Bifurcation Diagrams of Nonlinear Oscillatory Dynamical Systems: A Brief Review in 1D, 2D and 3D
<p>Five different 1D bifurcation diagrams for the <span class="html-italic">RLC</span> electric arc system [<a href="#B19-entropy-26-00770" class="html-bibr">19</a>] for the varying parameter <math display="inline"><semantics> <mrow> <mn>5</mn> <mo>≤</mo> <mi>R</mi> <mo>≤</mo> <mn>25</mn> </mrow> </semantics></math>. The integration fixed step-size in the Runge–Kutta IV method was <math display="inline"><semantics> <mrow> <mi>d</mi> <mi>t</mi> <mo>=</mo> <mn>0.01</mn> </mrow> </semantics></math>. (<b>a</b>) Bifurcation diagram with local maximum values of periodic and chaotic responses for the nonlinear electric arc system. (<b>b</b>) Diagram of the LLE corresponding to the diagram in (<b>a</b>). (<b>c</b>) Diagram of the 0–1 test values <span class="html-italic">K</span> corresponding to the diagram in (<b>a</b>). Parameters given in <a href="#app1-entropy-26-00770" class="html-app">Appendix A</a>. (<b>d</b>) Sample entropy values <math display="inline"><semantics> <mrow> <mi>S</mi> <mi>a</mi> <mi>E</mi> <mi>n</mi> </mrow> </semantics></math> corresponding to the diagram in (<b>a</b>). Parameters of the method were <math display="inline"><semantics> <mrow> <mi>r</mi> <mo>=</mo> <mn>0.05</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>m</mi> <mo>=</mo> <mn>100</mn> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mi>N</mi> <mo>=</mo> <mn>10,000</mn> </mrow> </semantics></math> (<a href="#app4-entropy-26-00770" class="html-app">Appendix D</a>). (<b>e</b>) Lyapunov dimension (LD) values corresponding to the diagram in (<b>a</b>). The LD values are greater than but close to 2 for the intervals marked with the red horizontal segments.</p> "> Figure 2
<p>Two diagrams for the electric arc circuit with varying <span class="html-italic">R</span> and <span class="html-italic">C</span> parameters. (<b>a</b>) Sample entropy diagram with <math display="inline"><semantics> <mrow> <mn>0</mn> <mo><</mo> <mi>S</mi> <mi>a</mi> <mi>E</mi> <mi>n</mi> <mo><</mo> <mn>0.076</mn> </mrow> </semantics></math>. (<b>b</b>) The 0–1 test diagram with <math display="inline"><semantics> <mrow> <mn>0</mn> <mo><</mo> <mi>K</mi> <mo><</mo> <mn>1</mn> </mrow> </semantics></math>.</p> "> Figure 3
<p>Two-parameter <math display="inline"><semantics> <mrow> <mn>1000</mn> <mo>×</mo> <mn>1000</mn> </mrow> </semantics></math> diagrams from the 0–1 test, each obtained using the Runge–Kutta IV solver with <math display="inline"><semantics> <mrow> <mi>d</mi> <mi>t</mi> <mo>=</mo> <mn>0.001</mn> </mrow> </semantics></math> for <math display="inline"><semantics> <mrow> <mn>0</mn> <mo>≤</mo> <mi>t</mi> <mo>≤</mo> <mn>500</mn> </mrow> </semantics></math> when the solution in the interval <math display="inline"><semantics> <mrow> <mn>300</mn> <mo>≤</mo> <mi>t</mi> <mo>≤</mo> <mn>500</mn> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mi>T</mi> <mo>=</mo> <mn>40</mn> </mrow> </semantics></math> was used in the 0–1 test. Obtaining each of the above two-parameter diagrams requires solving the nonlinear system in ref. [<a href="#B8-entropy-26-00770" class="html-bibr">8</a>] <math display="inline"><semantics> <msup> <mn>10</mn> <mn>6</mn> </msup> </semantics></math> times with additional computations (classification of the type of these solutions). A total of 256 gray levels were used for parameter <span class="html-italic">K</span> (along