Chaos in Discontinuous Differential Equations

  • Michal Fečkan
Part of the Nonlinear Physical Science book series (NPS)


This chapter is devoted to proving chaos for periodically perturbed piecewise smooth ODEs. We study two cases: firstly, when the homoclinic orbit of the unperturbed piecewise smooth ODE transversally crosses the discontinuity surface, and secondly, when a part of homoclinic orbit is sliding on the discontinuity surface.


Unstable Manifold Homoclinic Orbit Simple Zero Exponential Dichotomy Homoclinic Solution 
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  1. 1.
    B. BROGLIATO: Nonsmooth Impact Mechanics: Models, Dynamics, and Control, Lecture Notes in Control and Information Sciences 220, Springer-Verlag, Berlin, 1996.zbMATHGoogle Scholar
  2. 2.
    L.O. CHUA, M. KOMURO & T. MATSUMOTO: The double scroll family, IEEE Trans. CAS 33 (1986), 1072–1118.MathSciNetADSzbMATHCrossRefGoogle Scholar
  3. 3.
    B.F. FEENY & F.C. MOON: Empirical dry-friction modeling in a forced oscillator using chaos, Nonlinear Dynamics 47 (2007), 129–141.MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    U. GALVANETTO & C. KNUDSEN: Event maps in a stick-slip system, Nonlinear Dynamics 13 (1997), 99–115.MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    M. KUNZE & T. KÜPPER: Qualitative bifurcation analysis of a non-smooth friction-oscillator model, Z. Angew. Meth. Phys. (ZAMP) 48 (1997), 87–101.zbMATHCrossRefGoogle Scholar
  6. 6.
    YU. A. KUZNETSOV, S. RINALDI & A. GRAGNANI: One-parametric bifurcations in planar Filippov systems, Int. J. Bif. Chaos 13 (2003), 2157–2188.MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    R.I. LEINE & H. NIJMEIJER: Dynamics and Bifurcations of Non-smooth Mechanical Systems, Lecture Notes in Applied and Computational Mechanics 18, Springer, Berlin, 2004.zbMATHCrossRefGoogle Scholar
  8. 8.
    R.I. LEINE, D.H. VAN CAMPEN & B. L. VAN DE VRANDE: Bifurcations in nonlinear discontinuous systems, Nonl. Dynamics 23 (2000), 105–164.zbMATHCrossRefGoogle Scholar
  9. 9.
    M. KUNZE: Non-Smooth Dynamical Systems, LNM 1744, Springer, Berlin, 2000.zbMATHCrossRefGoogle Scholar
  10. 10.
    M. KUNZE & T. KÜPPER: Non-smooth dynamical systems: an overview, in: “Ergodic Theory, Analysis and Efficient Simulation of Dynamical Systems”, B. Fiedler ed., Springer, Berlin, 2001, 431–452.CrossRefGoogle Scholar
  11. 11.
    Y. LI & Z.C. FENG: Bifurcation and chaos in friction-induced vibration, Communications in Nonlinear Science and Numerical Simulation 9 (2004), 633–647.ADSzbMATHCrossRefGoogle Scholar
  12. 12.
    J. LLIBRE, E. PONCE & A.E. TERUEL: Horseshoes near homoclinic orbits for piecewise linear differential systems in ℝ3, Int. J. Bif. Chaos 17 (2007), 1171–1184.MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    J. AWREJCEWICZ & M.M. HOLICKE: Smooth and Nonsmooth High Dimensional Chaos and the Melnikov-Type Methods, World Scientific Publishing Co., Singapore, 2007.zbMATHGoogle Scholar
  14. 14.
    J. AWREJCEWICZ & C.H. LAMARQUE: Bifurcation and Chaos in Nonsmooth Mechanical Systems, World Scientific Publishing Co., Singapore, 2003.zbMATHGoogle Scholar
  15. 15.
    M. FEČKAN: Topological Degree Approach to Bifurcation Problems, Springer, Berlin, 2008.zbMATHCrossRefGoogle Scholar
  16. 16.
    Q. CAO, M. WIERCIGROCH, E.E. PAVLOVSKAIA, J.M.T. THOMPSON & C. GREBOGI: Piecewise linear approach to an archetypal oscillator for smooth and discontinuous dynamics, Phil. Trans. R. Soc. A 366 (2008), 635–652.MathSciNetADSzbMATHCrossRefGoogle Scholar
  17. 17.
    Z. DU & W. ZHANG: Melnikov method for homoclinic bifurcation in nonlinear impact oscillators, Computers Mathematics Applications 50 (2005), 445–458.MathSciNetzbMATHCrossRefGoogle Scholar
  18. 18.
