Nonlinear Dynamics

, Volume 85, Issue 2, pp 1091–1104 | Cite as

The Melnikov method of heteroclinic orbits for a class of planar hybrid piecewise-smooth systems and application

  • Shuangbao Li
  • Chao Shen
  • Wei Zhang
  • Yuxin Hao
Original Paper


In this work, we extend the well-known Melnikov method for smooth systems to a class of planar hybrid piecewise-smooth systems, defined in three zones separated by two switching manifolds \(x=-\alpha \) and \(x=\beta \). We suppose that the dynamic in each zone is governed by a smooth system. When a trajectory reaches the switching manifolds, then reset maps describing impacting rules on the switching manifolds will be applied instantaneously before the trajectory enters into the other zone. We also assume that the unperturbed system is a piecewise-defined continuous Hamiltonian system and possesses a pair of heteroclinic orbits transversally crossing the switching manifolds. Then, we study the persistence of the heteroclinic orbits under a non-autonomous periodic perturbation and the reset maps. In order to obtain this objective, we derive a Melnikov-type function by using the Hamiltonian function to measure the distance of the perturbed stable and unstable manifolds in this system. Finally, we employ the obtained Melnikov-type function to study the persistence of a heteroclinic cycle and complicated dynamics near the heteroclinic cycle for a concrete planar piecewise-smooth system.


Melnikov method Planar hybrid piecewise-smooth systems  Heteroclinic orbits Switching manifolds Reset maps 



The authors gratefully acknowledge the support of the National Natural Science Foundation of China (NNSFC) through Grant Nos. 11472298, 11290152, 11427801, 11272063, 11472056, the Funding Project for Academic Human Resources Development in Institutions of Higher Learning under the Jurisdiction of Beijing Municipality (PHRIHLB), the Natural Science Foundation of Tianjin City through Grant No. 13JCQNJC04400 and the Fundamental Research Funds for the Central Universities through Grant No. 3122013k005.


