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Dynamics
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Recent Advances in Dynamic Phenomena—2nd Edition
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Recent Advances in Dynamic Phenomena—2nd Edition

A special issue of Dynamics (ISSN 2673-8716).

Deadline for manuscript submissions: 31 December 2024 | Viewed by 2592

Special Issue Editor

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

Special Issue Information

Dear Colleagues,

The wonders of dynamic phenomena encapsulate the intricate motions and effects found in the fabric of nature. A closer examination reveals these dynamic occurrences across a spectrum of disciplines, encompassing physical, chemical, and biological systems. Their manifestation is attributed to a confluence of inertial forces and the distinctive characteristics inherent in various systems. Furthermore, the allure of intriguing dynamics extends to mechanical and electronic systems, integral components in applications such as robotics, aircraft, and vehicles.

This call invites contributions to a Special Issue dedicated to showcasing recent advances in understanding dynamic phenomena spanning from the minutest scales to the grandest. This includes the exploration of mechanism dynamics at the cellular level within biological systems, phenomena within the Earth's water or atmosphere, and those exhibited in mechanical and electronic systems. Researchers are encouraged to submit their work encompassing both theoretical and experimental findings.

Submissions are encouraged from diverse fields, including but not limited to:

  • Aerodynamics;
  • Biological systems and networks;
  • Cell dynamics;
  • Climate dynamics;
  • Dynamic cycles of birds and animals;
  • Dynamics in mechanics;
  • Fluid dynamics;
  • Gas dynamics;
  • Nonlinear dynamics and chaos;
  • Nuclear dynamics;
  • Quantum mechanics and electrodynamics;
  • Terrestrial dynamics.

Dr. Christos Volos
Guest Editor

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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. Dynamics is an international peer-reviewed open access quarterly journal published by MDPI.

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Keywords

  • chaos
  • dynamic phenomena
  • fluids
  • gas dynamics
  • mechanics
  • nonlinear systems
  • nuclear dynamics
  • quantum mechanics
  • electrodynamics
  • terrestrial dynamics

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Published Papers (3 papers)

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Research

Jump to: Review

16 pages, 8922 KiB  
Article
SPH Simulation of Molten Metal Flow Modeling Lava Flow Phenomena with Solidification
by Shingo Tomita, Joe Yoshikawa, Makoto Sugimoto, Hisaya Komen and Masaya Shigeta
Dynamics 2024, 4(2), 287-302; https://doi.org/10.3390/dynamics4020017 - 19 Apr 2024
Viewed by 840
Abstract
Characteristic dynamics in lava flows, such as the formation processes of lava levees, toe-like tips, and overlapped structures, were reproduced successfully through numerical simulation using the smoothed particle hydrodynamics (SPH) method. Since these specific phenomena have a great influence on the flow direction [...] Read more.
Characteristic dynamics in lava flows, such as the formation processes of lava levees, toe-like tips, and overlapped structures, were reproduced successfully through numerical simulation using the smoothed particle hydrodynamics (SPH) method. Since these specific phenomena have a great influence on the flow direction of lava flows, it is indispensable to elucidate them for accurate predictions of areas where lava strikes. At the first step of this study, lava was expressed using a molten metal with known physical properties. The computational results showed that levees and toe-like tips formed at the fringe of the molten metal flowing down on a slope, which appeared for actual lava flows as well. The dynamics of an overlapped structure formation were also simulated successfully; therein, molten metal flowed down, solidified, and changed the surface shape of the slope, and the second molten metal flowed over the changed surface shape. It was concluded that the computational model developed in this study using the SPH method is applicable for simulating and clarifying lava flow phenomena. Full article
(This article belongs to the Special Issue Recent Advances in Dynamic Phenomena—2nd Edition)
►▼ Show Figures