    A. KOVALEVA: The Melnikov criterion of instability for random rocking dynamics of a rigid block with an attached secondary structure, Nonlin. Anal., Real World Appl. 11 (2010), 472–479.MathSciNetzbMATHCrossRefGoogle Scholar
  19. 19.
    P. KUKUČKA: Melnikov method for discontinuous planar systems, Nonl. Anal., Th. Meth. Appl. 66 (2007), 2698–2719.zbMATHCrossRefGoogle Scholar
  20. 20.
    S LENCI & G. REGA: Heteroclinic bifurcations and optimal control in the nonlinear rocking dynamics of generic and slender rigid blocks, Int. J. Bif. Chaos 6 (2005), 1901–1918.MathSciNetGoogle Scholar
  21. 21.
    W. XU, J. FENG & H. RONG: Melnikov’s method for a general nonlinear vibro-impact oscillator, Nonlinear Analysis 71 (2009), 418–426.MathSciNetzbMATHCrossRefGoogle Scholar
  22. 22.
    P. COLLINS: Chaotic dynamics in hybrid systems, Nonlinear Dynamics Systems Theory 8 (2008), 169–194.zbMATHGoogle Scholar
  23. 23.
    A.L. FRADKOV, R.J. EVANS & B.R. ANDRIEVSKY: Control of chaos: methods and applications in mechanics, Phil. Trans. R. Soc. A 364 (2006), 2279–2307.MathSciNetADSzbMATHCrossRefGoogle Scholar
  24. 24.
    A.C.J. LUO: A theory for flow switchability in discontinuous dynamical systems, Nonl. Anal., Hyb. Sys. 2 (2008), 1030–1061.zbMATHCrossRefGoogle Scholar
  25. 25.
    A.C.J. LUO: Discontinuous Dynamical Systems on Time-varying Domains, Springer, 2008.Google Scholar
  26. 26.
    A.C.J. LUO: Singularity and Dynamics on Discontinuous Vector Fields, Elsevier Science, 2006.Google Scholar
  27. 27.
    J. AWREJCEWICZ, M. FEČKAN & P. OLEJNIK: On continuous approximation of discontinuous systems, Nonl. Anal., Th. Meth. Appl. 62 (2005), 1317–1331.zbMATHCrossRefGoogle Scholar
  28. 28.
    J. AWREJCEWICZ, M. FEČKAN & P. OLEJNIK: Bifurcations of planar sliding homoclinics, Mathematical Problems Engineering 2006 (2006), 1–13.Google Scholar
  29. 29.
    M.U. AKHMET: Perturbations and Hopf bifurcation of the planar discontinuous dynamical system, Nonlin. Anal., Th. Meth. Appl. 60 (2005), 163–178.MathSciNetzbMATHGoogle Scholar
  30. 30.
    M.U. AKHMET: Almost periodic solutions of differential equations with piecewise constant argument of generalized type, Nonlinear Anal., Hybird. Syst. 2 (2008), 456–467.MathSciNetzbMATHCrossRefGoogle Scholar
  31. 31.
    M.U. AKHMET & C. BÜYÜKADALI: On periodic solutions of differential equations with piecewise constant argument, Comp. Math. Appl. 56 (2008), 2034–2042.zbMATHCrossRefGoogle Scholar
  32. 32.
    M.U. AKHMET, C. BÜYÜKADALI & T. ERGENÇ: Periodic solutions of the hybrid system with small parameter, Nonl. Anal., Hyb. Sys. 2 (2008), 532–543.zbMATHCrossRefGoogle Scholar
  33. 33.
    M. FEČKAN & M. POSPÍŠIL: On the bifurcation of periodic orbits in discontinuous systems, Communications Mathematical Analysis 8 (2010), 87–108.zbMATHGoogle Scholar
  34. 34.
    F. BATTELLI & M. FEČKAN: Homoclinic trajectories in discontinuous systems, J. Dynamics Differential Equations 20 (2008), 337–376.ADSzbMATHCrossRefGoogle Scholar
  35. 35.
    F. BATTELLI & C. LAZZARI: Exponential dichotomies, heteroclinic orbits, and Melnikov functions J. Differential Equations 86 (1990), 342–366.MathSciNetzbMATHCrossRefGoogle Scholar
  36. 36.
    K.J. PALMER: Exponential dichotomies and transversal homoclinic points, J. Differential Equations 55 (1984), 225–256.MathSciNetzbMATHCrossRefGoogle Scholar
  37. 37.
    F. BATTELLI & M. FEČKAN: Subharmonic solutions in singular systems, J. Differential Equations 132 (1996), 21–45.MathSciNetzbMATHCrossRefGoogle Scholar
  38. 38.
    X.-B. LIN: Using Melnikov’s method to solve Silnikov’s problems, Proc. Roy. Soc. Edinburgh 116A (1990), 295–325.CrossRefGoogle Scholar
  39. 39.
    K. DEIMLING: Nonlinear Functional Analysis, Springer-Verlag, Berlin, 1985.zbMATHCrossRefGoogle Scholar
  40. 40.
