Homoclinic orbits in three-dimensional Shilnikov-type chaotic systems

doi:10.1088/1674-1056/22/9/090503

Abstract In this paper, the Padé approximant and analytic solution in the neighborhood of the initial value are introduced into the process of constructing the Shilnikov type homoclinic trajectories in three-dimensional nonlinear dynamical systems. The PID controller system with quadratic and cubic nonlinearities, the simplified solar-wind-driven-magnetosphere-ionosphere system, and the human DNA sequence system are considered. With the aid of presenting a new condition, the solutions of solving the boundary-value problems which are formulated for the trajectory and evaluating the initial amplitude values become available. At the same time, the value of the bifurcation parameter is obtained directly, which is almost consistent with the numerical result.

Keywords: chaos Shilnikov theorem homoclinic orbit Padé approximation

Received: 15 November 2012
Revised: 20 March 2013
Accepted manuscript online:

PACS:	05.45.Gg	(Control of chaos, applications of chaos)
	02.30.Oz	(Bifurcation theory)

Fund: Project supported by the National Natural Science Foundation of China (Grant Nos. 11072168 and 11102127), the Specialized Research Fund for the Doctoral Program of Higher Education of China (Grant No. 20100032120006), and the Research Program of Application Foundation and Advanced Technology of Tianjin, China (Grant Nos. 12JCYBJC12500 and 11JCYBJC05800).

Corresponding Authors: Wang Wei
E-mail: wangweifrancis@tju.edu.cn

Cite this article:

Feng Jing-Jing (冯晶晶), Zhang Qi-Chang (张琪昌), Wang Wei (王炜), Hao Shu-Ying (郝淑英) Homoclinic orbits in three-dimensional Shilnikov-type chaotic systems 2013 Chin. Phys. B 22 090503

[1]	Lorenz E N 1963 J. Atmos. Sci. 20 130
[2]	Ozoguz S, Elwakil A S and Kennedy M P 2002 Int. J. Bifurc. Chaos 12 1627
[3]	Rössler O E 1996 Phys. Lett. A 57 397
[4]	Chen G and Ueta T 1999 Int. J. Bifurc. Chaos 9 1465
[5]	Shilnikov L P 2000 Homoclinic Orbits: Since Poincaré till Today 571 WIAS
[6]	Uuml J H, Chen G R and Yu Y G 2002 Chin. Phys. Lett. 19 1260
[7]	Zhang R, Hu A H and Xu Z Y 2007 Acta Phys. Sin. 56 6851 (in Chinese)
[8]	Li Zhi and H C Z 2002 Chin. Phys 11 666
[9]	Leonov G A 2001 J. Appl. Math. Mech. 65 19
[10]	Li Z, Chen G R and Halang W A 2004 Inform. Sci. 165 235
[11]	Genesio R, Innocenti G and Gualdani F 2008 Phys. Lett. A 372 1799
[12]	Silva C P 1993 IEEE Trans. Circuits Syst. Regul. Pap. 40 675
[13]	Palmer K J 1996 Nonlinear Anal. Theory Methods Appl. 27 1075
[14]	Zhang Q C, Tian R L and Wang W 2008 Acta. Phys. Sin 57 2799 (in Chinese)
[15]	Zhou T S, Chen G R and Yang Q G 2004 Chaos, Solitons Fractals 19 985
[16]	Mikhlin Y V 2000 J. Sound Vib. 230 971
[17]	Vakakis A F and Azeez M F A 1998 Nonlinear Dyn. 15 245
[18]	Li Y H and Zhu S 2006 Chaos, Solitons Fractals 29 1155
[19]	Nandakumar K and Chatterjee A 2008 Nonlinear Dyn. 57 383
[20]	Chen B Y, Zhou T S and Chen G R 2009 Int. J. Bifurc. Chaos 19 1679
[21]	Emaci E, Vakakis A F, Andrianov I V and Mikhlin Y 1997 Nonlinear Dyn. 13 327
[22]	Manucharyan G V and Mikhlin Y V 2005 J. Appl. Math. Mech. 69 39
[23]	Mikhlin Y V 1995 J. Sound Vib. 182 577
[24]	Feng J J, Zhang Q C and Wang W 2012 Chaos, Solitons Fractals 45 950
[25]	Feng J J, Zhang Q C and Wang W 2011 Chin. Phys. B 20 090202
[26]	Wang W, Zhang Q C and Tian R L 2010 Chin. Phys. B 19 030517
[27]	Arneodo A, Coullet P and Tresser C 1981 Commun. Math. Phys. 79 573
[28]	Arneodo A, Coullet P H and Spiegel E A 1985 Geophys. Astrophys. Fluid Dyn. 31 1
[29]	Nicolay S, Argoul F, Touchon M, d’aubenton-Carafa Y, Thermes C and Arneodo A 2004 Phys. Rev. Lett. 93 108101

[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]	Formalism of rotating-wave approximation in high-spin system with quadrupole interaction Wen-Kui Ding(丁文魁) and Xiao-Guang Wang(王晓光). Chin. Phys. B, 2023, 32(3): 030301.
[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]	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.
[5]	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.
[6]	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.
[7]	Reflection and transmission of an Airy beam in a dielectric slab Xiaojin Yang(杨小锦), Tan Qu(屈檀), Zhensen Wu(吴振森), Haiying Li(李海英), Lu Bai(白璐), Lei Gong(巩蕾), and Zhengjun Li(李正军). Chin. Phys. B, 2022, 31(7): 074202.
[8]	Solutions and memory effect of fractional-order chaotic system: A review Shaobo He(贺少波), Huihai Wang(王会海), and Kehui Sun(孙克辉). Chin. Phys. B, 2022, 31(6): 060501.
[9]	Collective modes of type-II Weyl fermions with repulsive S-wave interaction Xun-Gao Wang(王勋高), Yuan Sun(孙远), Liang Liu(刘亮), and Wu-Ming Liu(刘伍明). Chin. Phys. B, 2022, 31(2): 026701.
[10]	Bifurcation and dynamics in double-delayed Chua circuits with periodic perturbation Wenjie Yang(杨文杰). Chin. Phys. B, 2022, 31(2): 020201.
[11]	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.
[12]	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.
[13]	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.
[14]	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.
[15]	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.

No Suggested Reading articles found!

Viewed

Full text

Abstract

Cited

Metrics
Related Articles

Homoclinic orbits in three-dimensional Shilnikov-type chaotic systems

Cite this article:

Online attention