Figure 1

Figure 1
<p>Schematic of cross-section of lava flow.</p> Full article ">Figure 2
<p>Computational domain.</p> Full article ">Figure 3
<p>The temperature distribution of gas at <math display="inline"><semantics> <mrow> <mi>z</mi> <mo>=</mo> <mn>5.1</mn> <mtext> </mtext> <mi mathvariant="normal">m</mi> <mi mathvariant="normal">m</mi> </mrow> </semantics></math> [<a href="#B32-dynamics-04-00017" class="html-bibr">32</a>].</p> Full article ">Figure 4
<p>The heat input distribution for generating molten metal, which is expressed as ion recombination under arc plasma conditions [<a href="#B31-dynamics-04-00017" class="html-bibr">31</a>].</p> Full article ">Figure 5
<p>Flow and solidification processes of molten metal with time evolution: (<b>a</b>) at <math display="inline"><semantics> <mrow> <mi>t</mi> <mo>=</mo> <mn>0.80</mn> </mrow> </semantics></math> s; (<b>b</b>) at <math display="inline"><semantics> <mrow> <mi>t</mi> <mo>=</mo> <mn>2.45</mn> </mrow> </semantics></math> s; (<b>c</b>) at <math display="inline"><semantics> <mrow> <mi>t</mi> <mo>=</mo> <mn>5.70</mn> </mrow> </semantics></math> s; (<b>d</b>) at <math display="inline"><semantics> <mrow> <mi>t</mi> <mo>=</mo> <mn>5.97</mn> </mrow> </semantics></math> s; (<b>e</b>) at <math display="inline"><semantics> <mrow> <mi>t</mi> <mo>=</mo> <mn>7.08</mn> </mrow> </semantics></math> s; (<b>f</b>) at <math display="inline"><semantics> <mrow> <mi>t</mi> <mo>=</mo> <mn>9.49</mn> </mrow> </semantics></math> s; (<b>g</b>) at <math display="inline"><semantics> <mrow> <mi>t</mi> <mo>=</mo> <mn>9.55</mn> </mrow> </semantics></math> s; (<b>h</b>) at <math display="inline"><semantics> <mrow> <mi>t</mi> <mo>=</mo> <mn>9.73</mn> </mrow> </semantics></math> s; (<b>i</b>) at <math display="inline"><semantics> <mrow> <mi>t</mi> <mo>=</mo> <mn>10.01</mn> </mrow> </semantics></math> s; (<b>j</b>) at <math display="inline"><semantics> <mrow> <mi>t</mi> <mo>=</mo> <mn>10.99</mn> </mrow> </semantics></math> s.</p> Full article ">Figure 5 Cont.
<p>Flow and solidification processes of molten metal with time evolution: (<b>a</b>) at <math display="inline"><semantics> <mrow> <mi>t</mi> <mo>=</mo> <mn>0.80</mn> </mrow> </semantics></math> s; (<b>b</b>) at <math display="inline"><semantics> <mrow> <mi>t</mi> <mo>=</mo> <mn>2.45</mn> </mrow> </semantics></math> s; (<b>c</b>) at <math display="inline"><semantics> <mrow> <mi>t</mi> <mo>=</mo> <mn>5.70</mn> </mrow> </semantics></math> s; (<b>d</b>) at <math display="inline"><semantics> <mrow> <mi>t</mi> <mo>=</mo> <mn>5.97</mn> </mrow> </semantics></math> s; (<b>e</b>) at <math display="inline"><semantics> <mrow> <mi>t</mi> <mo>=</mo> <mn>7.08</mn> </mrow> </semantics></math> s; (<b>f</b>) at <math display="inline"><semantics> <mrow> <mi>t</mi> <mo>=</mo> <mn>9.49</mn> </mrow> </semantics></math> s; (<b>g</b>) at <math display="inline"><semantics> <mrow> <mi>t</mi> <mo>=</mo> <mn>9.55</mn> </mrow> </semantics></math> s; (<b>h</b>) at <math display="inline"><semantics> <mrow> <mi>t</mi> <mo>=</mo> <mn>9.73</mn> </mrow> </semantics></math> s; (<b>i</b>) at <math display="inline"><semantics> <mrow> <mi>t</mi> <mo>=</mo> <mn>10.01</mn> </mrow> </semantics></math> s; (<b>j</b>) at <math display="inline"><semantics> <mrow> <mi>t</mi> <mo>=</mo> <mn>10.99</mn> </mrow> </semantics></math> s.</p> Full article ">Figure 6
<p>Molten metal flow at <math display="inline"><semantics> <mrow> <mi>t</mi> <mo>=</mo> <mn>5.97</mn> </mrow> </semantics></math> s: (<b>a</b>) <math display="inline"><semantics> <mrow> <mi>x</mi> </mrow> </semantics></math>-directional velocity distribution; (<b>b</b>) temperature distribution.</p> Full article ">Figure 7