the vertical bars on the right-hand side of each diagram). (<b>a</b>) Varying <math display="inline"><semantics> <msub> <mi>K</mi> <mrow> <mi>c</mi> <mi>h</mi> </mrow> </msub> </semantics></math> and <math display="inline"><semantics> <msub> <mi>k</mi> <mrow> <mi>E</mi> <mi>R</mi> <mo>,</mo> <mi>c</mi> <mi>h</mi> </mrow> </msub> </semantics></math> (constant <math display="inline"><semantics> <mrow> <msub> <mi>k</mi> <mrow> <mi>E</mi> <mi>R</mi> <mo>,</mo> <mi>p</mi> <mi>u</mi> <mi>m</mi> <mi>p</mi> </mrow> </msub> <mo>=</mo> <mn>20</mn> </mrow> </semantics></math>). (<b>b</b>) Varying <math display="inline"><semantics> <msub> <mi>K</mi> <mrow> <mi>c</mi> <mi>h</mi> </mrow> </msub> </semantics></math> and <math display="inline"><semantics> <msub> <mi>k</mi> <mrow> <mi>E</mi> <mi>R</mi> <mo>,</mo> <mi>p</mi> <mi>u</mi> <mi>m</mi> <mi>p</mi> </mrow> </msub> </semantics></math> (constant <math display="inline"><semantics> <mrow> <msub> <mi>k</mi> <mrow> <mi>E</mi> <mi>R</mi> <mo>,</mo> <mi>c</mi> <mi>h</mi> </mrow> </msub> <mo>=</mo> <mn>3500</mn> </mrow> </semantics></math>). (<b>c</b>) Varying <math display="inline"><semantics> <msub> <mi>K</mi> <mrow> <mi>c</mi> <mi>h</mi> </mrow> </msub> </semantics></math> and <math display="inline"><semantics> <msub> <mi>k</mi> <mrow> <mi>E</mi> <mi>R</mi> <mo>,</mo> <mi>p</mi> <mi>u</mi> <mi>m</mi> <mi>p</mi> </mrow> </msub> </semantics></math> (constant <math display="inline"><semantics> <mrow> <msub> <mi>k</mi> <mrow> <mi>E</mi> <mi>R</mi> <mo>,</mo> <mi>c</mi> <mi>h</mi> </mrow> </msub> <mo>=</mo> <mn>1000</mn> </mrow> </semantics></math>). (<b>d</b>) Diagram in area <span class="html-italic">A</span> in (<b>b</b>). (<b>e</b>) Diagram in area <span class="html-italic">B</span> in (<b>c</b>). (<b>f</b>) Diagram in area <span class="html-italic">C</span> in (<b>d</b>).</p> "> Figure 4
<p>Three-dimensional diagrams of test 0–1 for chaos for the electric arc system. A total of 256 gray levels were used for parameter <span class="html-italic">K</span> (vertical bars). (<b>a</b>) Parameters <math display="inline"><semantics> <mrow> <mi>R</mi> <mo>∈</mo> <mo>(</mo> <mn>3</mn> <mo>;</mo> <mn>40</mn> <mo>)</mo> <mo>,</mo> <mi>C</mi> <mo>∈</mo> <mo>(</mo> <mn>4.05</mn> <mo>;</mo> <mn>4.45</mn> <mo>)</mo> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>L</mi> <mo>∈</mo> <mo>(</mo> <mn>0.2</mn> <mo>;</mo> <mn>1.2</mn> <mo>)</mo> </mrow> </semantics></math>. Computations performed with <math display="inline"><semantics> <msup> <mn>10</mn> <mn>6</mn> </msup> </semantics></math> discrete points in the box of size <math display="inline"><semantics> <mrow> <mn>100</mn> <mo>×</mo> <mn>100</mn> <mo>×</mo> <mn>100</mn> </mrow> </semantics></math>. (<b>b</b>) Parameters <math display="inline"><semantics> <mrow> <mi>R</mi> <mo>∈</mo> <mo>(</mo> <mn>3</mn> <mo>;</mo> <mn>40</mn> <mo>)</mo> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>C</mi> <mo>∈</mo> <mo>(</mo> <mn>4.05</mn> <mo>;</mo> <mn>4.45</mn> <mo>)</mo> <mo>,</mo> <mi>L</mi> <mo>∈</mo> <mo>(</mo> <mn>0.2</mn> <mo>;</mo> <mn>0.4</mn> <mo>)</mo> </mrow> </semantics></math>. Computations performed with <math display="inline"><semantics> <mrow> <mn>8</mn> <mo>×</mo> <msup> <mn>10</mn> <mn>6</mn> </msup> </mrow> </semantics></math> discrete points in the small green box in (<b>a</b>) (size <math display="inline"><semantics> <mrow> <mn>200</mn> <mo>×</mo> <mn>200</mn> <mo>×</mo> <mn>200</mn> </mrow> </semantics></math>).</p> "> Figure 5
