一类新混沌系统的几何分析

数学物理学报 ›› 2019, Vol. 39 ›› Issue (2): 339-347.

一类新混沌系统的几何分析

黄燮桢^1,²,刘永建^2,*(),黄秋健³

¹ 闽南师范大学数学与统计学院福建漳州 363000
² 玉林师范学院; 广西高校复杂系统优化与大数据处理重点实验室广西玉林 537000
³ 广西民族大学理学院南宁 530006

收稿日期:2018-01-03 出版日期:2019-04-26 发布日期:2019-05-05
通讯作者: 刘永建 E-mail:liuyongjianmaths@126.com
基金资助:
国家自然科学基金(11561069);广西自然科学基金(2016GXNSFBA380170);广西自然科学基金(2017GXNSFAA198234);福建省数学类研究生教育创新基地项目(1013-313009);广西高等学校高水平创新团队及卓越学者计划项目([2018]35)

Geometric Analysis of a Class of the New Chaotic System

Xiezhen Huang^1,²,Yongjian Liu^2,*(),Qiujian Huang³

¹ School of Mathematics and Statistics, Minnan Normal University, Fujian Zhangzhou 363000
² Guangxi Colleges and Universities Key Laboratory of Complex System Optimization and Big Data Processing, Yulin Normal University, Guangxi Yulin 537000
³ School of Science, Guangxi University for Nationalities, Nanning 530006

Received:2018-01-03 Online:2019-04-26 Published:2019-05-05
Contact: Yongjian Liu E-mail:liuyongjianmaths@126.com
Supported by:
the NSFC(11561069);the Guangxi Natural Science Foundation of China(2016GXNSFBA380170);the Guangxi Natural Science Foundation of China(2017GXNSFAA198234);the Educational Innovation Base of the Graduate Students of Mathematics in Fujian(1013-313009);Guangxi University High Level Innovation Team and Distinguished Scholars Program of China([2018]35)

摘要/Abstract

摘要：

基于Poincaré紧致化技术，分析一类三维混沌系统的全局动力学行为.研究表明系统在无穷远处的奇点高度退化且不稳定.该文也通过设计一个不改变系统奇点结构的线性控制器，构造了一个受控系统，研究发现受控系统在特定参数组条件下，存在一簇退化奇异异宿轨.结合数值仿真结果，论文指出，当参数b和c发生微小扰动时，受控系统异宿环破裂，产生新的混沌吸引子.希冀这些研究对解释系统混沌几何机理能提供有益帮助.

关键词: Poincaré紧致化, 无穷远奇点, 奇异退化异宿轨, 混沌系统

Abstract:

Based on Poincaré compactification technology, the global dynamics behavior of three dimensional chaotic system is studied. The results show that the equilibria at infinity are unstable and highly degraded. The controlled system with a linear controller which does not change the singularity structure has a bunch of degenerate singular orbits. The chaotic attractors for the system in the case of small parameters b and c are found numerically, and thus the nearby singularly degenerate heteroclinic cycles. It is hoped that the investigation of this paper will be quite beneficial for further studies of the geometrical structure for the chaotic attractor.

Key words: Poincaré compactification, Singular point at infinity, Singularly degenerate heteroclinic cycles, Chaotic system

中图分类号:

O175

黄燮桢,刘永建,黄秋健. 一类新混沌系统的几何分析[J]. 数学物理学报, 2019, 39(2): 339-347.

Xiezhen Huang,Yongjian Liu,Qiujian Huang. Geometric Analysis of a Class of the New Chaotic System[J]. Acta mathematica scientia,Series A, 2019, 39(2): 339-347.