<p>Molten metal flow at <math display="inline"><semantics> <mrow> <mi>t</mi> <mo>=</mo> <mn>10.01</mn> </mrow> </semantics></math> s: (<b>a</b>) <math display="inline"><semantics> <mrow> <mi>x</mi> </mrow> </semantics></math>-directional velocity distribution; (<b>b</b>) temperature distribution.</p> Full article ">Figure 8
<p>Cross-sections at <math display="inline"><semantics> <mrow> <mi>x</mi> <mo>=</mo> <mn>18.0</mn> </mrow> </semantics></math> mm and <math display="inline"><semantics> <mrow> <mi>t</mi> <mo>=</mo> <mn>5.97</mn> </mrow> </semantics></math> s: (<b>a</b>) phase of particles; (<b>b</b>) <math display="inline"><semantics> <mrow> <mi>x</mi> </mrow> </semantics></math>-directional velocity distribution; (<b>c</b>) temperature distribution.</p> Full article ">Figure 9
<p>Cross-sections at <math display="inline"><semantics> <mrow> <mi>x</mi> <mo>=</mo> <mn>20.0</mn> </mrow> </semantics></math> mm and <math display="inline"><semantics> <mrow> <mi>t</mi> <mo>=</mo> <mn>10.01</mn> </mrow> </semantics></math> s: (<b>a</b>) phase of particles; (<b>b</b>) <math display="inline"><semantics> <mrow> <mi>x</mi> </mrow> </semantics></math>-directional velocity distribution; (<b>c</b>) temperature distribution.</p> Full article ">Figure 9 Cont.
<p>Cross-sections at <math display="inline"><semantics> <mrow> <mi>x</mi> <mo>=</mo> <mn>20.0</mn> </mrow> </semantics></math> mm and <math display="inline"><semantics> <mrow> <mi>t</mi> <mo>=</mo> <mn>10.01</mn> </mrow> </semantics></math> s: (<b>a</b>) phase of particles; (<b>b</b>) <math display="inline"><semantics> <mrow> <mi>x</mi> </mrow> </semantics></math>-directional velocity distribution; (<b>c</b>) temperature distribution.</p> Full article ">
18 pages, 5288 KiB  
Article
System Identification Using Self-Adaptive Filtering Applied to Second-Order Gradient Materials
by Thomas Kletschkowski
Dynamics 2024, 4(2), 254-271; https://doi.org/10.3390/dynamics4020015 - 7 Apr 2024
Viewed by 597
Abstract
For many engineering applications, it is sufficient to use the concept of simple materials. However, higher gradients of the kinematic variables are taken into account to model materials with internal length scales as well as to describe localization effects using gradient theories in [...] Read more.
For many engineering applications, it is sufficient to use the concept of simple materials. However, higher gradients of the kinematic variables are taken into account to model materials with internal length scales as well as to describe localization effects using gradient theories in finite plasticity or fluid mechanics. In many approaches, length scale parameters have been introduced that are related to a specific micro structure. An alternative approach is possible, if a thermodynamically consistent framework is used for material modeling, as shown in the present contribution. However, even if sophisticated and thermodynamically consistent material models can be established, there are still not yet standard experiments to determine higher order material constants. In order to contribute to this ongoing discussion, system identification based on the method of self-adaptive filtering is proposed in this paper. To evaluate the effectiveness of this approach, it has been applied to second-order gradient materials considering longitudinal vibrations. Based on thermodynamically consistent models that have been solved numerically, it has been possible to prove that system identification based on self-adaptive filtering can be used effectively for both narrow-band and broadband signals in the field of second-order gradient materials. It has also been found that the differences identified for simple materials and gradient materials allow for condition monitoring and detection of gradient effects in the material behavior. Full article
(This article belongs to the Special Issue Recent Advances in Dynamic Phenomena—2nd Edition)
►▼ Show Figures

