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On the existence of homoclinic orbits in some class of three-dimensional piecewise affine systems


In this paper, we investigate a class of three-dimensional piecewise affine systems with the matrix in each subsystem processing a pair of complex eigenvalues and a real eigenvalue. Furthermore, we obtain some sufficient and necessary conditions for the existence of homoclinic orbits under suitable assumptions. Finally, some concrete examples are presented to illustrate our results.

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  • Bernardo M, Budd C, Champneys AR, Kowalczyk P (2008) Piecewise-smooth dynamical systems: theory and applications, vol 163. Applied mathematical sciences. Springer, London

    MATH  Google Scholar 

  • Brogliato B (2000) Impacts in mechanical systems: analysis and modelling, vol 551. Lecture notes in physics. Springer, New York

    Book  Google Scholar 

  • Deng B, Hines G (2002) Food chain chaos due to Shil’nikovs orbit. Chaos 12(3):533–538

    MathSciNet  Article  Google Scholar 

  • Di Bernardo M, Budd C, Champneys A (2001) Grazing and border-collision in piecewise-smooth systems: a unified analytical framework. Phys Rev Lett 86(12):2553–2556

    Article  Google Scholar 

  • Huan S, Li Q, Yang XS (2012) Chaos in three-dimensional hybrid systems and design of chaos generators. Nonlinear Dyn. 69(4):1915–1927

    MathSciNet  Article  Google Scholar 

  • Kousaka T, Ueta T, Ma Y, Kawakami H (2006) Control of chaos in a piecewise smooth nonlinear system. Chaos, Solitons Fract 27(4):1019–1025

    Article  Google Scholar 

  • Leine R, Nijmeijer H (2013) Dynamics and bifurcations of non-smooth mechanical systems. Springer, Berlin

    MATH  Google Scholar 

  • Llibre J, Ponce E, Teruel AE (2007) Horseshoes near homoclinic orbits for piecewise linear differential systems in \(\mathbf{R^3}\). Int J Bifurc Chaos 17(04):1171–1184

    Article  Google Scholar 

  • Medrano-T RO, Baptista MS, Caldas IL (2003) Homoclinic orbits in a piecewise system and their relation with invariant sets. Physica D 186(3):133–147

    MathSciNet  Article  Google Scholar 

  • Nusse HE, Yorke JA (1995) Border-collision bifurcations for piecewise smooth one-dimensional maps. Int J Bifurc Chaos 5(01):189–207

    MathSciNet  Article  Google Scholar 

  • Shang D, Han M (2005) The existence of homoclinic orbits to saddle-focus. Appl Math Comput 163(2):621–631

    MathSciNet  MATH  Google Scholar 

  • Shil’nikov LP (1965) A case of the existence of a countable number of periodic motions. Sov Math Dokl 6:163–166

    Google Scholar 

  • Shil’nikov LP (1970) A contribution to the problem of the structure of an extended neighborhood of a rough equilibrium state of saddle-focus type. Math USSR-Sbornik 10:91–102

    Article  Google Scholar 

  • Shil’nikov LP, Shil’nikov AL, Turaev DV, Chua LO (2001) Methods of qualitative theory in nonlinear dynamics. World Scientific, Sigapore

    Book  Google Scholar 

  • Tresser C (1984) About some theorems by L.P. Shil’nikov. Inst H Poincaré Phys Thoré (4), 441–461

  • Watada K, Endo T, Seishi H (1998) Shil’nikov orbits in an autonomous third-order chaotic phase-locked loop. IEEE Trans Circuits Syst 45(9):979–983

    Article  Google Scholar 

  • Wiggins S (2003) Introduction to applied nonlinear dynamical systems and chaos, 2nd edn. Springer, New York

    MATH  Google Scholar 

  • Wilczak D (2006) The existence of Shilnikov homoclinic orbits in the Michelson system: a computer assisted proof. Found Comput Math 6(4):495–535

    MathSciNet  Article  Google Scholar 

  • Wu T, Yang XS (2016) A new class of 3-dimensional piecewise affine systems with homoclinic orbits. Discrete Contin Dyn Syst 36(9):5119–5129

    MathSciNet  Article  Google Scholar 

  • Zhusubaliyev ZT, Mosekilde E (2003) Bifurcations and Chaos in piecewise-smooth dynamical systems: applications to power converters, relay and pulse-width modulated control systems, and human decision-making behavior. World Scientific, Singapore

    Book  Google Scholar 

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The authors are grateful to the editors and the anonymous reviewers for their careful reading and valuable suggestions. This work is supported by National Natural Science Foundation of China (11472111).

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Correspondence to Xiaosong Yang.

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Communicated by Maria do Rosário de Pinho.

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Chen, Y., Wu, T. & Yang, X. On the existence of homoclinic orbits in some class of three-dimensional piecewise affine systems. Comp. Appl. Math. 37, 6022–6033 (2018).

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  • Homoclinic orbit
  • Chaos
  • Piecewise affine systems

Mathematics Subject Classification

  • 37D45
  • 37G20