Dynamics of a New Four-Thirds-Degree Sub-Quadratic Lorenz-like System
Aiming to explore the subtle connection between the number of nonlinear terms in Lorenz-like systems and hidden attractors, this paper introduces a new simple sub-quadratic four-thirds-degree Lorenz-like system, where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML"...
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2024-09-01
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| author | Guiyao Ke Jun Pan Feiyu Hu Haijun Wang |
| author_facet | Guiyao Ke Jun Pan Feiyu Hu Haijun Wang |
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| description | Aiming to explore the subtle connection between the number of nonlinear terms in Lorenz-like systems and hidden attractors, this paper introduces a new simple sub-quadratic four-thirds-degree Lorenz-like system, where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mover accent="true"><mi>x</mi><mo>˙</mo></mover><mo>=</mo><mi>a</mi><mrow><mo>(</mo><mi>y</mi><mo>−</mo><mi>x</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mover accent="true"><mi>y</mi><mo>˙</mo></mover><mo>=</mo><mi>c</mi><mi>x</mi><mo>−</mo><mroot><mi>x</mi><mn>3</mn></mroot><mi>z</mi></mrow></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mover accent="true"><mi>z</mi><mo>˙</mo></mover><mo>=</mo><mo>−</mo><mi>b</mi><mi>z</mi><mo>+</mo><mroot><mi>x</mi><mn>3</mn></mroot><mi>y</mi></mrow></semantics></math></inline-formula>, and uncovers the following property of these systems: decreasing the powers of the nonlinear terms in a quadratic Lorenz-like system where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mover accent="true"><mi>x</mi><mo>˙</mo></mover><mo>=</mo><mi>a</mi><mrow><mo>(</mo><mi>y</mi><mo>−</mo><mi>x</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mover accent="true"><mi>y</mi><mo>˙</mo></mover><mo>=</mo><mi>c</mi><mi>x</mi><mo>−</mo><mi>x</mi><mi>z</mi></mrow></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mover accent="true"><mi>z</mi><mo>˙</mo></mover><mo>=</mo><mo>−</mo><mi>b</mi><mi>z</mi><mo>+</mo><mi>x</mi><mi>y</mi></mrow></semantics></math></inline-formula>, may narrow, or even eliminate the range of the parameter <i>c</i> for hidden attractors, but enlarge it for self-excited attractors. By combining numerical simulation, stability and bifurcation theory, most of the important dynamics of the Lorenz system family are revealed, including self-excited Lorenz-like attractors, Hopf bifurcation and generic pitchfork bifurcation at the origin, singularly degenerate heteroclinic cycles, degenerate pitchfork bifurcation at non-isolated equilibria, invariant algebraic surface, heteroclinic orbits and so on. The obtained results may verify the generalization of the second part of the celebrated Hilbert’s sixteenth problem to some degree, showing that the number and mutual disposition of attractors and repellers may depend on the degree of chaotic multidimensional dynamical systems. |
| format | Article |
| id | doaj-art-afbb264caafe4ef2bf7b4574affb51e5 |
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| publishDate | 2024-09-01 |
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| spelling | doaj-art-afbb264caafe4ef2bf7b4574affb51e52025-08-20T01:56:10ZengMDPI AGAxioms2075-16802024-09-0113962510.3390/axioms13090625Dynamics of a New Four-Thirds-Degree Sub-Quadratic Lorenz-like SystemGuiyao Ke0Jun Pan1Feiyu Hu2Haijun Wang3School of Information, Zhejiang Guangsha Vocational and Technical University of Construction, Dongyang 322100, ChinaDepartment of Big Data Science, School of Science, Zhejiang University of Science and Technology, Hangzhou 310023, ChinaCollege of Sustainability and Tourism, Ritsumeikan Asia Pacific University, Beppu 874-8577, Oita, JapanSchool of Electronic and Information Engineering, Taizhou University, Taizhou 318000, ChinaAiming to explore the subtle connection between the number of nonlinear terms in Lorenz-like systems and hidden attractors, this paper introduces a new simple sub-quadratic four-thirds-degree Lorenz-like system, where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mover accent="true"><mi>x</mi><mo>˙</mo></mover><mo>=</mo><mi>a</mi><mrow><mo>(</mo><mi>y</mi><mo>−</mo><mi>x</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mover accent="true"><mi>y</mi><mo>˙</mo></mover><mo>=</mo><mi>c</mi><mi>x</mi><mo>−</mo><mroot><mi>x</mi><mn>3</mn></mroot><mi>z</mi></mrow></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mover accent="true"><mi>z</mi><mo>˙</mo></mover><mo>=</mo><mo>−</mo><mi>b</mi><mi>z</mi><mo>+</mo><mroot><mi>x</mi><mn>3</mn></mroot><mi>y</mi></mrow></semantics></math></inline-formula>, and uncovers the following property of these systems: decreasing the powers of the nonlinear terms in a quadratic Lorenz-like system where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mover accent="true"><mi>x</mi><mo>˙</mo></mover><mo>=</mo><mi>a</mi><mrow><mo>(</mo><mi>y</mi><mo>−</mo><mi>x</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mover accent="true"><mi>y</mi><mo>˙</mo></mover><mo>=</mo><mi>c</mi><mi>x</mi><mo>−</mo><mi>x</mi><mi>z</mi></mrow></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mover accent="true"><mi>z</mi><mo>˙</mo></mover><mo>=</mo><mo>−</mo><mi>b</mi><mi>z</mi><mo>+</mo><mi>x</mi><mi>y</mi></mrow></semantics></math></inline-formula>, may narrow, or even eliminate the range of the parameter <i>c</i> for hidden attractors, but enlarge it for self-excited attractors. By combining numerical simulation, stability and bifurcation theory, most of the important dynamics of the Lorenz system family are revealed, including self-excited Lorenz-like attractors, Hopf bifurcation and generic pitchfork bifurcation at the origin, singularly degenerate heteroclinic cycles, degenerate pitchfork bifurcation at non-isolated equilibria, invariant algebraic surface, heteroclinic orbits and so on. The obtained results may verify the generalization of the second part of the celebrated Hilbert’s sixteenth problem to some degree, showing that the number and mutual disposition of attractors and repellers may depend on the degree of chaotic multidimensional dynamical systems.https://www.mdpi.com/2075-1680/13/9/625generalization of hilbert’s 16th problemsub-quadratic Lorenz-like systemheteroclinic orbitLyapunov function |
| spellingShingle | Guiyao Ke Jun Pan Feiyu Hu Haijun Wang Dynamics of a New Four-Thirds-Degree Sub-Quadratic Lorenz-like System Axioms generalization of hilbert’s 16th problem sub-quadratic Lorenz-like system heteroclinic orbit Lyapunov function |
| title | Dynamics of a New Four-Thirds-Degree Sub-Quadratic Lorenz-like System |
| title_full | Dynamics of a New Four-Thirds-Degree Sub-Quadratic Lorenz-like System |
| title_fullStr | Dynamics of a New Four-Thirds-Degree Sub-Quadratic Lorenz-like System |
| title_full_unstemmed | Dynamics of a New Four-Thirds-Degree Sub-Quadratic Lorenz-like System |
| title_short | Dynamics of a New Four-Thirds-Degree Sub-Quadratic Lorenz-like System |
| title_sort | dynamics of a new four thirds degree sub quadratic lorenz like system |
| topic | generalization of hilbert’s 16th problem sub-quadratic Lorenz-like system heteroclinic orbit Lyapunov function |
| url | https://www.mdpi.com/2075-1680/13/9/625 |
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