Figure 1

Figure 1
<p>Coordinate system and schematic configuration of grid points.</p> Full article ">Figure 2
<p>Behavior of simple material. (<b>Left</b>): impulse response. (<b>Right</b>): frequency response.</p> Full article ">Figure 3
<p>Behavior of gradient material. (<b>Left</b>): impulse response. (<b>Right</b>): frequency response.</p> Full article ">Figure 4
<p>Time-harmonic excitation applied to simple material. (<b>Left</b>): system response and steady-state solution. (<b>Right</b>): system response and fully identified model of system response.</p> Full article ">Figure 5
<p>Adaption process of filter for simple material. (<b>Left</b>): learning curve. (<b>Right</b>): development of filter coefficients.</p> Full article ">Figure 6
<p>Time-harmonic excitation applied to second-order gradient material. (<b>Left</b>): system response and steady-state solution. (<b>Right</b>): system response and fully identified model of system response.</p> Full article ">Figure 7
<p>Adaption process of filter for second-order material. (<b>Left</b>): Learning curve. (<b>Right</b>): development of filter coefficients.</p> Full article ">Figure 8
<p>Random excitation applied to simple material. (<b>Left</b>): time domain response. (<b>Right</b>): system response and fully identified model of system response.</p> Full article ">Figure 9
<p>Adaption process of filter for simple material. (<b>Left</b>): learning curve. (<b>Right</b>): development of two filter coefficients.</p> Full article ">Figure 10
<p>Random excitation applied to second-order gradient material. (<b>Left</b>): time domain response. (<b>Right</b>): system response and fully identified model of system response.</p> Full article ">Figure 11
<p>Adaption process of filter for second-order material. (<b>Left</b>): learning curve. (<b>Right</b>): development of two filter coefficients.</p> Full article ">Figure 12
<p>Normalized filter weights. (<b>Left</b>): simple material. (<b>Right</b>): second-order gradient material.</p> Full article ">

Review

Jump to: Research

31 pages, 5192 KiB  
Review
Cupolets: History, Theory, and Applications
by Matthew A. Morena and Kevin M. Short
Dynamics 2024, 4(2), 394-424; https://doi.org/10.3390/dynamics4020022 - 13 May 2024
Viewed by 604
Abstract
In chaos control, one usually seeks to stabilize the unstable periodic orbits (UPOs) that densely inhabit the attractors of many chaotic dynamical systems. These orbits collectively play a significant role in determining the dynamics and properties of chaotic systems and are said to [...] Read more.
In chaos control, one usually seeks to stabilize the unstable periodic orbits (UPOs) that densely inhabit the attractors of many chaotic dynamical systems. These orbits collectively play a significant role in determining the dynamics and properties of chaotic systems and are said to form the skeleton of the associated attractors. While UPOs are insightful tools for analysis, they are naturally unstable and, as such, are difficult to find and computationally expensive to stabilize. An alternative to using UPOs is to approximate them using cupolets. Cupolets, a name derived from chaotic, unstable, periodic, orbit-lets, are a relatively new class of waveforms that represent highly accurate approximations to the UPOs of chaotic systems, but which are generated via a particular control scheme that applies tiny perturbations along Poincaré sections. Originally discovered in an application of secure chaotic communications, cupolets have since gone on to play pivotal roles in a number of theoretical and practical applications. These developments include using cupolets as wavelets for image compression, targeting in dynamical systems, a chaotic analog to quantum entanglement, an abstract reducibility classification, a basis for audio and video compression, and, most recently, their detection in a chaotic neuron model. This review will detail the historical development of cupolets, how they are generated, and their successful integration into theoretical and computational science and will also identify some unanswered questions and future directions for this work. Full article
(This article belongs to the Special Issue Recent Advances in Dynamic Phenomena—2nd Edition)
►▼ Show Figures