<p>Three-dimensional bifurcation diagrams for the cytosolic calcium oscillation model. Computations performed with <math display="inline"><semantics> <msup> <mn>10</mn> <mn>6</mn> </msup> </semantics></math> (<b>a</b>) and <math display="inline"><semantics> <mrow> <mn>8</mn> <mo>×</mo> <msup> <mn>10</mn> <mn>6</mn> </msup> </mrow> </semantics></math> (<b>b</b>,<b>c</b>) discrete points. (<b>a</b>) <math display="inline"><semantics> <mrow> <msub> <mi>k</mi> <mrow> <mi>E</mi> <mi>R</mi> <mo>,</mo> <mi>p</mi> <mi>u</mi> <mi>m</mi> <mi>p</mi> </mrow> </msub> <mo>∈</mo> <mrow> <mo>(</mo> <mn>16</mn> <mo>;</mo> <mn>26</mn> <mo>)</mo> </mrow> <mo>,</mo> <msub> <mi>K</mi> <mrow> <mi>c</mi> <mi>h</mi> </mrow> </msub> <mo>∈</mo> <mrow> <mo>(</mo> <mn>3.8</mn> <mo>;</mo> <mn>5.8</mn> <mo>)</mo> </mrow> <mo>,</mo> <msub> <mi>k</mi> <mrow> <mi>E</mi> <mi>R</mi> <mo>,</mo> <mi>c</mi> <mi>h</mi> </mrow> </msub> <mo>∈</mo> <mrow> <mo>(</mo> <mn>3500</mn> <mo>;</mo> <mn>5500</mn> <mo>)</mo> </mrow> </mrow> </semantics></math>. (<b>b</b>) <math display="inline"><semantics> <mrow> <msub> <mi>k</mi> <mrow> <mi>E</mi> <mi>R</mi> <mo>,</mo> <mi>p</mi> <mi>u</mi> <mi>m</mi> <mi>p</mi> </mrow> </msub> <mo>∈</mo> <mrow> <mo>(</mo> <mn>16</mn> <mo>;</mo> <mn>26</mn> <mo>)</mo> </mrow> <mo>,</mo> <msub> <mi>K</mi> <mrow> <mi>c</mi> <mi>h</mi> </mrow> </msub> <mo>∈</mo> <mrow> <mo>(</mo> <mn>3.8</mn> <mo>;</mo> <mn>5.8</mn> <mo>)</mo> </mrow> <mo>,</mo> <msub> <mi>k</mi> <mrow> <mi>E</mi> <mi>R</mi> <mo>,</mo> <mi>c</mi> <mi>h</mi> </mrow> </msub> <mo>∈</mo> <mrow> <mo>(</mo> <mn>3500</mn> <mo>;</mo> <mn>5500</mn> <mo>)</mo> </mrow> </mrow> </semantics></math>. (<b>c</b>) <math display="inline"><semantics> <mrow> <msub> <mi>k</mi> <mrow> <mi>E</mi> <mi>R</mi> <mo>,</mo> <mi>p</mi> <mi>u</mi> <mi>m</mi> <mi>p</mi> </mrow> </msub> <mo>∈</mo> <mrow> <mo>(</mo> <mn>16</mn> <mo>;</mo> <mn>21</mn> <mo>)</mo> </mrow> <mo>,</mo> <msub> <mi>K</mi> <mrow> <mi>c</mi> <mi>h</mi> </mrow> </msub> <mo>∈</mo> <mrow> <mo>(</mo> <mn>3.8</mn> <mo>;</mo> <mn>4.8</mn> <mo>)</mo> </mrow> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msub> <mi>k</mi> <mrow> <mi>E</mi> <mi>R</mi> <mo>,</mo> <mi>c</mi> <mi>h</mi> </mrow> </msub> <mo>∈</mo> <mrow> <mo>(</mo> <mn>3500</mn> <mo>;</mo> <mn>4500</mn> <mo>)</mo> </mrow> </mrow> </semantics></math> (the green box in (<b>b</b>)).</p> "> Figure 6
<p>Three-dimensional diagrams of size <math display="inline"><semantics> <mrow> <mn>100</mn> <mo>×</mo> <mn>100</mn> <mo>×</mo> <mn>100</mn> </mrow> </semantics></math> of the 0–1 test for the arc system with parameters <math display="inline"><semantics> <mrow> <mi>R</mi> <mo>∈</mo> <mo>[</mo> <mn>3</mn> <mo>,</mo> <mn>40</mn> <mo>]</mo> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>C</mi> <mo>∈</mo> <mo>[</mo> <mn>4.05</mn> <mo>,</mo> <mn>4.45</mn> <mo>]</mo> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mi>L</mi> <mo>∈</mo> <mo>[</mo> <mn>0.2</mn> <mo>,</mo> <mn>0.4</mn> <mo>]</mo> </mrow> </semantics></math>. Points representing chaotic responses with <span class="html-italic">K</span> values close to 1 are shown in (<b>a</b>,<b>b</b>). Points representing periodic responses with <span class="html-italic">K</span> values close to 0 are shown in (<b>c</b>,<b>d</b>). (<b>a</b>) Points in the cube with <math display="inline"><semantics> <mrow> <mi>K</mi> <mo>∈</mo> <mo>(</mo> <mn>0.9</mn> <mo>;</mo> <mn>1.0</mn> <mo>)</mo> </mrow> </semantics></math>. (<b>b</b>) Points