Heteroclinic cycles in a new class of four-dimensional discontinuous piecewise affine systems

doi:10.1088/1674-1056/27/11/110201

Abstract

It is a huge challenge to give an existence theorem for heteroclinic cycles in the high-dimensional discontinuous piecewise systems (DPSs). This paper first provides a new class of four-dimensional (4D) two-zone discontinuous piecewise affine systems (DPASs), and then gives a useful criterion to ensure the existence of heteroclinic cycles in the systems by rigorous mathematical analysis. To illustrate the feasibility and efficiency of the theory, two numerical examples, exhibiting chaotic behaviors in a small neighborhood of heteroclinic cycles, are discussed.

Keywords: heteroclinic cycle chaos discontinuous piecewise affine system

Received: 20 June 2018
Revised: 05 September 2018
Accepted manuscript online:

PACS:	02.60.-x	(Numerical approximation and analysis)
	05.45.-a	(Nonlinear dynamics and chaos)

Fund:

Project supported by the National Natural Science Foundation of China (Grant Nos. 11472212 and 11532011).

Corresponding Authors: Wei Xu
E-mail: weixu@nwpu.edu.cn

Cite this article:

Wenjing Xu(徐文静), Wei Xu(徐伟), Li Cai(蔡力) Heteroclinic cycles in a new class of four-dimensional discontinuous piecewise affine systems 2018 Chin. Phys. B 27 110201

[1]	Voorsluijs V and Decker Y D 2016 Phys. D 335 1
[2]	Schiff S J, Jerger K, Duong D H, Chang T, Spano M L and Ditto W L1994 Nature 370 615
[3]	Hou Y Y 2017 ISA Trans. 70 260
[4]	Wang T C, He X and Huang T W 2016 Neurocomputing 190 95
[5]	Fotsa R T and Woafo P 2016 Chaos Solit. Fract. 93 48
[6]	Tacha O I, Volos C K, Kyprianidis I M, Stouboulos I N, VaidyanathanS and Pham V T 2016 Appl. Math. Comput. 26 95
[7]	Zaher A A and Abdulnasser A R 2011 Commun. Nonlinear Sci. Numer. Simul. 16 3721
[8]	Luo S H, Wu S L and Gao R Z 2015 Chaos 25 073102
[9]	Sakai K, Upadhyaya S K, Sanchez P A and Sviridova N V 2017 Chaos 27 033115
[10]	Yang J Q, Chen Y T and Zhu F L 2015 Neurocomputing 167 587
[11]	Wang X, Akgul A, Cicek S, Pham V T and Hoang D V 2017 Int. J. Bifurcation and Chaos 27 1750130
[12]	Zhang X D, Liu X D, Zhen Y and Liu C 2013 Chinese Physics B 22 030509
[13]	Shi P M, Han D Y and Liu B 2010 Chinese Physics B 19 090306
[14]	Lorenz E 1963 J. Atmos. Sci. 20 130
[15]	Chua L and Ying R 1983 IEEE Trans. Circuits Syst. 30 125
[16]	Chen G R and Ueta T 1999 Int. J. Bifurcation and Chaos 9 1465
[17]	Lü J H and Chen G R 2002 Int. J. Bifurcation and Chaos 12 659
[18]	Yang Q G, Chen G R and Zhou T S 2006 Int. J. Bifurcation and Chaos 16 2855
[19]	Tigan G and Opriş D 2008 Chaos Solit. Fract. 36 1315
[20]	Wei Z C and Yang Q G 2011 Nonlin. Anal.:Real World Appl. 12 106
[21]	Yang Q G and Chen Y M 2014 Int. J. Bifurcation and Chaos 24 1450055
[22]	Shil'nikov L P, Shil'nikov A, Turaev D and Chua L 1998 Methods of Qualitative Theory in Nonlinear Dynamics (Part I) (Singapore:WorldScientific)
[23]	Shil'nikov L P, Shil'nikov A, Turaev D and Chua L 2001 Methods of Qualitative Theory in Nonlinear Dynamics (Part Ⅱ) (Singapore:WorldScientific)
[24]	Carmona V, Fernández-Sáaacute F and Teruel N E 2008 Siam J. Appl. Dyn. Syst. 7 1032
[25]	Li G and Chen X 2009 Commun. Nonlinear Sci. Numer. Simul. 14 194
[26]	Bao J and Yang Q 2011 Appl. Math. Comput. 217 6526
[27]	Leonov G A 2014 Nonlinear Dyn. 78 2751
[28]	Han C, Yuan F and Wang X 2015 J. Eng. 2 615187
[29]	Wu T T, Wang L and Yang X S 2016 Nonlinear Dyn. 84 817
[30]	Wang L and Yang X S 2017 Nonlinear Anal. Hybrid Syst. 23 44
[31]	Carmona V, Fernández-Sánchez F and García-Medina E 2017 Elsevier Science Inc. 296 33
[32]	Chen Y L, Wang L and Yang X S 2018 Nonlinear Dyn. 91 67
[33]	Yang Q G and Lu K 2018 Nonlinear Dyn. 93 2445
[34]	Wu T T and Yang X S 2016 Chaos 26 053104
[35]	Wu T T and Yang X S 2018 Nonlinear Anal. Hybrid Syst. 27 366
[36]	Tresser C 1984 Inst. H. Poincaré Phys. Thoré 40 441
[37]	Wiggins S and Mazel D S 1990 Computers in Phy. 4
[38]	Wu T T, Li Q D and Yang X S 2016 Int. J. Bifurcation and Chaos 26 1650154

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Heteroclinic cycles in a new class of four-dimensional discontinuous piecewise affine systems

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