Figure 1

Figure 1
<p>Double scroll attractor projected into the <math display="inline"><semantics> <mrow> <msub> <mi>v</mi> <msub> <mi>C</mi> <mn>1</mn> </msub> </msub> <mo>−</mo> <msub> <mi>i</mi> <mi>L</mi> </msub> </mrow> </semantics></math> plane, with control planes intersecting lobes 0 and 1 [<a href="#B11-dynamics-04-00022" class="html-bibr">11</a>]. Reproduced from Kevin M. Short and Matthew A. Morena, “Signatures of quantum mechanics in chaotic systems”, Entropy 21(6), 618 (2019) <a href="https://doi.org/10.3390/e21060618" target="_blank">https://doi.org/10.3390/e21060618</a> (accessed on 9 March 2024), with the permission of MDPI.</p> Full article ">Figure 2
<p>Comparing the <math display="inline"><semantics> <mrow> <mi>r</mi> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> </semantics></math> function for <math display="inline"><semantics> <mrow> <mi>P</mi> <mo>=</mo> <mn>5</mn> </mrow> </semantics></math> (red) future intersections with Poincaré surfaces and <math display="inline"><semantics> <mrow> <mi>P</mi> <mo>=</mo> <mn>12</mn> </mrow> </semantics></math> (blue) future intersections with the Poincaré surfaces.</p> Full article ">Figure 3
<p>Plot of the simplest cupolet, <b>C</b>00. Phase space plot in left column; time series plots for <math display="inline"><semantics> <mrow> <mi>x</mi> <mo>−</mo> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>y</mi> <mo>−</mo> </mrow> </semantics></math>, and <math display="inline"><semantics> <mrow> <mi>z</mi> <mo>−</mo> </mrow> </semantics></math> coordinates in center column; Magnitude FFT spectrum of four periods of the corresponding <math display="inline"><semantics> <mrow> <mi>x</mi> <mo>−</mo> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>y</mi> <mo>−</mo> </mrow> </semantics></math>, and <math display="inline"><semantics> <mrow> <mi>z</mi> <mo>−</mo> </mrow> </semantics></math> time series in right column.</p> Full article ">Figure 4
<p>Plot of cupolet <b>C</b>000101. Phase space plot in left column; time series plots for <math display="inline"><semantics> <mrow> <mi>x</mi> <mo>−</mo> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>y</mi> <mo>−</mo> </mrow> </semantics></math>, and <math display="inline"><semantics> <mrow> <mi>z</mi> <mo>−</mo> </mrow> </semantics></math> coordinates in center column; Magnitude FFT spectrum of four periods of the corresponding <math display="inline"><semantics> <mrow> <mi>x</mi> <mo>−</mo> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>y</mi> <mo>−</mo> </mrow> </semantics></math>, and <math display="inline"><semantics> <mrow> <mi>z</mi> <mo>−</mo> </mrow> </semantics></math> time series in right column.</p> Full article ">Figure 5
<p>Plot of cupolet <b>C</b>00101. Phase space plot in left column; time series plots for <math display="inline"><semantics> <mrow> <mi>x</mi> <mo>−</mo> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>y</mi> <mo>−</mo> </mrow> </semantics></math>, and <math display="inline"><semantics> <mrow> <mi>z</mi> <mo>−</mo> </mrow> </semantics></math> coordinates in center column; Magnitude FFT spectrum of four periods of the corresponding <math display="inline"><semantics> <mrow> <mi>x</mi> <mo>−</mo> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>y</mi> <mo>−</mo> </mrow> </semantics></math>, and <math display="inline"><semantics> <mrow> <mi>z</mi> <mo>−</mo> </mrow> </semantics></math> time series in right column.</p> Full article ">Figure 6
<p>Plot of cupolet <b>C</b>01011. Phase space plot in left column; time series plots for <math display="inline"><semantics> <mrow> <mi>x</mi> <mo>−</mo> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>y</mi> <mo>−</mo> </mrow> </semantics></math>, and <math display="inline"><semantics> <mrow> <mi>z</mi> <mo>−</mo> </mrow> </semantics></math> coordinates in center column; Magnitude FFT spectrum of four periods of the corresponding <math display="inline"><semantics> <mrow> <mi>x</mi> <mo>−</mo> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>y</mi> <mo>−</mo> </mrow> </semantics></math>, and <math display="inline"><semantics> <mrow> <mi>z</mi> <mo>−</mo> </mrow> </semantics></math> time series in right column.</p> Full article ">Figure 7
