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一类三维保守系统中的同宿轨异宿环和混沌
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Abstract:
本文研究了一个三维保守系统的同宿轨、异宿环和混沌。当系统有一条平衡点曲线时，我们得到连接原点的两条同宿轨并且给出了解析表达式。在另一条件下，我们发现系统存在一个异宿环，并且给出了解析表达式。当系统有无穷多孤立平衡点时，我们分析了这些平衡点的稳定性，发现系统具有暂态混沌现象。此外，系统展示了一些共存的轨道，包括混沌和周期轨共存，拟周期轨和周期轨共存，混沌和拟周期轨共存。
This paper investigates the homoclinic orbits, heteroclinic cycle and chaos in a three-dimensional conservative system. First, if the system has a curve of equilibria, we get two homoclinic orbits associating with the origin and give the analytical expression. Under another condition, the system has a heteroclinic cycle and the analytical expression is also given. Second, if the system has infinitely many isolated equilibria, we analyse the stabilities of the equilibria. In this case, we find that the system could show transient chaos phenomenon. Moreover, several coexisting orbits are found in the system, including coexisting chaotic orbit and periodic orbits, coexisting quasi-periodic orbits and periodic orbits, coexisting chaotic orbit and quasi-periodic orbits.
| [1] | 王高雄, 朱思铭, 等. 常微分方程[M]. 北京: 高等教育出版社, 2007. |
| [2] | 杨廷. 三维非线性自治系统的复杂动力学研究[D]: [博士学位论文]. 广州: 华南理工大学, 2018. |
| [3] | Lorenz, E.N. (1963) Deterministic Nonperiodic Flow. Journal of the Atmospheric Sciences, 20, 130-141. https://doi.org/10.1175/1520-0469(1963)020<0130:dnf>2.0.co;2 |
| [4] | Strogatz, S.H. (2018) Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry, and Engineering. CRC Press. |
| [5] | Zaher, A.A. and Abu-Rezq, A. (2011) On the Design of Chaos-Based Secure Communication Systems. Communications in Nonlinear Science and Numerical Simulation, 16, 3721-3737. https://doi.org/10.1016/j.cnsns.2010.12.032 |
| [6] | Shil’nikov, L.P. (2001) Methods of Qualitative Theory in Nonlinear Dynamics. World Scientific. |
| [7] | Yang, T. (2020) Dynamical Analysis on a Finance System with Nonconstant Elasticity of Demand. International Journal of Bifurcation and Chaos, 30, Article ID: 2050148. https://doi.org/10.1142/s0218127420501485 |
| [8] | Sahu, S.R. and Raw, S.N. (2023) Appearance of Chaos and Bi-Stability in a Fear Induced Delayed Predator-Prey System: A Mathematical Modeling Study. Chaos, Solitons & Fractals, 175, Article ID: 114008. https://doi.org/10.1016/j.chaos.2023.114008 |
| [9] | Dong, E., Yuan, M., Du, S. and Chen, Z. (2019) A New Class of Hamiltonian Conservative Chaotic Systems with Multistability and Design of Pseudo-Random Number Generator. Applied Mathematical Modelling, 73, 40-71. https://doi.org/10.1016/j.apm.2019.03.037 |
| [10] | Lai, Y.-C. and Tel, T. (2011) Transient Chaos: Complex Dynamics on Finite Time Scales. Springer Science & Business Media. |
| [11] | Wiggins, S. (2013) Global Bifurcations and Chaos: Analytical Methods. Springer Science & Business Media, Berlin. |
| [12] | Champneys, A.R., Kirk, V., Knobloch, E., Oldeman, B.E. and Rademacher, J.D.M. (2009) Unfolding a Tangent Equilibrium-to-Periodic Heteroclinic Cycle. SIAM Journal on Applied Dynamical Systems, 8, 1261-1304. https://doi.org/10.1137/080734923 |
| [13] | Hastings, S.P. and Troy, W.C. (1994) A Proof That the Lorenz Equations Have a Homoclinic Orbit. Journal of Differential Equations, 113, 166-188. https://doi.org/10.1006/jdeq.1994.1119 |
| [14] | Kokubu, H. and Roussarie, R. (2004) Existence of a Singularly Degenerate Heteroclinic Cycle in the Lorenz System and Its Dynamical Consequences: Part I. Journal of Dynamics and Differential Equations, 16, 513-557. https://doi.org/10.1007/s10884-004-4290-4 |
