Shilnikov sense chaos in a simple three-dimensional system

doi:10.1088/1674-1056/19/3/030517

Abstract The Shilnikov sense Smale horseshoe chaos in a simple 3D nonlinear system is studied. The proportional integral derivative (PID) controller is improved by introducing the quadratic and cubic nonlinearities into the governing equations. For the discussion of chaos, the bifurcate parameter value is selected in a reasonable regime at the requirement of the Shilnikov theorem. The analytic expression of the Shilnikov type homoclinic orbit is accomplished. It depends on the series form of the manifolds surrounding the saddle-focus equilibrium. Then the methodology is extended to research the dynamical behaviours of the simplified solar-wind-driven-magnetosphere-ionosphere system. As is illustrated, the Lyapunov characteristic exponent spectra of the two systems indicate the existence of chaotic attractor under some specific parameter conditions.

Keywords: chaos Shilnikov theorem homoclinic orbit manifold

Received: 02 January 2009
Revised: 21 September 2009
Accepted manuscript online:

PACS:	96.50.Ci	(Solar wind plasma; sources of solar wind)
	94.30.-d	(Physics of the magnetosphere)
	94.20.-y	(Physics of the ionosphere)

Fund: Project supported by the National Natural Science Foundation of China (Grant No.~10872141).

Cite this article:

Wang Wei(王炜), Zhang Qi-Chang(张琪昌), and Tian Rui-Lan(田瑞兰) Shilnikov sense chaos in a simple three-dimensional system 2010 Chin. Phys. B 19 030517

[1]	Shilnikov L P 1970 Math. USSR-Sbornik 10 91
[2]	Zhang Q C, Tian R L and Wang W 2008 Acta Phys. Sin. 57 2799 (in Chinese)
[3]	Zhou T S, Chen G R and Yang Q G 2004 Chaos, Solitons and Fractals 19 985
[4]	Li Z, Chen G R and Halang W A 2004 Information Sciences 165 235
[5]	Genesio R, Innocenti G and Gualdani F 2008 Phys. Lett. A 372 1799
[6]	Xu P C and Jing Z J 2000 Chaos, Solitons and Fractals 11 853
[7]	Mikhlin Y V 2000 J. Sound Vib. 230 971
[8]	Vakakis A F and Azeez M F A 1998 Nonlinear DynAm. 15 245
[9]	Li Y H and Zhu S M 2006 Chaos, Solitons and Fractals 29 1155
[10]	Silva C P 1993 IEEE Trans.Circuits Syst. I Fundam. Theor. Appl. 40 675
[11]	Cui F S, Chew C H, Xu J X and Cai Y L 1997 Nonlinear DynAm. 12 251
[12]	Stenfanski A, Dabrowski A and Kapitaniak T 2005 Chaos, Solitons and Fractals 23 1651
[13]	Chen Z M, Djidjdli K and Price W G 2006 Appl. Math. Comput. 174 982
[14]	Smith J P, Thiffeault J L and Horton W 2000 J. Geophys Res. 105 12
[15]	Horton W, Weigel R S and Sprott J C 2001 Phys. Plasmas 8 2946
[16]	Vassiliadis D, Klimas A J, Baker D N and RobertsD A 1996 J. Geophys. Res. 101 19779

