Dynamics of a New Four-Thirds-Degree Sub-Quadratic Lorenz-like System
<p>For <math display="inline"><semantics> <mrow> <mo>(</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo>)</mo> <mo>=</mo> <mo>(</mo> <mn>4</mn> <mo>,</mo> <mn>1</mn> <mo>)</mo> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>c</mi> <mo>∈</mo> <mo>[</mo> <mn>0</mn> <mo>,</mo> <mn>5</mn> <mo>]</mo> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mrow> <mo>(</mo> <msubsup> <mi>x</mi> <mrow> <mn>0</mn> </mrow> <mn>1</mn> </msubsup> <mo>,</mo> <msubsup> <mi>y</mi> <mrow> <mn>0</mn> </mrow> <mn>1</mn> </msubsup> <mo>,</mo> <msubsup> <mi>z</mi> <mrow> <mn>0</mn> </mrow> <mn>1</mn> </msubsup> <mo>)</mo> </mrow> <mo>=</mo> <mrow> <mo>(</mo> <mn>0.13</mn> <mo>,</mo> <mn>1.3</mn> <mo>,</mo> <mn>1.6</mn> <mo>)</mo> </mrow> <mo>×</mo> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>7</mn> </mrow> </msup> </mrow> </semantics></math>; (<b>a</b>–<b>c</b>) bifurcation diagrams; (<b>d</b>) Lyapunov exponents versus <span class="html-italic">c</span> of system (<a href="#FD1-axioms-13-00625" class="html-disp-formula">1</a>). In contrast to system (<a href="#FD2-axioms-13-00625" class="html-disp-formula">2</a>) [<a href="#B22-axioms-13-00625" class="html-bibr">22</a>] (Figure 3, p. 363), these figures suggest that the solutions for the system in (<a href="#FD1-axioms-13-00625" class="html-disp-formula">1</a>) display stable equilibria and period orbits, rather than the self-excited and hidden attractors shown in the system in (<a href="#FD2-axioms-13-00625" class="html-disp-formula">2</a>).</p> "> Figure 2
<p>For <math display="inline"><semantics> <mrow> <mo>(</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo>)</mo> <mo>=</mo> <mo>(</mo> <mn>3</mn> <mo>,</mo> <mn>1.5</mn> <mo>)</mo> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>c</mi> <mo>∈</mo> <mo>[</mo> <mn>0.1</mn> <mo>,</mo> <mn>599.1</mn> <mo>]</mo> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mrow> <mo>(</mo> <msubsup> <mi>x</mi> <mrow> <mn>0</mn> </mrow> <mn>2</mn> </msubsup> <mo>,</mo> <msubsup> <mi>y</mi> <mrow> <mn>0</mn> </mrow> <mn>2</mn> </msubsup> <mo>,</mo> <msubsup> <mi>z</mi> <mrow> <mn>0</mn> </mrow> <mn>2</mn> </msubsup> <mo>)</mo> </mrow> <mo>=</mo> <mrow> <mo>(</mo> <mn>1.314</mn> <mo>,</mo> <mn>2.236</mn> <mo>,</mo> <mn>4.669</mn> <mo>)</mo> </mrow> </mrow> </semantics></math>; (<b>a</b>–<b>c</b>) bifurcation diagrams; (<b>d</b>) Lyapunov exponents versus <span class="html-italic">c</span> of system (<a href="#FD2-axioms-13-00625" class="html-disp-formula">2</a>). The subfigures (<b>a</b>–<b>c</b>) are consistent with the subfigure (<b>d</b>), showing that system (<a href="#FD2-axioms-13-00625" class="html-disp-formula">2</a>) mainly experiences periodic behaviors.</p> "> Figure 3
