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

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

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.
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.

[1]

W.-J. Beyn, Y.-K Zou. Discretizations of dynamical systems with a saddle-node homoclinic orbit. Discrete and Continuous Dynamical Systems, 1996, 2 (3) : 351-365. doi: 10.3934/dcds.1996.2.351

[2]

Alexei Pokrovskii, Oleg Rasskazov, Daniela Visetti. Homoclinic trajectories and chaotic behaviour in a piecewise linear oscillator. Discrete and Continuous Dynamical Systems - B, 2007, 8 (4) : 943-970. doi: 10.3934/dcdsb.2007.8.943

[3]

Nina Lebedeva, Vladimir Matveev, Anton Petrunin, Vsevolod Shevchishin. Smoothing 3-dimensional polyhedral spaces. Electronic Research Announcements, 2015, 22: 12-19. doi: 10.3934/era.2015.22.12

[4]

Andy Hammerlindl. Partial hyperbolicity on 3-dimensional nilmanifolds. Discrete and Continuous Dynamical Systems, 2013, 33 (8) : 3641-3669. doi: 10.3934/dcds.2013.33.3641

[5]

Lingling Liu, Bo Gao, Dongmei Xiao, Weinian Zhang. Identification of focus and center in a 3-dimensional system. Discrete and Continuous Dynamical Systems - B, 2014, 19 (2) : 485-522. doi: 10.3934/dcdsb.2014.19.485

[6]

Rafel Prohens, Antonio E. Teruel. Canard trajectories in 3D piecewise linear systems. Discrete and Continuous Dynamical Systems, 2013, 33 (10) : 4595-4611. doi: 10.3934/dcds.2013.33.4595

[7]

Chen-Chang Peng, Kuan-Ju Chen. Existence of transversal homoclinic orbits in higher dimensional discrete dynamical systems. Discrete and Continuous Dynamical Systems - B, 2010, 14 (3) : 1181-1197. doi: 10.3934/dcdsb.2010.14.1181

[8]

Jaume Llibre, Claudio A. Buzzi, Paulo R. da Silva. 3-dimensional Hopf bifurcation via averaging theory. Discrete and Continuous Dynamical Systems, 2007, 17 (3) : 529-540. doi: 10.3934/dcds.2007.17.529

[9]

Shinobu Hashimoto, Shin Kiriki, Teruhiko Soma. Moduli of 3-dimensional diffeomorphisms with saddle-foci. Discrete and Continuous Dynamical Systems, 2018, 38 (10) : 5021-5037. doi: 10.3934/dcds.2018220

[10]

Philippe Jouan, Ronald Manríquez. Solvable approximations of 3-dimensional almost-Riemannian structures. Mathematical Control and Related Fields, 2022, 12 (2) : 303-326. doi: 10.3934/mcrf.2021023

[11]

Lijun Wei, Xiang Zhang. Limit cycle bifurcations near generalized homoclinic loop in piecewise smooth differential systems. Discrete and Continuous Dynamical Systems, 2016, 36 (5) : 2803-2825. doi: 10.3934/dcds.2016.36.2803

[12]

Victoriano Carmona, Emilio Freire, Soledad Fernández-García. Periodic orbits and invariant cones in three-dimensional piecewise linear systems. Discrete and Continuous Dynamical Systems, 2015, 35 (1) : 59-72. doi: 10.3934/dcds.2015.35.59

[13]

P. Adda, J. L. Dimi, A. Iggidir, J. C. Kamgang, G. Sallet, J. J. Tewa. General models of host-parasite systems. Global analysis. Discrete and Continuous Dynamical Systems - B, 2007, 8 (1) : 1-17. doi: 10.3934/dcdsb.2007.8.1

[14]

Wade Hindes. Orbit counting in polarized dynamical systems. Discrete and Continuous Dynamical Systems, 2022, 42 (1) : 189-210. doi: 10.3934/dcds.2021112

[15]

Tiago de Carvalho, Rodrigo Donizete Euzébio, Jaume Llibre, Durval José Tonon. Detecting periodic orbits in some 3D chaotic quadratic polynomial differential systems. Discrete and Continuous Dynamical Systems - B, 2016, 21 (1) : 1-11. doi: 10.3934/dcdsb.2016.21.1

[16]

Sebastian Springer, Heikki Haario, Vladimir Shemyakin, Leonid Kalachev, Denis Shchepakin. Robust parameter estimation of chaotic systems. Inverse Problems and Imaging, 2019, 13 (6) : 1189-1212. doi: 10.3934/ipi.2019053

[17]

M. L. Bertotti, Sergey V. Bolotin. Chaotic trajectories for natural systems on a torus. Discrete and Continuous Dynamical Systems, 2003, 9 (5) : 1343-1357. doi: 10.3934/dcds.2003.9.1343

[18]

Matteo Petrera, Yuri B. Suris. Geometry of the Kahan discretizations of planar quadratic Hamiltonian systems. Ⅱ. Systems with a linear Poisson tensor. Journal of Computational Dynamics, 2019, 6 (2) : 401-408. doi: 10.3934/jcd.2019020

[19]

Marcelo R. R. Alves. Positive topological entropy for Reeb flows on 3-dimensional Anosov contact manifolds. Journal of Modern Dynamics, 2016, 10: 497-509. doi: 10.3934/jmd.2016.10.497

[20]

Vadim Kaloshin, Maria Saprykina. Generic 3-dimensional volume-preserving diffeomorphisms with superexponential growth of number of periodic orbits. Discrete and Continuous Dynamical Systems, 2006, 15 (2) : 611-640. doi: 10.3934/dcds.2006.15.611

2020 Impact Factor: 1.392

Metrics

  • PDF downloads (105)
  • HTML views (0)
  • Cited by (7)

Other articles
by authors

[Back to Top]