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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 and 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.

[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.

[3]

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

[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.

[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.

[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.

[7]

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

[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.

[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.

[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.

[11]

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

[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.

[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.

[14]

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

[15]

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

[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.

[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.

[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.

[19]

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

[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.

[21]

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

[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.

[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.

[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.

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.

[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.

[3]

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

[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.

[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.

[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.

[7]

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

[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.

[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.

[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.

[11]

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

[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.

[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.

[14]

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

[15]

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

[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.

[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.

[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.

[19]

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

[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.

[21]

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

[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.

[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.

[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.

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