<p>Plot of cupolet <b>C</b>01011011. Phase space plot in left column; time series plots for <span class="html-italic">x</span>-, <span class="html-italic">y</span>-, and <span class="html-italic">z</span>-coordinates in center column; Magnitude FFT spectrum of 4 periods of the corresponding <span class="html-italic">x</span>-, <span class="html-italic">y</span>-, and <span class="html-italic">z</span>-time series in right column.</p> Full article ">Figure 8
<p>Illustrating the diversity in cupolets with (<b>a</b>) their spectral variation among cupolets and (<b>b</b>) their time-domain variations [<a href="#B52-dynamics-04-00022" class="html-bibr">52</a>].</p> Full article ">Figure 9
<p>(<b>a</b>) Original <math display="inline"><semantics> <mrow> <mn>256</mn> <mo>×</mo> <mn>256</mn> </mrow> </semantics></math> image and incremental resolution levels. The number of basis elements in each resolution level is (<b>b</b>) 24 (<b>c</b>) 56 and (<b>d</b>) 120 per window [<a href="#B60-dynamics-04-00022" class="html-bibr">60</a>]. Reproduced from Kourosh Zarringhalam and Kevin M. Short, “Generating an adaptive multiresolution image analysis with compact cupolets”, Nonlinear Dynamics, 52, 51-70 (2008) <a href="https://doi.org/10.1007/s11071-007-9257-7" target="_blank">https://doi.org/10.1007/s11071-007-9257-7</a> (accessed on 9 March 2024), with the permission of Springer Publishing.</p> Full article ">Figure 9 Cont.
<p>(<b>a</b>) Original <math display="inline"><semantics> <mrow> <mn>256</mn> <mo>×</mo> <mn>256</mn> </mrow> </semantics></math> image and incremental resolution levels. The number of basis elements in each resolution level is (<b>b</b>) 24 (<b>c</b>) 56 and (<b>d</b>) 120 per window [<a href="#B60-dynamics-04-00022" class="html-bibr">60</a>]. Reproduced from Kourosh Zarringhalam and Kevin M. Short, “Generating an adaptive multiresolution image analysis with compact cupolets”, Nonlinear Dynamics, 52, 51-70 (2008) <a href="https://doi.org/10.1007/s11071-007-9257-7" target="_blank">https://doi.org/10.1007/s11071-007-9257-7</a> (accessed on 9 March 2024), with the permission of Springer Publishing.</p> Full article ">Figure 10
<p>(<b>a</b>) Cupolet <b>C</b>00 (blue orbit) destabilizes to a chaotic transient (purple orbit) when the implementation of its control sequence is disrupted, (<b>b</b>) a smooth, transient−free transition from <b>C</b>00 to <b>C</b>01 (red orbit), (<b>c</b>) a lengthy transition from <b>C</b>00 to <b>C</b>01 that involves multiple intermediary loops around the attractor, (<b>d</b>) time series showing the transition seen in (<b>c</b>). Reproduced from Matthew A. Morena, Kevin M. Short, and Erica E. Cooke, “Controlled transitions between cupolets of chaotic systems”, Chaos 24, 013111 (2014) <a href="https://doi.org/10.1063/1.4862668" target="_blank">https://doi.org/10.1063/1.4862668</a> (accessed on 9 March 2024), with the permission of AIP Publishing.</p> Full article ">Figure 11
<p>A weighted digraph showing the interconnectivity of cupolets as edges and control bins as vertices. Reproduced from Matthew A. Morena, Kevin M. Short, and Erica E. Cooke, “Controlled transitions between cupolets of chaotic systems”, Chaos 24, 013111 (2014) <a href="https://doi.org/10.1063/1.4862668" target="_blank">https://doi.org/10.1063/1.4862668</a> (accessed on 9 March 2024), with the permission of AIP Publishing.</p> Full article ">Figure 12
<p>Illustrating the amalgamation of composite cupolet <b>C</b>0001111101110111 (period-16) by simpler, fundamental cupolets <b>C</b>000110111 (period-9), <b>C</b>01111 (period-5), and <b>C</b>11 (period-2). Reproduced from Matthew A. Morena and Kevin M. Short, “Fundamental cupolets of chaotic systems”, Chaos, 30, 093114 (2020) <a href="https://doi.org/10.1063/5.0003443" target="_blank">https://doi.org/10.1063/5.0003443</a> (accessed on 9 March 2024), with the permission of AIP Publishing.</p> Full article ">Figure 13
