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September  2016, 36(9): 5119-5129. doi: 10.3934/dcds.2016022

A new class of 3-dimensional piecewise affine systems with homoclinic orbits

Tiantian Wu 1, and  Xiao-Song Yang 1,

1. 

School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan, 430074, China, China

Received  July 2015 Revised  December 2015 Published  May 2016

Based on mathematical analysis, this paper proves the existence of homoclinic orbits in a new class of 3-dimensional piecewise affine systems, and gives an example to illustrate the effectiveness of the method.
Keywords: 3-dimensional piecewise systems., Homoclinic orbit, chaotic systems.
Mathematics Subject Classification: Primary: 37D45; Secondary: 37G2.
Citation: Tiantian Wu, Xiao-Song Yang. A new class of 3-dimensional piecewise affine systems with homoclinic orbits. Discrete & Continuous Dynamical Systems, 2016, 36 (9) : 5119-5129. doi: 10.3934/dcds.2016022
References:
[1]

G. F. V. Amaral, C. Letellier and L. A. Aguirre, Piecewise affine models of chaotic attractors: The Rossler and Lorenz systems, Chaos, 16 (2006), 013115, 14pp. doi: 10.1063/1.2149527.  Google Scholar

[2]

M. L. Barakat, A. S. Mansingka, A. G. Radwan and K. N. Salama, Hardware stream cipher with controllable chaos generator for colour image encryption, IET Image Process., 8 (2014), 33-43. doi: 10.1049/iet-ipr.2012.0586.  Google Scholar

[3]

S. Chakraborty and S. K. Dana, Shil'nikov chaos and mixed-mode oscillation in Chua's circuit, Chaos, 20 (2010), 23107, 7pp. Google Scholar

[4]

T. Chien and T. Liao, Design of secure digital communication systems using chaotic modulation, cryptography and chaotic synchronization, Chaos Solitons Fract., 24 (2005), 241-255. doi: 10.1016/S0960-0779(04)00542-9.  Google Scholar

[5]

V. Carmona, F. Fernández-Sánchez and A. E. Teruel, Existence of a reversible T-point heteroclinic cycle in a piecewise linear version of the Michelson system, SIAM J. Appl. Dyn. Syst., 7 (2008), 1032-1048. doi: 10.1137/070709542.  Google Scholar

[6]

V. Carmona, F. Fernández-Sánchez, E. García-Medina and A. E. Teruel, Existence of homoclinic connections in continuous piecewise linear systems, Chaos, 20 (2010), 013124, 8pp. doi: 10.1063/1.3339819.  Google Scholar

[7]

M. di Bernardo and C. K. Tse, Chaos in Power Electronics: An Overview, Chaos in Circuits and Systems, World Scientific, 2002. Google Scholar

[8]

S. M. Huan, Q. D. Li and X.-S. Yang, Chaos in three-dimensional hybrid systems and design of chaos generators, Nonlinear Dyn., 69 (2012), 1915-1927. doi: 10.1007/s11071-012-0396-0.  Google Scholar

[9]

S. M. Huan and X.-S. Yang, Existence of chaotic invariant set in a class of 4-dimensional piecewise linear dynamical systems, Int. J. Bifurc. Chaos, 24 (2014), 1450158, 16pp. doi: 10.1142/S0218127414501582.  Google Scholar

[10]

T. Kousaka, T. Ueta and H. Kawakami, Chaos in a simple hybrid system and its control, Electron Lett., 37 (2001), p1. doi: 10.1049/el:20010033.  Google Scholar

[11]

J. Lü, T. Zhou, G. Chen and X.-S. Yang, Generating chaos with a switching piecewise-linear controller, Chaos, 12 (2002), 344-349. Google Scholar

[12]

R. O. Medrano-T., M. S. Baptista and I. L. Caldas, Homoclinic orbits in a piecewise system and their relation with invariant sets, Physica D, 186 (2003), 133-147. doi: 10.1016/j.physd.2003.08.002.  Google Scholar

[13]

V. Nair and R. I. Sujith, Identifying homoclinic orbits in the dynamics of intermittent signals through recurrence quantification, Chaos, 23 (2013), 033136, 6pp. doi: 10.1063/1.4821475.  Google Scholar