  1. 1.
    Brogliato, B.: Nonsmooth Mechanics. Springer, London (1999)CrossRefMATHGoogle Scholar
  2. 2.
    Bernardo, M.D., Kowalczyk, P., Nordmark, A.B.: Sliding bifurcations: a novel mechanism for the sudden onset of chaos in dry friction oscillators. Int. J. Bifurc. Chaos Appl. Sci. Eng. 13, 2935–2948 (2003)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Banerjee, S., Verghese, G.: Nonlinear Phenomena in Power Electronics: Attractors, Bifurcations. Chaos and Nonlinear Control. Wiley-IEEE Press, New York (2001)CrossRefGoogle Scholar
  4. 4.
    Garcia, M., Chatterjee, A., Ruina, A., Coleman, M.: The simplest walking model:stability, complexity and scaling ASME. J. Biomech. Eng. 120, 281–288 (1998)CrossRefGoogle Scholar
  5. 5.
    Bernardo, M.D., Garofalo, L., Vasca, F.: Bifurcations in piecewise-smooth feedback systems. Int. J. Control 75, 1243–1259 (2002)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Kunze, M.: Non-smooth Dynamical Systems. Springer, Berlin (2000)CrossRefMATHGoogle Scholar
  7. 7.
    Awrejcewicz, J., Holicke, M.M.: Smooth and Nonsmooth High Dimensional Chaos and Melnikov-Type Method. World Scientific, Singapore (2007)MATHGoogle Scholar
  8. 8.
    Bernardo, M.D., Budd, C.J., Champneys, A.R., Kowalczyk, P.: Piecewise-Smooth Dynamical Systems: Theory and Application. Springer, London (2008)MATHGoogle Scholar
  9. 9.
    Leine, R.I., Van Campen, D.H., Van de Vrande, B.L.: Bifurcations in nonlinear discontinuous systems. Nonlinear Dyn. 23, 105–164 (2000)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Makarenkov, O., Lamb, J.S.W.: Dynamics and bifurcations of nonsmooth systems: a survey. Phys. D 241, 1826–1844 (2012)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Melnikov, V.K.: On the stability of the center for time periodic perturbations. Tans. Moscow Math. Soc. 12, 1–57 (1963)MathSciNetGoogle Scholar
  12. 12.
    Guckenheimer, J., Holmes, P.: Nonlinear Oscillations, Dynamical System and Bifurcations of Vector Fields. Springer, New York (1983)CrossRefMATHGoogle Scholar
  13. 13.
    Wiggins, S.: Global Bifurcations and Chaos-Analytical Methods. Springer, New York (1988)CrossRefMATHGoogle Scholar
  14. 14.
    Kukučka, P.: Melnikov method for discontinuous planar systems. Nonlinear Anal. 66, 2698–2719 (2007)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Battelli, F., Fečkan, M.: Homoclinic trajectories in discontinuous systems. J. Dyn. Differ. Equ. 20, 337–376 (2008)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Battelli, F., Fečkan, M.: Bifurcation and chaos near sliding homoclinics. J. Differ. Equ. 248, 2227–2262 (2010)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Battelli, F., Fečkan, M.: Nonsmooth homoclinic orbits, Melnikov functions and chaos in discontinuous systems. Phys. D 241, 1962–1975 (2012)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Li, S.B., Zhang, W., Hao, Y.X.: Melnikov-type method for a class of discontinuous planar systems and applications. Int. J. Bifurc. Chaos 24(1450022), 1–18 (2014)MathSciNetGoogle Scholar
  19. 19.
    Du, Z., Zhang, W.: Melnikov method for homoclinic bifurcations in nonlinear impact oscillators. Comput. Math. Appl. 50, 445–458 (2005)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Granados, A., Hogan, S.J., Seara, T.M.: The Melnikov method and subharmonic orbits in a piecewise-smooth system. SIAM J. Appl. Dyn. Syst. 11, 801–830 (2012)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Carmona, V., Fernández-García, S., Freire, E., Torres, F.: Melnikov theory for a class of planar hybrid systems. Phys. D 248, 44–54 (2013)MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Granados, A., Hogan, S.J., Seara, T.M.: The scattering map in two coupled piecewise-smooth systems, with numerical application to rocking blocks. Phys. D 269, 1–20 (2014)MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Li, S.B., Shen, C., Zhang, W., Hao, Y.X.: Homoclinic bifurcations and chaotic dynamics for a piecewise linear system under a periodic excitation and a viscous damping. Nonlinear Dyn. 79, 2395–2406 (2015)MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Gao, J., Du, Z.: Homoclinic bifurcation in a quasiperiodically excited impact inverted pendulum. Nonlinear Dyn. 79, 445–458 (2015)MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Castro, J., Alvarez, J.: Melnikov-type chaos of planar systems with two discontinuities. Int. J. Bifurc. Chaos 25, 1550027 (2015)MathSciNetCrossRefMATHGoogle Scholar
  26. 26.
    Tian, R.L., Zhou, Y.F., Zhang, B.L., Yang, X.W.: Chaotic threshold for a class of impulsive differential system. Nonlinear Dyn. 79, 445–458 (2015)Google Scholar
  27. 27.
    Li, S.B., Ma, W.S., Zhang, W., Hao, Y.X.: Melnikov method for a three-zonal planar hybrid piecewise-smooth system and application. Int. J. Bifurc. Chaos 26(1650014), 1–13 (2016)MathSciNetMATHGoogle Scholar
  28. 28.
    Li, S.B., Ma, W.S., Zhang, W., Hao, Y.X.: Melnikov method for a class of planar hybrid piecewise-smooth systems. Int. J. Bifurc. Chaos 26(1650030), 1–12 (2016)MathSciNetMATHGoogle Scholar
  29. 29.
    Bertozzi, A.L.: Heteroclinic orbits and chaotic dynamics in planar fluid flow. SIAM J. Math. Anal. 19, 1271–1294 (1988)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2016

Authors and Affiliations

  1. 1.College of ScienceCivil Aviation University of ChinaTianjinChina
  2. 2.Beijing Key Laboratory on Nonlinear Vibrations and Strength of Mechanical Structures, College of Mechanical EngineeringBeijing University of TechnologyBeijingChina
  3. 3.College of Mechanical EngineeringBeijing Information Science and Technology UniversityBeijingChina

Personalised recommendations