<p>Step-by-step formation of chaotic entanglement for the double scroll system [<a href="#B11-dynamics-04-00022" class="html-bibr">11</a>]. (<b>a</b>) An external control is applied to System I. (<b>b</b>) This subsequently stabilizes System I onto cupolet <b>C</b>000011111, which then begins producing symbolic dynamics information in the form of its visitation sequence. (<b>c</b>) An interaction function converts the visitation sequence into the emitted sequence <b>E</b>011101111 and transmits this information as a control to System II. This induces System II to stabilize onto cupolet <b>C</b>011101111. (<b>d</b>) Cupolet <b>C</b>011101111’s visitation sequence is then converted to the emitted sequence <b>E</b>000011111 and transmitted as control information back to System I, replacing the external control. Each emitted sequence exactly matches the corresponding cupolet’s control sequence, which locks both systems onto the persistent, periodic orbits of their cupolets. The cupolets’ entanglement will perpetuate as long as the exchange function continues mediating the exchange of control information between the systems. Reproduced from Kevin M. Short and Matthew A. Morena, “Signatures of quantum mechanics in chaotic systems”, Entropy 21(6), 618 (2019) <a href="https://doi.org/10.3390/e21060618" target="_blank">https://doi.org/10.3390/e21060618</a> (accessed on 9 March 2024), with the permission of MDPI.</p> Full article ">Figure 14
<p>Cupolets (<b>a</b>) <b>C</b>000011 (period 24) and (<b>b</b>) <b>C</b>000111 (period 18) can be induced into chaotic entanglement via an integrate−and−fire interaction function. The cupolets’ visitation and emitted sequences are listed in <a href="#dynamics-04-00022-t002" class="html-table">Table 2</a>.</p> Full article ">Figure 15
<p>Numerical integration of Equation (<a href="#FD5-dynamics-04-00022" class="html-disp-formula">5</a>) with <math display="inline"><semantics> <mrow> <mi>r</mi> <mo>=</mo> <mn>0.006</mn> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mi>I</mi> <mo>=</mo> <mn>3.25</mn> </mrow> </semantics></math> so that the model is in a chaotic region [<a href="#B93-dynamics-04-00022" class="html-bibr">93</a>]. (<b>A</b>) 3-Dimensional phase space plot of the <span class="html-italic">x</span>, <span class="html-italic">y</span>, and <span class="html-italic">z</span> dynamics. Projection of (<b>A</b>) onto (<b>B</b>) the <span class="html-italic">x</span>-<span class="html-italic">y</span> plane, (<b>C</b>) the <span class="html-italic">x</span>-<span class="html-italic">z</span> plane, and (<b>D</b>) the <span class="html-italic">y</span>-<span class="html-italic">z</span> plane. Reproduced from John E. Parker and Kevin M. Short, “Cupolets in a chaotic neuron model”, Chaos 32, 113104 (2022) <a href="https://doi.org/10.1063/5.0101667" target="_blank">https://doi.org/10.1063/5.0101667</a> (accessed on 9 March 2024), with the permission of AIP Publishing.</p> Full article ">Figure 16
<p>One dimensional time series of each dynamical variable (<b>A</b>) <span class="html-italic">x</span>, (<b>B</b>) <span class="html-italic">y</span>, and (<b>C</b>) <span class="html-italic">z</span> in <a href="#dynamics-04-00022-f015" class="html-fig">Figure 15</a> and Equation (<a href="#FD5-dynamics-04-00022" class="html-disp-formula">5</a>). Reproduced from John E. Parker and Kevin M. Short, “Cupolets in a chaotic neuron model”, Chaos 32, 113104 (2022) <a href="https://doi.org/10.1063/5.0101667" target="_blank">https://doi.org/10.1063/5.0101667</a> (accessed on 9 March 2024), with the permission of AIP Publishing.</p> Full article ">Figure 17
<p>Plot of the Hindmarsh-Rose dynamical system in chaotic regime with control planes [<a href="#B93-dynamics-04-00022" class="html-bibr">93</a>]. Reproduced from John E. Parker and Kevin M. Short, “Cupolets in a chaotic neuron model”, Chaos 32, 113104 (2022) <a href="https://doi.org/10.1063/5.0101667" target="_blank">https://doi.org/10.1063/5.0101667</a> (accessed on 9 March 2024), with the permission of AIP Publishing.</p> Full article ">Figure 18
<p>Selection of Hindmarsh-Rose cupolets: (<b>A</b>) <b>C</b>11, (<b>B</b>) <b>C</b>0110, (<b>C</b>) <b>C</b>1010010, and (<b>D</b>) <b>C</b>01010010 [<a href="#B93-dynamics-04-00022" class="html-bibr">93</a>]. Reproduced from John E. Parker and Kevin M. Short, “Cupolets in a chaotic neuron model”, Chaos 32, 113104 (2022) <a href="https://doi.org/10.1063/5.0101667" target="_blank">https://doi.org/10.1063/5.0101667</a> (accessed on 9 March 2024), with the permission of AIP Publishing.</p> Full article ">Figure 19
<p>(<b>a</b>–<b>d</b>) Homologous cupolets from the Hindmarsh-Rose system. All four cupolets derive from the control code 11100010, and the resulting cupolet is dependent on the initial condition.</p> Full article ">
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