参考文献

1	Lorenz E N . Deterministic nonperiodic flow. Journal of Atmospheric Sciences, 1963, 20: 130- 141
2	Chen G R , Ueta T . Yet another chaotic attractor. International Journal of Bifurcation and Chaos, 1999, 9: 1465- 1466
3	Lü J H , Chen G R . A new chaotic attractor coined. International Journal of Bifurcation and Chaos, 2002, 12: 659- 661
4	Vaně?ek A , ?elikovsky S . Control Systems:From Linear Analysis to Synthesis of Chaos. London:Prentice Hall, 1996, 1479- 1480
5	Sprott J C . Some simple chaotic flows. Physical Review E, 1994, 50: R647- R650
6	R?sler O E . An equation for continuous chaos. Physics Letters A, 1976, 57: 397- 398
7	Chua L O , Matsumoto T , Komuro M . The double scroll family. IEEE Transactions on Circuits and Systems, 1986, 33: 1072- 1118
8	Qi G Y , Chen G R , Du S Z , et al. Analysis of a new chaotic system. Physica A Statistical Mechanics and Its Applications, 2005, 352: 295- 308
9	Sundarapandian V , Pehlivan I . Analysis, control, synchronization, and circuit design of a novel chaotic system. Mathematical and Computer Modelling, 2012, 55: 1904- 1915
10	Pehlivan I , Uyaroglu Y . A new 3D chaotic system with golden proportion equilibria:Analysis and electronic circuit realization. Computers and Electrical Engineering, 2012, 38: 1777- 1784
11	?elikovsky S , Chen G R . On a generalized Lorenz canonical form of chaotic systems. International Journal of Bifurcation and Chaos, 2002, 12: 1789- 1812
12	Yang Q G , Chen G R . A chaotic system with one saddle and two stable node-foci. International Journal of Bifurcation and Chaos, 2008, 18: 1393- 1414
13	Yang Q G , Wei Z C , Chen G R . An unusual 3D autonomous quadratic chaotic system with two stable node-foci. International Journal of Bifurcation and Chaos, 2010, 20: 1061- 1083
14	Wei Z C , Yang Q G . Dynamical analysis of a new autonomous 3-D chaotic system only with stable equilibria. Nonlinear Analysis:Real World Applications, 2011, 12: 106- 118
15	Liu Y J , Pang S Q , Chen D Y . An unusual chaotic system and its control. Mathematical and Computer Modelling, 2013, 57: 2473- 2493
16	Gambino G , Sciacca V . Intermittent and passivity based control strategies for a hyperchaotic system. Applied Mathematics and Computation, 2013, 221: 367- 382
17	Geng F J , Li X Y . Singular orbits and dynamics at infinity of a conjugate Lorenz-like system. Mathematical Modelling and Analysis, 2015, 20: 148- 167
18	Wang H J , Li X Y . New results to a three-dimensional chaotic system with two different kinds of nonisolated equilibria. Journal of Computational and Nonlinear Dynamics, 2015, 10: 161- 169
19	Ott E , Grebogi C , Yorke J A . Controlling chaos. Physical Review Letters, 1990, 64: 1196- 1199
20	Goedgebuer J P , Larger L , Port H . Optical cryptosystem based on synchronization of hyper-chaos generated by a delayed feedback laser diode. Physical Review Letters, 1998, 80: 2249- 2254
21	Chen G R , Dong X N . On feedback control of chaotic continuous-time systems. IEEE Transactions on Circuits and Systems I Fundamental Theory and Applications, 1993, 40: 591- 601
22	Suykens J A K , Curran P F , Vanderwalle J , et al. Nonlinear H.∞ synchronization of Lur'e systems:dynamic output feedback case. IEEE Transactions on Circuits and Systems I:Fundamental Theory and Applications, 1997, 44: 1089- 1092
23	Zhu C X . Feedback control methods for stabilizing unstable equilibrium points in a new chaotic system. Nonlinear Analysis, 2009, 71: 2441- 2446
24	Zhu C X . Adaptive synchronization of two novel different hyperchaotic systems with partly uncertain parameters. Applied Mathematics and Computation, 2009, 215: 557- 561
25	陈卫田, 颜世田, 张正强, 等. 不确定非线性系统的鲁棒自适应控制器. 数学物理学报, 2002, 22: 194- 202
25	Chen W T , Yan S T , Zhang Z Q , et al. Robust adaptive controller for uncertain nonlinear system. Acta Mathmatica Scientia, 2002, 22: 194- 202
26	Liu Y J , Yang Q G . Dynamics of the Lü system on the invariant algebraic surface and at infinity. International Journal of Bifurcation and Chaos, 2011, 21: 2559- 2582
27	刘正荣. 微分方程定性方法和数值模拟. 广州: 华南理工大学出版社, 2013
27	Liu Z R . Qualitative Method and Numerical Simulation of Differential Equation. Guangzhou: South China University of Technology Press, 2013