    B.M. LEVITAN & V.V. ZHIKOV: Almost Periodic Functions and Differential Equations, Cambridge University Press, New York, 1983.Google Scholar
  41. 41.
    J. K. HALE: Ordinary Differential Equations, 2nd ed., Robert E. Krieger Pub. Co., New York, 1980.zbMATHGoogle Scholar
  42. 42.
    K.R. MEYER & G. R. SELL: Melnikov transforms, Bernoulli bundles, and almost periodic perturbations, Trans. Amer. Math. Soc. 314 (1989), 63–105.MathSciNetzbMATHGoogle Scholar
  43. 43.
    K.J. PALMER & D. STOFFER: Chaos in almost periodic systems, Zeit. Ang. Math. Phys. (ZAMP) 40 (1989), 592–602.MathSciNetzbMATHCrossRefGoogle Scholar
  44. 44.
    D. STOFFER: Transversal homoclinic points and hyperbolic sets for non-autonomous maps I, II, Zeit. ang. Math. Phys. (ZAMP) 39 (1988), 518–549, 783–812.MathSciNetzbMATHCrossRefGoogle Scholar
  45. 45.
    S. WIGGINS: Chaos in the dynamics generated by sequences of maps, with applications to chaotic advection in flows with aperiodic time dependence, Z. Angew. Math. Phys. (ZAMP) 50 (1999), 585–616.MathSciNetzbMATHCrossRefGoogle Scholar
  46. 46.
    S. WIGGINS: Chaotic Transport in Dynamical Systems, Springer-Verlag, New York, 1992.zbMATHCrossRefGoogle Scholar
  47. 47.
    J. GUCKENHEIMER & P. HOLMES: Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Springer, New York, 1983.zbMATHGoogle Scholar
  48. 48.
    F. BATTELLI & M. FEČKAN: Nonsmooth homoclinic orbits, Melnikov functions and chaos in discontinuous systems, submitted.Google Scholar
  49. 49.
    M. DI BERNARDO, C.J. BUDD, A.R. CHAMPNEYS & P. KOWALCZYK: Piecewise-smooth Dynamical Systems: Theory and Applications, Appl. Math. Scien. 163, Springer, Berlin, 2008.Google Scholar
  50. 50.
    A. FIDLIN: Nonlinear Oscillations in Mechanical Engineering, Springer, Berlin, 2006.Google Scholar
  51. 51.
    F. GIANNAKOPOULOS & K. PLIETE: Planar systems of piecewise linear differential equations with a line of discontinuity, Nonlinearity 14 (2001), 1611–1632.MathSciNetADSzbMATHCrossRefGoogle Scholar
  52. 52.
    K. POPP: Some model problems showing stick-slip motion and chaos, in: “ASME WAM, Proc. Symp. Friction-Induced Vibration, Chatter, Squeal and Chaos”, R.A. Ibrahim and A. Soom, Eds., 49, ASME New York, 1992, 1–12.Google Scholar
  53. 53.
    K. POPP, N. HINRICHS & M. OESTREICH: Dynamical behaviour of a friction oscillator with simultaneous self and external excitation in: “Sadhana”: Academy Proceedings in Engineering Sciences 20, Part 2–4, Indian Academy of Sciences, Bangalore, India, 1995, 627–654.Google Scholar
  54. 54.
    K. POPP & P. STELTER: Stick-slip vibrations and chaos, Philos. Trans. R. Soc. London A 332 (1990), 89–105.ADSzbMATHCrossRefGoogle Scholar
  55. 55.
    F. BATTELLI & M. FEČKAN: Bifurcation and chaos near sliding homoclinics, J. Differential Equations 248 (2010), 2227–2262.MathSciNetzbMATHCrossRefGoogle Scholar
  56. 56.
    F. DUMORTIER, R. ROUSSARIE, J. SOTOMAYOR & H. ZOLADEK: Bifurcations of Planar Vector Fields, Nilpotent Singularities and Abelian Integrals, LNM 1480, Springer-Verlag, Berlin, 1991.zbMATHGoogle Scholar
  57. 57.
    F. DERCOLE, A. GRAGNANI, YU. A. KUZNETSOV & S. RINALDI: Numerical sliding bifurcation analysis: an application to a relay control system, IEEE Tran. Cir. Sys.-I: Fund. Th. Appl. 50 (2003), 1058–1063.MathSciNetCrossRefGoogle Scholar
  58. 58.
    A.B. NORDMARK & P. KOWALCZYK: A codimension-two scenario of sliding solutions in grazing-sliding bifurcations, Nonlinearity 19 (2006), 1–26.MathSciNetADSzbMATHCrossRefGoogle Scholar

Copyright information

© Higher Education Press, Beijing and Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Michal Fečkan
    • 1
  1. 1.Department of Mathematical Analysis and Numerical Mathematics Faculty of Mathematics, Physics and InformaticsComenius University Mlynská dolinaBratislavaSlovakia

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