10.3390/e17096289 - 10 Sep 2015
Cited by 145 | Viewed by 37266
Abstract
In this paper, the fractional equations of the mass-spring-damper system with Caputo and Caputo–Fabrizio derivatives are presented. The physical units of the system are preserved by introducing an auxiliary parameter ?. The input of the resulting equations is a constant and periodic source; [...] Read more.
In this paper, the fractional equations of the mass-spring-damper system with Caputo and Caputo–Fabrizio derivatives are presented. The physical units of the system are preserved by introducing an auxiliary parameter ?. The input of the resulting equations is a constant and periodic source; for the Caputo case, we obtain the analytical solution, and the resulting equations are given in terms of the Mittag–Leffler function; for the Caputo–Fabrizio approach, the numerical solutions are obtained by the numerical Laplace transform algorithm. Our results show that the mechanical components exhibit viscoelastic behaviors producing temporal fractality at different scales and demonstrate the existence of Entropy 2015, 17 6290 material heterogeneities in the mechanical components. The Markovian nature of the model is recovered when the order of the fractional derivatives is equal to one. Full article
(This article belongs to the Special Issue Complex and Fractional Dynamics)
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<p>Mass-spring-damper system.</p> Full article ">Figure 2
<p>Mass-spring system with a constant source, Caputo derivative approach.</p> Full article ">Figure 3
<p>Mass-spring system with a constant source, Caputo–Fabrizio derivative approach.</p> Full article ">Figure 4
<p>Mass-spring system with a periodic source, Caputo derivative approach.</p> Full article ">Figure 5
<p>Mass-spring system with a periodic source, Caputo–Fabrizio derivative approach.</p> Full article ">Figure 6
<p>Damper-spring system with a constant source, Caputo derivative approach.</p> Full article ">Figure 7
<p>Damper-spring system with a constant source, Caputo–Fabrizio derivative approach.</p> Full article ">Figure 8
<p>Damper-spring system with a periodic source, Caputo derivative approach.</p> Full article ">Figure 9
<p>Damper-spring system with periodic source, Caputo–Fabrizio derivative approach.</p> Full article ">Figure 10
<p>Mass-spring-damper system without a source, Caputo derivative approach.</p> Full article ">Figure 11
<p>Mass-spring-damper system without a source, Caputo–Fabrizio derivative approach.</p> Full article ">
2847 KiB  
Article
Fractional State Space Analysis of Economic Systems
by J. A. Tenreiro Machado, Maria Eugénia Mata and António M. Lopes
Entropy 2015, 17(8), 5402-5421; https://doi.org/10.3390/e17085402 - 29 Jul 2015
Cited by 78 | Viewed by 7731
Abstract
This paper examines modern economic growth according to the multidimensional scaling (MDS) method and state space portrait (SSP) analysis. Electing GDP per capita as the main indicator for economic growth and prosperity, the long-run perspective from 1870 to 2010 identifies the main similarities [...] Read more.
This paper examines modern economic growth according to the multidimensional scaling (MDS) method and state space portrait (SSP) analysis. Electing GDP per capita as the main indicator for economic growth and prosperity, the long-run perspective from 1870 to 2010 identifies the main similarities among 34 world partners’ modern economic growth and exemplifies the historical waving mechanics of the largest world economy, the USA. MDS reveals two main clusters among the European countries and their old offshore territories, and SSP identifies the Great Depression as a mild challenge to the American global performance, when compared to the Second World War and the 2008 crisis. Full article
(This article belongs to the Special Issue Complex and Fractional Dynamics)
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Figure 1

Figure 1