[14]

I. Pehlivan and Y. Uyaroglu, Simplified chaotic diffusionless Lorenz attractor and its application to secure communication systems, IET Commun., 1 (2007), 1015-1022. Google Scholar

[15]

L. P. Shil'nikov, A case of the existence of a countable number of periodic motions, Sov. Math.Dokl., 6 (1965), 163-166. Google Scholar

[16]

L. P. Shil'nikov, A contribution of the problem of the structure of an extended neighborhood of rough equilibrium state of saddle-focus type, Math. USSR Sb., 10 (1970), 91-102. Google Scholar

[17]

L. P. Shil'nikov, A. Shil'nikov, D. Turaev and L. Chua, Methods of Qualitative theory in Nonlinear Dynamics, Part I, World Scientific, Singapore, 1998. doi: 10.1142/9789812798596.  Google Scholar

[18]

L. P. Shil'nikov, A. Shil'nikov, D. Turaev and L. Chua, Methods of Qualitative Theory in Nonlinear Dynamics, Part II, World Scientific, Singapore, 2001. doi: 10.1142/9789812798558_0001.  Google Scholar

[19]

C. Tresser, About some theorems by L. P. Shil'nikov, Inst. H. Poincare Phys. Thoré., 40 (1984), 441-461.  Google Scholar

[20]

K. Watada, T. Endo and H. Seishi, Shilnikov orbits in an autonomous third-order chaotic phase-locked loop, IEEE Trans. Circuits Syst., 45 (1998), 979-983. Google Scholar

[21]

S. Wiggins, Introduction to Applied Nonlinear Dynamical Systems and Chaos, $2^{nd}$ edition, Springer-Verlag, New York, 2003.  Google Scholar

[22]

D. Wilczak, The existence of Shilnikov homoclinic orbits in the Michelson system: A computer assisted proof, Found Comput. Math., 6 (2006), 495-535. doi: 10.1007/s10208-005-0201-2.  Google Scholar

[23]

X.-S. Yang and Q. D. Li, Chaos generator via Wien-bridge oscillator, Electron. Lett., 38 (2002), 623-625. doi: 10.1049/el:20020456.  Google Scholar

[24]

X.-S. Yang and Q. D. Li, Generate n-scroll attractor in linear system by scalar output feedback, Chaos Solitons Fract., 18 (2003), 25-29. doi: 10.1016/S0960-0779(02)00638-0.  Google Scholar

show all references

References:
[1]

G. F. V. Amaral, C. Letellier and L. A. Aguirre, Piecewise affine models of chaotic attractors: The Rossler and Lorenz systems, Chaos, 16 (2006), 013115, 14pp. doi: 10.1063/1.2149527.  Google Scholar

[2]

M. L. Barakat, A. S. Mansingka, A. G. Radwan and K. N. Salama, Hardware stream cipher with controllable chaos generator for colour image encryption, IET Image Process., 8 (2014), 33-43. doi: 10.1049/iet-ipr.2012.0586.  Google Scholar

[3]

S. Chakraborty and S. K. Dana, Shil'nikov chaos and mixed-mode oscillation in Chua's circuit, Chaos, 20 (2010), 23107, 7pp. Google Scholar

[4]

T. Chien and T. Liao, Design of secure digital communication systems using chaotic modulation, cryptography and chaotic synchronization, Chaos Solitons Fract., 24 (2005), 241-255. doi: 10.1016/S0960-0779(04)00542-9.  Google Scholar

[5]

V. Carmona, F. Fernández-Sánchez and A. E. Teruel, Existence of a reversible T-point heteroclinic cycle in a piecewise linear version of the Michelson system, SIAM J. Appl. Dyn. Syst., 7 (2008), 1032-1048. doi: 10.1137/070709542.  Google Scholar

[6]

V. Carmona, F. Fernández-Sánchez, E. García-Medina and A. E. Teruel, Existence of homoclinic connections in continuous piecewise linear systems, Chaos, 20 (2010), 013124, 8pp. doi: 10.1063/1.3339819.  Google Scholar