<p>For <math display="inline"><semantics> <mrow> <mo>(</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo>)</mo> <mo>=</mo> <mo>(</mo> <mn>3</mn> <mo>,</mo> <mn>1.5</mn> <mo>)</mo> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>c</mi> <mo>∈</mo> <mo>[</mo> <mn>0.1</mn> <mo>,</mo> <mn>599.1</mn> <mo>]</mo> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mrow> <mo>(</mo> <msubsup> <mi>x</mi> <mrow> <mn>0</mn> </mrow> <mn>2</mn> </msubsup> <mo>,</mo> <msubsup> <mi>y</mi> <mrow> <mn>0</mn> </mrow> <mn>2</mn> </msubsup> <mo>,</mo> <msubsup> <mi>z</mi> <mrow> <mn>0</mn> </mrow> <mn>2</mn> </msubsup> <mo>)</mo> </mrow> <mo>=</mo> <mrow> <mo>(</mo> <mn>1.314</mn> <mo>,</mo> <mn>2.236</mn> <mo>,</mo> <mn>4.669</mn> <mo>)</mo> </mrow> </mrow> </semantics></math>; (<b>a</b>–<b>c</b>) bifurcation diagrams; (<b>d</b>) Lyapunov exponents versus <span class="html-italic">c</span> of system (<a href="#FD1-axioms-13-00625" class="html-disp-formula">1</a>). In contrast with <a href="#axioms-13-00625-f002" class="html-fig">Figure 2</a>, the four subfigures show that system (<a href="#FD1-axioms-13-00625" class="html-disp-formula">1</a>) mainly behaves in a similar way to self-excited attractors, verifying the introduced property, i.e., a decrease in powers of nonlinear terms of the quadratic Lorenz-like system (<a href="#FD2-axioms-13-00625" class="html-disp-formula">2</a>) may narrow or even eliminate the range of the parameter <span class="html-italic">c</span> for hidden attractors, but enlarge it for self-excited attractors.</p> "> Figure 4
<p>Phase portraits of system (<a href="#FD1-axioms-13-00625" class="html-disp-formula">1</a>) for <math display="inline"><semantics> <mrow> <mo>(</mo> <mi>a</mi> <mo>,</mo> <mi>c</mi> <mo>,</mo> <mi>b</mi> <mo>)</mo> <mo>=</mo> <mo>(</mo> <mn>3</mn> <mo>,</mo> <mn>100</mn> <mo>,</mo> <mn>1.5</mn> <mo>)</mo> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mrow> <mo>(</mo> <msubsup> <mi>x</mi> <mrow> <mn>0</mn> </mrow> <mn>2</mn> </msubsup> <mo>,</mo> <msubsup> <mi>y</mi> <mrow> <mn>0</mn> </mrow> <mn>2</mn> </msubsup> <mo>,</mo> <msubsup> <mi>z</mi> <mrow> <mn>0</mn> </mrow> <mn>2</mn> </msubsup> <mo>)</mo> </mrow> <mo>=</mo> <mrow> <mo>(</mo> <mn>1.314</mn> <mo>,</mo> <mn>2.236</mn> <mo>,</mo> <mn>4.669</mn> <mo>)</mo> </mrow> </mrow> </semantics></math> illustrating the existence of two-scroll self-excited attractor suggested in <a href="#axioms-13-00625-f003" class="html-fig">Figure 3</a>.</p> "> Figure 4 Cont.
<p>Phase portraits of system (<a href="#FD1-axioms-13-00625" class="html-disp-formula">1</a>) for <math display="inline"><semantics> <mrow> <mo>(</mo> <mi>a</mi> <mo>,</mo> <mi>c</mi> <mo>,</mo> <mi>b</mi> <mo>)</mo> <mo>=</mo> <mo>(</mo> <mn>3</mn> <mo>,</mo> <mn>100</mn> <mo>,</mo> <mn>1.5</mn> <mo>)</mo> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mrow> <mo>(</mo> <msubsup> <mi>x</mi> <mrow> <mn>0</mn> </mrow> <mn>2</mn> </msubsup> <mo>,</mo> <msubsup> <mi>y</mi> <mrow> <mn>0</mn> </mrow> <mn>2</mn> </msubsup> <mo>,</mo> <msubsup> <mi>z</mi> <mrow> <mn>0</mn> </mrow> <mn>2</mn> </msubsup> <mo>)</mo> </mrow> <mo>=</mo> <mrow> <mo>(</mo> <mn>1.314</mn> <mo>,</mo> <mn>2.236</mn> <mo>,</mo> <mn>4.669</mn> <mo>)</mo> </mrow> </mrow> </semantics></math> illustrating the existence of two-scroll self-excited attractor suggested in <a href="#axioms-13-00625-f003" class="html-fig">Figure 3</a>.</p> "> Figure 5