in the cube with <math display="inline"><semantics> <mrow> <mi>K</mi> <mo>∈</mo> <mo>(</mo> <mn>0.99</mn> <mo>;</mo> <mn>1.0</mn> <mo>)</mo> </mrow> </semantics></math>. (<b>c</b>) Points in the cube with <math display="inline"><semantics> <mrow> <mi>K</mi> <mo>∈</mo> <mo>(</mo> <mn>0</mn> <mo>;</mo> <mn>0.1</mn> <mo>)</mo> </mrow> </semantics></math>. (<b>d</b>) Points in the cube with <math display="inline"><semantics> <mrow> <mi>K</mi> <mo>∈</mo> <mo>(</mo> <mn>0</mn> <mo>;</mo> <mn>0.01</mn> <mo>)</mo> </mrow> </semantics></math>.</p> "> Figure 7
<p>Three-dimensional diagrams of the 0–1 test and sample entropy methods for the electric arc system. A total of 256 gray levels were used for the values of sample entropy (vertical gray bars in (<b>b</b>,<b>c</b>). (<b>a</b>) Parameters <math display="inline"><semantics> <mrow> <mi>R</mi> <mo>∈</mo> <mo>(</mo> <mn>15</mn> <mo>;</mo> <mn>16</mn> <mo>)</mo> <mo>,</mo> <mi>C</mi> <mo>∈</mo> <mo>(</mo> <mn>4.36</mn> <mo>;</mo> <mn>4.52</mn> <mo>)</mo> <mo>,</mo> <mi>L</mi> <mo>∈</mo> <mo>(</mo> <mn>0.11</mn> <mo>;</mo> <mn>0.16</mn> <mo>)</mo> </mrow> </semantics></math>. Computations performed with <math display="inline"><semantics> <msup> <mn>10</mn> <mn>6</mn> </msup> </semantics></math> points using the 0–1 test method. (<b>b</b>) Parameters as in (<b>a</b>). Computations performed with <math display="inline"><semantics> <msup> <mn>10</mn> <mn>6</mn> </msup> </semantics></math> points using the sample entropy method. (<b>c</b>) Parameters as in (<b>a</b>). Computations performed with <math display="inline"><semantics> <mrow> <mn>8</mn> <mo>×</mo> <msup> <mn>10</mn> <mn>6</mn> </msup> </mrow> </semantics></math> points using the sample entropy method.</p> "> Figure A1
<p>Electric arc circuits. Circuit B is also described by Equation (A6) with a suitable change in variables. (<b>a</b>) Circuit A. (<b>b</b>) Circuit B.</p> ">
Abstract
:1. Introduction
2. One-Dimensional Bifurcation Diagrams
3. Moving from 1D to 2D Bifurcation Diagrams
4. Three-Dimensional Bifurcation Diagrams
5. Conclusions
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Conflicts of Interest
Appendix A. The 0–1 Test (For Chaos)
Appendix B. The Electric Arc System
Appendix C. Model of Cytosolic Calcium Oscillations
Appendix D. Sample Entropy Concept [9]
Appendix E. Computational Environment
- Figure 4a, the 0–1 test with points: 25,007 s.
- Figure 4b, the 0–1 test with points: 200,891 s.
- Figure 5a, the 0–1 test with points: 4717 s.
- Figure 5b, the 0–1 test with points: 36,989 s.
- Figure 5c, the 0–1 test with points: 37,303 s.
- Figure 7a, the 0–1 test diagram with points: 28,275 s.
- Figure 7b, the sample entropy method with points: 65,212 s.
- Figure 7c, the sample entropy method with points: 477,603 s.
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Marszalek, W.; Walczak, M. Bifurcation Diagrams of Nonlinear Oscillatory Dynamical Systems: A Brief Review in 1D, 2D and 3D. Entropy 2024, 26, 770. https://doi.org/10.3390/e26090770
Marszalek W, Walczak M. Bifurcation Diagrams of Nonlinear Oscillatory Dynamical Systems: A Brief Review in 1D, 2D and 3D. Entropy. 2024; 26(9):770. https://doi.org/10.3390/e26090770
Chicago/Turabian StyleMarszalek, Wieslaw, and Maciej Walczak. 2024. "Bifurcation Diagrams of Nonlinear Oscillatory Dynamical Systems: A Brief Review in 1D, 2D and 3D" Entropy 26, no. 9: 770. https://doi.org/10.3390/e26090770