[7]

M. di Bernardo and C. K. Tse, Chaos in Power Electronics: An Overview, Chaos in Circuits and Systems, World Scientific, 2002. Google Scholar

[8]

S. M. Huan, Q. D. Li and X.-S. Yang, Chaos in three-dimensional hybrid systems and design of chaos generators, Nonlinear Dyn., 69 (2012), 1915-1927. doi: 10.1007/s11071-012-0396-0.  Google Scholar

[9]

S. M. Huan and X.-S. Yang, Existence of chaotic invariant set in a class of 4-dimensional piecewise linear dynamical systems, Int. J. Bifurc. Chaos, 24 (2014), 1450158, 16pp. doi: 10.1142/S0218127414501582.  Google Scholar

[10]

T. Kousaka, T. Ueta and H. Kawakami, Chaos in a simple hybrid system and its control, Electron Lett., 37 (2001), p1. doi: 10.1049/el:20010033.  Google Scholar

[11]

J. Lü, T. Zhou, G. Chen and X.-S. Yang, Generating chaos with a switching piecewise-linear controller, Chaos, 12 (2002), 344-349. Google Scholar

[12]

R. O. Medrano-T., M. S. Baptista and I. L. Caldas, Homoclinic orbits in a piecewise system and their relation with invariant sets, Physica D, 186 (2003), 133-147. doi: 10.1016/j.physd.2003.08.002.  Google Scholar

[13]

V. Nair and R. I. Sujith, Identifying homoclinic orbits in the dynamics of intermittent signals through recurrence quantification, Chaos, 23 (2013), 033136, 6pp. doi: 10.1063/1.4821475.  Google Scholar

[14]

I. Pehlivan and Y. Uyaroglu, Simplified chaotic diffusionless Lorenz attractor and its application to secure communication systems, IET Commun., 1 (2007), 1015-1022. Google Scholar

[15]

L. P. Shil'nikov, A case of the existence of a countable number of periodic motions, Sov. Math.Dokl., 6 (1965), 163-166. Google Scholar

[16]

L. P. Shil'nikov, A contribution of the problem of the structure of an extended neighborhood of rough equilibrium state of saddle-focus type, Math. USSR Sb., 10 (1970), 91-102. Google Scholar

[17]

L. P. Shil'nikov, A. Shil'nikov, D. Turaev and L. Chua, Methods of Qualitative theory in Nonlinear Dynamics, Part I, World Scientific, Singapore, 1998. doi: 10.1142/9789812798596.  Google Scholar

[18]

L. P. Shil'nikov, A. Shil'nikov, D. Turaev and L. Chua, Methods of Qualitative Theory in Nonlinear Dynamics, Part II, World Scientific, Singapore, 2001. doi: 10.1142/9789812798558_0001.  Google Scholar

[19]

C. Tresser, About some theorems by L. P. Shil'nikov, Inst. H. Poincare Phys. Thoré., 40 (1984), 441-461.  Google Scholar

[20]

K. Watada, T. Endo and H. Seishi, Shilnikov orbits in an autonomous third-order chaotic phase-locked loop, IEEE Trans. Circuits Syst., 45 (1998), 979-983. Google Scholar

[21]

S. Wiggins, Introduction to Applied Nonlinear Dynamical Systems and Chaos, $2^{nd}$ edition, Springer-Verlag, New York, 2003.  Google Scholar

[22]

D. Wilczak, The existence of Shilnikov homoclinic orbits in the Michelson system: A computer assisted proof, Found Comput. Math., 6 (2006), 495-535. doi: 10.1007/s10208-005-0201-2.  Google Scholar

[23]

X.-S. Yang and Q. D. Li, Chaos generator via Wien-bridge oscillator, Electron. Lett., 38 (2002), 623-625. doi: 10.1049/el:20020456.  Google Scholar

[24]

X.-S. Yang and Q. D. Li, Generate n-scroll attractor in linear system by scalar output feedback, Chaos Solitons Fract., 18 (2003), 25-29. doi: 10.1016/S0960-0779(02)00638-0.  Google Scholar

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