<p>The multidimensional scaling (MDS) maps for the Pearson correlation <math display="inline"> <msub> <mi>ρ</mi> <mrow> <mi>i</mi> <mi>j</mi> </mrow> </msub> </math> between the GDP <span class="html-italic">per capita</span> time series 1870–2010 of <span class="html-italic">n</span> = 34 countries: (<b>a</b>) two-dimensional; (<b>b</b>) three-dimensional.</p> Full article ">Figure 2
<p>Shepard plots for the Pearson correlation <math display="inline"> <msub> <mi>ρ</mi> <mrow> <mi>i</mi> <mi>j</mi> </mrow> </msub> </math> between the GDP <span class="html-italic">per capita</span> time series 1870–2010 of <span class="html-italic">n</span> = 34 countries for representations: (<b>a</b>) two-dimensional; (<b>b</b>) three-dimensional.</p> Full article ">Figure 3
<p>Stress plot for the Pearson correlation <math display="inline"> <msub> <mi>ρ</mi> <mrow> <mi>i</mi> <mi>j</mi> </mrow> </msub> </math> between the GDP <span class="html-italic">per capita</span> time series 1870–2010 of <math display="inline"> <mrow> <mi>n</mi> <mo>=</mo> <mn>34</mn> </mrow> </math> countries.</p> Full article ">Figure 4
<p>The MDS maps for the normalized mutual information <math display="inline"> <mrow> <msub> <mi>I</mi> <mi>N</mi> </msub> <mrow> <mo>(</mo> <msub> <mi>y</mi> <mi>i</mi> </msub> <mo>,</mo> <msub> <mi>y</mi> <mi>j</mi> </msub> <mo>)</mo> </mrow> </mrow> </math> between the GDP <span class="html-italic">per capita</span> time series 1870–2010 of <span class="html-italic">n</span> = 34 countries: (<b>a</b>) two-dimensional; (<b>b</b>) three-dimensional.</p> Full article ">Figure 5
<p>The GDP <span class="html-italic">per capita</span> of the USA during 1870–2010.</p> Full article ">Figure 6
<p>The state space portrait (SSP) for the USA GDP <span class="html-italic">per capita</span> time series 1872–2008: (<b>a</b>) two-dimensional map; (<b>b</b>) three-dimensional map.</p> Full article ">Figure 7
<p>Maps of <math display="inline"> <msub> <mi>ρ</mi> <mrow> <mi>p</mi> <mi>q</mi> </mrow> </msub> </math> for the GDP <span class="html-italic">per capita</span> of USA during 1870–2010: (<b>a</b>) contour plot; (<b>b</b>) contour plot showing a few isoclines.</p> Full article ">Figure 8
<p>The two-dimensional SSP for the GDP <span class="html-italic">per capita</span> of the USA during 1870–2010.</p> Full article ">Figure 9
<p>Locus of points <math display="inline"> <mrow> <mo>(</mo> <msubsup> <mi>α</mi> <mi>p</mi> <mi>c</mi> </msubsup> <mo>,</mo> <msubsup> <mi>α</mi> <mi>q</mi> <mi>c</mi> </msubsup> <mo>)</mo> </mrow> </math> corresponding to the maximum curvatures of the isoclines that contain the point <math display="inline"> <mrow> <mo>(</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo>)</mo> </mrow> </math>, obtained with the Pearson correlation.</p> Full article ">Figure 10
<p>Contour plot of <math display="inline"> <msub> <mi>I</mi> <mrow> <mi>p</mi> <mi>q</mi> </mrow> </msub> </math> for the GDP <span class="html-italic">per capita</span> of the USA during 1870–2010.</p> Full article ">Figure 11
<p>Locus of points <math display="inline"> <mrow> <mo>(</mo> <msubsup> <mi>α</mi> <mi>p</mi> <mi>c</mi> </msubsup> <mo>,</mo> <msubsup> <mi>α</mi> <mi>q</mi> <mi>c</mi> </msubsup> <mo>)</mo> </mrow> </math> corresponding to the maximum curvatures of the isoclines that contain the point <math display="inline"> <mrow> <mo>(</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo>)</mo> </mrow> </math>, obtained with the mutual information, <math display="inline"> <msub> <mi>I</mi> <mrow> <mi>p</mi> <mi>q</mi> </mrow> </msub> </math>.</p> Full article ">
872 KiB  
Article
Fractional Differential Texture Descriptors Based on the Machado Entropy for Image Splicing Detection
by Rabha W. Ibrahim, Zahra Moghaddasi, Hamid A. Jalab and Rafidah Md Noor
Entropy 2015, 17(7), 4775-4785; https://doi.org/10.3390/e17074775 - 8 Jul 2015
Cited by 33 | Viewed by 5646
Abstract
Image splicing is a common operation in image forgery. Different techniques of image splicing detection have been utilized to regain people’s trust. This study introduces a texture enhancement technique involving the use of fractional differential masks based on the Machado entropy. The masks [...] Read more.