<p>Poincaré cross-sections of system (<a href="#FD1-axioms-13-00625" class="html-disp-formula">1</a>) for <math display="inline"><semantics> <mrow> <mo>(</mo> <mi>a</mi> <mo>,</mo> <mi>c</mi> <mo>,</mo> <mi>b</mi> <mo>)</mo> <mo>=</mo> <mo>(</mo> <mn>3</mn> <mo>,</mo> <mn>100</mn> <mo>,</mo> <mn>1.5</mn> <mo>)</mo> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mrow> <mo>(</mo> <msubsup> <mi>x</mi> <mrow> <mn>0</mn> </mrow> <mn>2</mn> </msubsup> <mo>,</mo> <msubsup> <mi>y</mi> <mrow> <mn>0</mn> </mrow> <mn>2</mn> </msubsup> <mo>,</mo> <msubsup> <mi>z</mi> <mrow> <mn>0</mn> </mrow> <mn>2</mn> </msubsup> <mo>)</mo> </mrow> <mo>=</mo> <mrow> <mo>(</mo> <mn>1.314</mn> <mo>,</mo> <mn>2.236</mn> <mo>,</mo> <mn>4.669</mn> <mo>)</mo> </mrow> </mrow> </semantics></math> showing the geometrical structure of the Lorenz-like attractor depicted in <a href="#axioms-13-00625-f004" class="html-fig">Figure 4</a>.</p> "> Figure 6
<p>Phase portraits of system (<a href="#FD1-axioms-13-00625" class="html-disp-formula">1</a>) for <math display="inline"><semantics> <mrow> <mo>(</mo> <mi>a</mi> <mo>,</mo> <mi>c</mi> <mo>)</mo> <mo>=</mo> <mo>(</mo> <mn>1</mn> <mo>,</mo> <mn>36</mn> <mo>)</mo> </mrow> </semantics></math>, (<b>a</b>) <math display="inline"><semantics> <mrow> <mi>b</mi> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math>, (<b>b</b>) <math display="inline"><semantics> <mrow> <mi>b</mi> <mo>=</mo> <mn>0.07</mn> </mrow> </semantics></math>, and <math display="inline"><semantics> <mrow> <mrow> <mo>(</mo> <msubsup> <mi>x</mi> <mrow> <mn>0</mn> </mrow> <mrow> <mn>1</mn> <mo>,</mo> <mn>3</mn> </mrow> </msubsup> <mo>,</mo> <msubsup> <mi>y</mi> <mrow> <mn>0</mn> </mrow> <mrow> <mn>1</mn> <mo>,</mo> <mn>3</mn> </mrow> </msubsup> <mo>,</mo> <msubsup> <mi>z</mi> <mrow> <mn>0</mn> </mrow> <mn>3</mn> </msubsup> <mo>)</mo> </mrow> <mo>=</mo> <mrow> <mo>(</mo> <mo>±</mo> <mn>1.3</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>8</mn> </mrow> </msup> <mo>,</mo> <mo>±</mo> <mn>1.3</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>7</mn> </mrow> </msup> <mo>,</mo> <mo>−</mo> <mn>1</mn> <mo>)</mo> </mrow> </mrow> </semantics></math>. Both figures imply that collapsing singularly degenerate heteroclinic cycles in system (<a href="#FD1-axioms-13-00625" class="html-disp-formula">1</a>) create limited cycles rather than strange attractors.</p> "> Figure 7
<p>Phase portraits of system (<a href="#FD1-axioms-13-00625" class="html-disp-formula">1</a>) for <math display="inline"><semantics> <mrow> <mo>(</mo> <mi>a</mi> <mo>,</mo> <mi>c</mi> <mo>,</mo> <mi>b</mi> <mo>)</mo> <mo>=</mo> <mo>(</mo> <mn>4</mn> <mo>,</mo> <mn>1.8182</mn> <mo>,</mo> <mn>1</mn> <mo>)</mo> </mrow> </semantics></math> and (<b>a</b>) <math display="inline"><semantics> <mrow> <mrow> <mo>(</mo> <msubsup> <mi>x</mi> <mrow> <mn>0</mn> </mrow> <mrow> <mn>1</mn> <mo>,</mo> <mn>3</mn> </mrow> </msubsup> <mo>,</mo> <msubsup> <mi>y</mi> <mrow> <mn>0</mn> </mrow> <mrow> <mn>1</mn> <mo>,</mo> <mn>3</mn> </mrow> </msubsup> <mo>,</mo> <msubsup> <mi>z</mi> <mrow> <mn>0</mn> </mrow> <mn>4</mn> </msubsup> <mo>)</mo> </mrow> <mo>=</mo> <mrow> <mo>(</mo> <mo>±</mo> <mn>0.13</mn> <mo>,</mo> <mo>±</mo> <mn>1.3</mn> <mo>,</mo> <mn>1.6</mn> <mo>)</mo> </mrow> <mo>×</mo> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>7</mn> </mrow> </msup> