Complex dynamics of a sub-quadratic Lorenz-like system
Motivated by the generic dynamical property of most quadratic Lorenz-type systems that the unstable manifolds of the origin tending to the stable manifold of nontrivial symmetrical equilibria forms a pair of heteroclinic orbits, this technical note reports a new 3D sub-quadratic Lorenz-like system:...
| Main Authors: | , , , , , |
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| Format: | Article |
| Language: | English |
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De Gruyter
2023-07-01
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| Series: | Open Physics |
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| Online Access: | https://doi.org/10.1515/phys-2022-0251 |
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| author | Li Zhenpeng Ke Guiyao Wang Haijun Pan Jun Hu Feiyu Su Qifang |
| author_facet | Li Zhenpeng Ke Guiyao Wang Haijun Pan Jun Hu Feiyu Su Qifang |
| author_sort | Li Zhenpeng |
| collection | DOAJ |
| description | Motivated by the generic dynamical property of most quadratic Lorenz-type systems that the unstable manifolds of the origin tending to the stable manifold of nontrivial symmetrical equilibria forms a pair of heteroclinic orbits, this technical note reports a new 3D sub-quadratic Lorenz-like system: x˙=a(y−x)\dot{x}=a(y-x), y˙=cx3+dy−x3z\dot{y}=c\sqrt[3]{x}+{\rm{d}}y-\sqrt[3]{x}z and z˙=−bz+x3y\dot{z}=-bz+\sqrt[3]{x}y. Instead, the unstable manifolds of nontrivial symmetrical equilibria tending to the stable manifold of the origin creates a pair of heteroclinic orbits. This drives one to further investigate it and reveal its other hidden dynamics: Hopf bifurcation, invariant algebraic surfaces, ultimate bound sets, globally exponentially attractive sets, existence of homoclinic and heteroclinic orbits, singularly degenerate heteroclinic cycles, and so on. The main contributions of this work are summarized as follows: First, the ultimate boundedness of that system yields the globally exponentially attractive sets of it. Second, the existence of another heteroclinic orbits is also proved by utilizing two different Lyapunov functions. Finally, on the invariant algebraic surface z=34ax43z=\frac{3}{4a}\sqrt[3]{{x}^{4}}, the existence of a pair of homoclinic orbits to the origin, and two pairs of heteroclinic orbits to two pairs of nontrivial symmetrical equilibria is also proved by utilizing a Hamiltonian function. In addition, the correctness of the theoretical results is illustrated via numerical examples. |
| first_indexed | 2024-03-08T13:49:29Z |
| format | Article |
| id | doaj.art-9f9b91a40c9245f3835fd1a050c9e6fa |
| institution | Directory Open Access Journal |
| issn | 2391-5471 |
| language | English |
| last_indexed | 2024-03-08T13:49:29Z |
| publishDate | 2023-07-01 |
| publisher | De Gruyter |
| record_format | Article |
| series | Open Physics |
| spelling | doaj.art-9f9b91a40c9245f3835fd1a050c9e6fa2024-01-16T07:19:11ZengDe GruyterOpen Physics2391-54712023-07-0121130354110.1515/phys-2022-0251Complex dynamics of a sub-quadratic Lorenz-like systemLi Zhenpeng0Ke Guiyao1Wang Haijun2Pan Jun3Hu Feiyu4Su Qifang5School of Electronic and Information Engineering (School of Big Data Science), Taizhou University, Taizhou, Zhejiang, 318000, ChinaSchool of Information, Zhejiang Guangsha Vocational and Technical University of Construction, Dongyang, Zhejiang 322100, ChinaSchool of Electronic and Information Engineering (School of Big Data Science), Taizhou University, Taizhou, Zhejiang, 318000, 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, Jumonjibaru, Beppu, Oita, 874-8577, JapanSchool of Electronic and Information Engineering (School of Big Data Science), Taizhou University, Taizhou, Zhejiang, 318000, ChinaMotivated by the generic dynamical property of most quadratic Lorenz-type systems that the unstable manifolds of the origin tending to the stable manifold of nontrivial symmetrical equilibria forms a pair of heteroclinic orbits, this technical note reports a new 3D sub-quadratic Lorenz-like system: x˙=a(y−x)\dot{x}=a(y-x), y˙=cx3+dy−x3z\dot{y}=c\sqrt[3]{x}+{\rm{d}}y-\sqrt[3]{x}z and z˙=−bz+x3y\dot{z}=-bz+\sqrt[3]{x}y. Instead, the unstable manifolds of nontrivial symmetrical equilibria tending to the stable manifold of the origin creates a pair of heteroclinic orbits. This drives one to further investigate it and reveal its other hidden dynamics: Hopf bifurcation, invariant algebraic surfaces, ultimate bound sets, globally exponentially attractive sets, existence of homoclinic and heteroclinic orbits, singularly degenerate heteroclinic cycles, and so on. The main contributions of this work are summarized as follows: First, the ultimate boundedness of that system yields the globally exponentially attractive sets of it. Second, the existence of another heteroclinic orbits is also proved by utilizing two different Lyapunov functions. Finally, on the invariant algebraic surface z=34ax43z=\frac{3}{4a}\sqrt[3]{{x}^{4}}, the existence of a pair of homoclinic orbits to the origin, and two pairs of heteroclinic orbits to two pairs of nontrivial symmetrical equilibria is also proved by utilizing a Hamiltonian function. In addition, the correctness of the theoretical results is illustrated via numerical examples.https://doi.org/10.1515/phys-2022-0251sub-quadratic lorenz-like systemglobally exponentially attractive sethomoclinic orbitheteroclinic orbitlyapunov function |
| spellingShingle | Li Zhenpeng Ke Guiyao Wang Haijun Pan Jun Hu Feiyu Su Qifang Complex dynamics of a sub-quadratic Lorenz-like system Open Physics sub-quadratic lorenz-like system globally exponentially attractive set homoclinic orbit heteroclinic orbit lyapunov function |
| title | Complex dynamics of a sub-quadratic Lorenz-like system |
| title_full | Complex dynamics of a sub-quadratic Lorenz-like system |
| title_fullStr | Complex dynamics of a sub-quadratic Lorenz-like system |
| title_full_unstemmed | Complex dynamics of a sub-quadratic Lorenz-like system |
| title_short | Complex dynamics of a sub-quadratic Lorenz-like system |
| title_sort | complex dynamics of a sub quadratic lorenz like system |
| topic | sub-quadratic lorenz-like system globally exponentially attractive set homoclinic orbit heteroclinic orbit lyapunov function |
| url | https://doi.org/10.1515/phys-2022-0251 |
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