Image splicing is a common operation in image forgery. Different techniques of image splicing detection have been utilized to regain people’s trust. This study introduces a texture enhancement technique involving the use of fractional differential masks based on the Machado entropy. The masks slide over the tampered image, and each pixel of the tampered image is convolved with the fractional mask weight window on eight directions. Consequently, the fractional differential texture descriptors are extracted using the gray-level co-occurrence matrix for image splicing detection. The support vector machine is used as a classifier that distinguishes between authentic and spliced images. Results prove that the achieved improvements of the proposed algorithm are compatible with other splicing detection methods. Full article
(This article belongs to the Special Issue Complex and Fractional Dynamics)
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<p>Standard deviation distributions of extracted features. Rows indicate the standard deviation distributions of features extracted from gray-scale images. The first column indicates the original features. The second column shows the features after applying kernel PCA.</p> Full article ">
<p>Samples of DVMM image dataset.</p> Full article ">
<p>Selection of α value.</p> Full article ">
<p>Comparison between the features with 1764-D and features with Kernel PCA in 40-D.</p> Full article ">
436 KiB  
Article
H? Control for Markov Jump Systems with Nonlinear Noise Intensity Function and Uncertain Transition Rates
by Xiaonian Wang and Yafeng Guo
Entropy 2015, 17(7), 4762-4774; https://doi.org/10.3390/e17074762 - 6 Jul 2015
Cited by 2 | Viewed by 4107
Abstract
The problem of robust H? control is investigated for Markov jump systems with nonlinear noise intensity function and uncertain transition rates. A robust H? performance criterion is developed for the given systems for the first time. Based on the developed performance [...] Read more.
The problem of robust H? control is investigated for Markov jump systems with nonlinear noise intensity function and uncertain transition rates. A robust H? performance criterion is developed for the given systems for the first time. Based on the developed performance criterion, the desired H? state-feedback controller is also designed, which guarantees the robust H? performance of the closed-loop system. All the conditions are in terms of linear matrix inequalities (LMIs), and hence they can be readily solved by any LMI solver. Finally, a numerical example is provided to demonstrate the effectiveness of the proposed methods. Full article
(This article belongs to the Special Issue Complex and Fractional Dynamics)
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<p>State response of the open-loop system with 1000 random samplings.</p> Full article ">
<p>State response of the closed-loop system with 1000 random samplings.</p> Full article ">
<p>Functional cost <span class="html-italic">J</span>(<span class="html-italic">T<sub>f</sub></span>) with 1000 random samplings.</p> Full article ">
729 KiB  
Article
Generalized Boundary Conditions for the Time-Fractional Advection Diffusion Equation
by Yuriy Povstenko
Entropy 2015, 17(6), 4028-4039; https://doi.org/10.3390/e17064028 - 12 Jun 2015
Cited by 23 | Viewed by 5681
Abstract
The different kinds of boundary conditions for standard and fractional diffusion and advection diffusion equations are analyzed. Near the interface between two phases there arises a transition region which state differs from the state of contacting media owing to the different material particle [...] Read more.
The different kinds of boundary conditions for standard and fractional diffusion and advection diffusion equations are analyzed. Near the interface between two phases there arises a transition region which state differs from the state of contacting media owing to the different material particle interaction conditions. Particular emphasis has been placed on the conditions of nonperfect diffusive contact for the time-fractional advection diffusion equation. When the reduced characteristics of the interfacial region are equal to zero, the conditions of perfect contact are obtained as a particular case. Full article
(This article belongs to the Special Issue Complex and Fractional Dynamics)
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<p>Thin intermediate layer between two media.</p> Full article ">
<p>A contact surface Σ having its own physical characteristics.</p> Full article ">
202 KiB  
Article
Existence of Ulam Stability for Iterative Fractional Differential Equations Based on Fractional Entropy
by Rabha W. Ibrahim and Hamid A. Jalab
Entropy 2015, 17(5), 3172-3181; https://doi.org/10.3390/e17053172 - 13 May 2015
Cited by 36 | Viewed by 5015
Abstract
In this study, we introduce conditions for the existence of solutions for an iterative functional differential equation of fractional order. We prove that the solutions of the above class of fractional differential equations are bounded by Tsallis entropy. The method depends on the [...] Read more.
In this study, we introduce conditions for the existence of solutions for an iterative functional differential equation of fractional order. We prove that the solutions of the above class of fractional differential equations are bounded by Tsallis entropy. The method depends on the concept of Hyers-Ulam stability. The arbitrary order is suggested in the sense of Riemann-Liouville calculus. Full article
(This article belongs to the Special Issue Complex and Fractional Dynamics)
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