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mrow> <mo>(</mo> <msubsup> <mi>x</mi> <mrow> <mn>0</mn> </mrow> <mrow> <mn>3</mn> <mo>,</mo> <mn>4</mn> </mrow> </msubsup> <mo>,</mo> <msubsup> <mi>y</mi> <mrow> <mn>0</mn> </mrow> <mrow> <mn>3</mn> <mo>,</mo> <mn>4</mn> </mrow> </msubsup> <mo>,</mo> <msubsup> <mi>z</mi> <mrow> <mn>0</mn> </mrow> <mn>4</mn> </msubsup> <mo>)</mo> </mrow> <mo>=</mo> <mrow> <mo>(</mo> <mo>±</mo> <mn>2.4</mn> <mo>,</mo> <mo>±</mo> <mn>2.41</mn> <mo>,</mo> <mn>3.3</mn> <mo>)</mo> </mrow> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mrow> <mo>(</mo> <msubsup> <mi>x</mi> <mrow> <mn>0</mn> </mrow> <mrow> <mn>5</mn> <mo>,</mo> <mn>6</mn> </mrow> </msubsup> <mo>,</mo> <msubsup> <mi>y</mi> <mrow> <mn>0</mn> </mrow> <mrow> <mn>5</mn> <mo>,</mo> <mn>6</mn> </mrow> </msubsup> <mo>,</mo> <msubsup> <mi>z</mi> <mrow> <mn>0</mn> </mrow> <mn>5</mn> </msubsup> <mo>)</mo> </mrow> <mo>=</mo> <mrow> <mo>(</mo> <mo>±</mo> <mn>2.39</mn> <mo>,</mo> <mo>±</mo> <mn>2.4</mn> <mo>,</mo> <mn>3.32</mn> <mo>)</mo> </mrow> </mrow> </semantics></math>, (<b>b</b>) <math display="inline"><semantics> <mrow> <mrow> <mo>(</mo> <msubsup> <mi>x</mi> <mrow> <mn>0</mn> </mrow> <mn>3</mn> </msubsup> <mo>,</mo> <msubsup> <mi>y</mi> <mrow> <mn>0</mn> </mrow> <mn>3</mn> </msubsup> <mo>,</mo> <msubsup> <mi>z</mi> <mrow> <mn>0</mn> </mrow> <mn>4</mn> </msubsup> <mo>)</mo> </mrow> <mo>=</mo> <mrow> <mo>(</mo> <mn>2.4</mn> <mo>,</mo> <mn>2.41</mn> <mo>,</mo> <mn>3.3</mn> <mo>)</mo> </mrow> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mrow> <mo>(</mo> <msubsup> <mi>x</mi> <mrow> <mn>0</mn> </mrow> <mn>5</mn> </msubsup> <mo>,</mo> <msubsup> <mi>y</mi> <mrow> <mn>0</mn> </mrow> <mn>5</mn> </msubsup> <mo>,</mo> <msubsup> <mi>z</mi> <mrow> <mn>0</mn> </mrow> <mn>5</mn> </msubsup> <mo>)</mo> </mrow> <mo>=</mo> <mrow> <mo>(</mo> <mn>2.39</mn> <mo>,</mo> <mn>2.4</mn> <mo>,</mo> <mn>3.32</mn> <mo>)</mo> </mrow> </mrow> </semantics></math>, (<b>c</b>) <math display="inline"><semantics> <mrow> <mrow> <mo>(</mo> <msubsup> <mi>x</mi> <mrow> <mn>0</mn> </mrow> <mn>4</mn> </msubsup> <mo>,</mo> <msubsup> <mi>y</mi> <mrow> <mn>0</mn> </mrow> <mn>4</mn> </msubsup> <mo>,</mo> <msubsup> <mi>z</mi> <mrow> <mn>0</mn> </mrow> <mn>4</mn> </msubsup> <mo>)</mo> </mrow> <mo>=</mo> <mrow> <mo>(</mo> <mo>−</mo> <mn>2.4</mn> <mo>,</mo> <mo>−</mo> <mn>2.41</mn> <mo>,</mo> <mn>3.3</mn> <mo>)</mo> </mrow> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mrow> <mo>(</mo> <msubsup> <mi>x</mi> <mrow> <mn>0</mn> </mrow> <mn>6</mn> </msubsup> <mo>,</mo> <msubsup> <mi>y</mi> <mrow> <mn>0</mn> </mrow> <mn>6</mn> </msubsup> <mo>,</mo> <msubsup> <mi>z</mi> <mrow> <mn>0</mn> </mrow> <mn>5</mn> </msubsup> <mo>)</mo> </mrow> <mo>=</mo> <mrow> <mo>(</mo> <mo>−</mo> <mn>2.39</mn> <mo>,</mo> <mo>−</mo> <mn>2.4</mn> <mo>,</mo> <mn>3.32</mn> <mo>)</mo> </mrow> </mrow> </semantics></math>, showing at least five limit cycles for system (<a href="#FD1-axioms-13-00625" class="html-disp-formula">1</a>) when Hopf bifurcation occurs at <math display="inline"><semantics> <msub> <mi>E</mi> <mo>±</mo> </msub> <mo>,</mo> </semantics></math> i.e., two around <math display="inline"><semantics> <msub> <mi>E</mi> <mo>+</mo> </msub> </semantics></math>, two around <math display="inline"><semantics> <msub> <mi>E</mi> <mo>−</mo> </msub> </semantics></math> and one around <math display="inline"><semantics> <msub> <mi>E</mi> <mo>±</mo> </msub> </semantics></math>.</p> "> Figure 8