| [15] | Leonov, G.A. (2014) Fishing Principle for Homoclinic and Heteroclinic Trajectories. Nonlinear Dynamics, 78, 2751-2758. https://doi.org/10.1007/s11071-014-1622-8 |
| [16] | Coomes, B.A., Koçak, H. and Palmer, K.J. (2015) A Computable Criterion for the Existence of Connecting Orbits in Autonomous Dynamics. Journal of Dynamics and Differential Equations, 28, 1081-1114. https://doi.org/10.1007/s10884-015-9437-y |
| [17] | Yang, Q. and Yang, T. (2017) Complex Dynamics in a Generalized Langford System. Nonlinear Dynamics, 91, 2241-2270. https://doi.org/10.1007/s11071-017-4012-1 |
| [18] | Yang, T. (2020) Homoclinic Orbits and Chaos in the Generalized Lorenz System. Discrete & Continuous Dynamical Systems B, 25, 1097-1108. https://doi.org/10.3934/dcdsb.2019210 |
| [19] | Lu, K., Yang, Q. and Chen, G. (2019) Singular Cycles and Chaos in a New Class of 3D Three-Zone Piecewise Affine Systems. Chaos: An Interdisciplinary Journal of Nonlinear Science, 29, Article ID: 043124. https://doi.org/10.1063/1.5089662 |
| [20] | Lu, K. and Xu, W. (2022) Coexisting Singular Cycles in a Class of Three-Dimensional Three-Zone Piecewise Affine Systems. Discrete and Continuous Dynamical Systems B, 27, 7315-7349. https://doi.org/10.3934/dcdsb.2022045 |
| [21] | Lu, K., Xu, W., Yang, T. and Xiang, Q. (2022) Chaos Emerges from Coexisting Homoclinic Cycles for a Class of 3D Piecewise Systems. Chaos, Solitons & Fractals, 162, Article ID: 112470. https://doi.org/10.1016/j.chaos.2022.112470 |
| [22] | Vieira, R.S.S. and Mosna, R.A. (2022) Homoclinic Chaos in the Hamiltonian Dynamics of Extended Test Bodies. Chaos, Solitons & Fractals, 163, Article ID: 112541. https://doi.org/10.1016/j.chaos.2022.112541 |
| [23] | Ashtari, O. and Schneider, T.M. (2023) Jacobian-Free Variational Method for Computing Connecting Orbits in Nonlinear Dynamical Systems. Chaos: An Interdisciplinary Journal of Nonlinear Science, 33, Article ID: 073134. https://doi.org/10.1063/5.0143923 |
| [24] | Hale, J.K., Peletier, L.A. and Troy, W.C. (2000) Exact Homoclinic and Heteroclinic Solutions of the Gray-Scott Model for Autocatalysis. SIAM Journal on Applied Mathematics, 61, 102-130. https://doi.org/10.1137/s0036139998334913 |
| [25] | Bao, J. and Yang, Q. (2011) A New Method to Find Homoclinic and Heteroclinic Orbits. Applied Mathematics and Computation, 217, 6526-6540. https://doi.org/10.1016/j.amc.2011.01.032 |
| [26] | Li, J. and Zhao, X. (2011) Exact Heteroclinic Cycle Family and Quasi-Periodic Solutions for the Three-Dimensional Systems Determined by Chazy Class IX. International Journal of Bifurcation and Chaos, 21, 1357-1367. https://doi.org/10.1142/s0218127411029227 |
| [27] | Li, J. and Chen, F. (2011) Exact Homoclinic Orbits and Heteroclinic Families for a Third-Order System in the Chazy Class XI (N = 3). International Journal of Bifurcation and Chaos, 21, 3305-3322. https://doi.org/10.1142/s0218127411030544 |
| [28] | Algaba, A., Freire, E., Gamero, E. and Rodríguez-Luis, A.J. (2015) An Exact Homoclinic Orbit and Its Connection with the Rössler System. Physics Letters A, 379, 1114-1121. https://doi.org/10.1016/j.physleta.2015.02.017 |
| [29] | Zhang, T. and Li, J. (2017) Exact Torus Knot Periodic Orbits and Homoclinic Orbits in a Class of Three-Dimensional Flows Generated by a Planar Cubic System. International Journal of Bifurcation and Chaos, 27, Article ID: 1750205. https://doi.org/10.1142/s0218127417502054 |