[1]	An incommensurate fractional discrete macroeconomic system: Bifurcation, chaos, and complexity Abderrahmane Abbes, Adel Ouannas, and Nabil Shawagfeh. Chin. Phys. B, 2023, 32(3): 030203.
[2]	A novel algorithm to analyze the dynamics of digital chaotic maps in finite-precision domain Chunlei Fan(范春雷) and Qun Ding(丁群). Chin. Phys. B, 2023, 32(1): 010501.
[3]	Memristor hyperchaos in a generalized Kolmogorov-type system with extreme multistability Xiaodong Jiao(焦晓东), Mingfeng Yuan(袁明峰), Jin Tao(陶金), Hao Sun(孙昊), Qinglin Sun(孙青林), and Zengqiang Chen(陈增强). Chin. Phys. B, 2023, 32(1): 010507.
[4]	Synchronously scrambled diffuse image encryption method based on a new cosine chaotic map Xiaopeng Yan(闫晓鹏), Xingyuan Wang(王兴元), and Yongjin Xian(咸永锦). Chin. Phys. B, 2022, 31(8): 080504.
[5]	Multi-target ranging using an optical reservoir computing approach in the laterally coupled semiconductor lasers with self-feedback Dong-Zhou Zhong(钟东洲), Zhe Xu(徐喆), Ya-Lan Hu(胡亚兰), Ke-Ke Zhao(赵可可), Jin-Bo Zhang(张金波),Peng Hou(侯鹏), Wan-An Deng(邓万安), and Jiang-Tao Xi(习江涛). Chin. Phys. B, 2022, 31(7): 074205.
[6]	Bifurcation and dynamics in double-delayed Chua circuits with periodic perturbation Wenjie Yang(杨文杰). Chin. Phys. B, 2022, 31(2): 020201.
[7]	Complex dynamic behaviors in hyperbolic-type memristor-based cellular neural network Ai-Xue Qi(齐爱学), Bin-Da Zhu(朱斌达), and Guang-Yi Wang(王光义). Chin. Phys. B, 2022, 31(2): 020502.
[8]	Energy spreading, equipartition, and chaos in lattices with non-central forces Arnold Ngapasare, Georgios Theocharis, Olivier Richoux, Vassos Achilleos, and Charalampos Skokos. Chin. Phys. B, 2022, 31(2): 020506.
[9]	Resonance and antiresonance characteristics in linearly delayed Maryland model Hsinchen Yu(于心澄), Dong Bai(柏栋), Peishan He(何佩珊), Xiaoping Zhang(张小平), Zhongzhou Ren(任中洲), and Qiang Zheng(郑强). Chin. Phys. B, 2022, 31(12): 120502.
[10]	An image encryption algorithm based on spatiotemporal chaos and middle order traversal of a binary tree Yining Su(苏怡宁), Xingyuan Wang(王兴元), and Shujuan Lin(林淑娟). Chin. Phys. B, 2022, 31(11): 110503.
[11]	Nonlinear dynamics analysis of cluster-shaped conservative flows generated from a generalized thermostatted system Yue Li(李月), Zengqiang Chen(陈增强), Zenghui Wang(王增会), and Shijian Cang(仓诗建). Chin. Phys. B, 2022, 31(1): 010501.
[12]	Dynamics analysis in a tumor-immune system with chemotherapy Hai-Ying Liu(刘海英), Hong-Li Yang(杨红丽), and Lian-Gui Yang(杨联贵). Chin. Phys. B, 2021, 30(5): 058201.
[13]	Control of chaos in Frenkel-Kontorova model using reinforcement learning You-Ming Lei(雷佑铭) and Yan-Yan Han(韩彦彦). Chin. Phys. B, 2021, 30(5): 050503.
[14]	Resistance fluctuations in superconducting K_xFe_2-ySe₂ single crystals studied by low-frequency noise spectroscopy Hai Zi(子海), Yuan Yao(姚湲), Ming-Chong He(何明冲), Di Ke(可迪), Hong-Xing Zhan(詹红星), Yu-Qing Zhao(赵宇清), Hai-Hu Wen(闻海虎), and Cong Ren(任聪). Chin. Phys. B, 2021, 30(4): 047402.
[15]	A multi-directional controllable multi-scroll conservative chaos generator: Modelling, analysis, and FPGA implementation En-Zeng Dong(董恩增), Rong-Hao Li(李荣昊), and Sheng-Zhi Du(杜升之). Chin. Phys. B, 2021, 30(2): 020505.

No Suggested Reading articles found!

Viewed

Full text

Abstract

Cited

Metrics
Related Articles

Shilnikov sense chaos in a simple three-dimensional system

Cite this article:

Online attention