<p>For <math display="inline"><semantics> <mrow> <mo>(</mo> <mi>a</mi> <mo>,</mo> <mi>c</mi> <mo>)</mo> <mo>=</mo> <mo>(</mo> <mn>6</mn> <mo>,</mo> <mn>100</mn> <mo>)</mo> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>b</mi> <mo>=</mo> <mn>8</mn> <mo>,</mo> <mn>9</mn> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mrow> <mo>(</mo> <msubsup> <mi>x</mi> <mrow> <mn>0</mn> </mrow> <mrow> <mn>1</mn> <mo>,</mo> <mn>3</mn> </mrow> </msubsup> <mo>,</mo> <msubsup> <mi>y</mi> <mrow> <mn>0</mn> </mrow> <mrow> <mn>1</mn> <mo>,</mo> <mn>3</mn> </mrow> </msubsup> <mo>,</mo> <msubsup> <mi>z</mi> <mrow> <mn>0</mn> </mrow> <mn>3</mn> </msubsup> <mo>)</mo> </mrow> <mo>=</mo> <mrow> <mo>(</mo> <mo>±</mo> <mn>0.13</mn> <mo>,</mo> <mo>±</mo> <mn>1.3</mn> <mo>,</mo> <mn>1.6</mn> <mo>)</mo> </mrow> <mo>×</mo> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>7</mn> </mrow> </msup> </mrow> </semantics></math>, phase portraits of system (<a href="#FD1-axioms-13-00625" class="html-disp-formula">1</a>), verifying the existence of a pair of heteroclinic orbits to unstable <math display="inline"><semantics> <msub> <mi>E</mi> <mn>0</mn> </msub> </semantics></math> and stable <math display="inline"><semantics> <msub> <mi>E</mi> <mo>±</mo> </msub> </semantics></math> when <math display="inline"><semantics> <mrow> <mi>c</mi> <mo>></mo> <mn>0</mn> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mn>3</mn> <mi>b</mi> <mo>≥</mo> <mn>4</mn> <mi>a</mi> <mo>></mo> <mn>0</mn> </mrow> </semantics></math>.</p> ">
Abstract
:1. Introduction
2. New Four-Thirds-Degree Lorenz-like System and Its Main Dynamics
3. Hopf Bifurcation
4. Existence of Heteroclinic Orbit
4.1.
- 1.
- Assume , and . is one of the stationary points.
- 2.
- If and , , then we arrive at and , . Namely, .
4.2.
5. Conclusions
Author Contributions
Funding
Data Availability Statement
Conflicts of Interest
References
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b | a | c | Property of |
---|---|---|---|
<0 | <0 | <0 | A 1D and a 2D |
>0 | A 3D | ||
>0 | <0 | A 2D and a 1D | |
>0 | A 1D and a 2D | ||
>0 | <0 | <0 | A 2D and a 1D |
>0 | A 1D and a 2D | ||
>0 | <0 | A 3D | |
>0 | A 2D and a 1D |
z | a | Property of |
---|---|---|
>0 | <0 | A 1D , a 1D and a 1D |
>0 | A 2D and a 1D | |
<0 | <0 | A 2D and a 1D |
>0 | A 1D , a 1D and a 1D |
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Ke, G.; Pan, J.; Hu, F.; Wang, H. Dynamics of a New Four-Thirds-Degree Sub-Quadratic Lorenz-like System. Axioms 2024, 13, 625. https://doi.org/10.3390/axioms13090625
Ke G, Pan J, Hu F, Wang H. Dynamics of a New Four-Thirds-Degree Sub-Quadratic Lorenz-like System. Axioms. 2024; 13(9):625. https://doi.org/10.3390/axioms13090625
Chicago/Turabian StyleKe, Guiyao, Jun Pan, Feiyu Hu, and Haijun Wang. 2024. "Dynamics of a New Four-Thirds-Degree Sub-Quadratic Lorenz-like System" Axioms 13, no. 9: 625. https://doi.org/10.3390/axioms13090625