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On the number of limit cycles in small perturbations of a piecewise linear Hamiltonian system with a heteroclinic loop

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Abstract

In this paper, the authors consider limit cycle bifurcations for a kind of nonsmooth polynomial differential systems by perturbing a piecewise linear Hamiltonian system with a center at the origin and a heteroclinic loop around the origin. When the degree of perturbing polynomial terms is n (n ≥ 1), it is obtained that n limit cycles can appear near the origin and the heteroclinic loop respectively by using the first Melnikov function of piecewise near-Hamiltonian systems, and that there are at most n + [n+1/2] limit cycles bifurcating from the periodic annulus between the center and the heteroclinic loop up to the first order in ε. Especially, for n = 1, 2, 3 and 4, a precise result on the maximal number of zeros of the first Melnikov function is derived.

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References

  1. [1]

    di Bernardo, M., Budd, C. J., Champneys, A. R. and Kowalczyk, P., Piecewise-Smooth Dynamical Systems, Theory and Applications, Springer-Verlag, London, 2008.

  2. [2]

    Andronov, A. A., Khaikin, S. E. and Vitt, A. A., Theory of Oscillators, Pergamon Press, Oxford, 1965.

  3. [3]

    Kunze, M., Non-smooth Dynamical Systems, Springer-Verlag, Berlin, 2000.

  4. [4]

    Filippov, A. F., Differential Equations with Discontinuous Righthand Sides, Kluwer Academic, Netherlands, 1988.

  5. [5]

    Bernardo, M. D., Budd, C. J. and Champneys, A. R., Grazing, skipping and sliding: Analysis of the nonsmooth dynamics of the DC/DC buck converter, Nonlinearity, 11, 1998, 859–890.

  6. [6]

    Budd, C. J., Non-smooth dynamical systems and the grazing bifurcation, Nonlinear Mathematics and Its Applications, Cambridge Univ. Press, Cambridge, 1996.

  7. [7]

    Bernardo, M. D., Kowalczyk, P. and Nordmark, A. B., Sliding bifurcations: A novel mechanism for the sudden onset of chaos in dry friction oscillators, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 13(10), 2003, 2935–2948.

  8. [8]

    Nusse, H. E. and Yorke, J. A., Border-collision bifurcations including “period two to period three” for piecewise smooth systems, Physica D, 57(1–2), 1992, 39–57.

  9. [9]

    Han, M. and Zhang, W., On Hopf bifurcation in nonsmooth planar systems, J. Differential Equations, 248, 2010, 2399–2416.

  10. [10]

    Coll, B., Gasull, A. and Prohens R., Degenerate Hopf bifurcations in discontinuous planar systems, J. Math. Anal. Appl., 253(2), 2001, 671–690.

  11. [11]

    Gasull, A. and Torregrosa, J., Center-focus problem for discontinuous planar differential equations, Int. J. Bifur. Chaos, 13(7), 2003, 1755–1765.

  12. [12]

    Chen, X. and Du, Z., Limit cycles bifurcate from centers of discontinuous quadratic systems, Comput. Math. Appl., 59(12), 2010, 3836–3848.

  13. [13]

    Liu, X. and Han, M., Bifurcation of limit cycles by perturbing piecewise Hamiltonian systems, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 20(5), 2010, 1–12.

  14. [14]

    Liang, F., Han, M. and Romanovski, V. G., Bifurcation of limit cycles by perturbing a piecewise linear Hamiltonian system with a homoclinic loop, Nonlinear Analysis, 75(11), 2012, 4355–4374.

  15. [15]

    Karlin, S. and Studden, W., Tchebycheff Systems: With Applications in Analysis and Statistics, Interscience Publishers, New York, 1966.

  16. [16]

    Du, Z. and Zhang, W., Melnikov method for homoclinic bifurcation in nonlinear impact oscillators, Comput. Math. Appl., 50(3–4), 2005, 445–458.

  17. [17]

    Battelli, F. and Feckan, M., Bifurcation and chaos near sliding homoclinics, J. Differential Equations, 248(9), 2010, 2227–2262.

  18. [18]

    Llibre, J. and Makhlonf, A., Bifurcation of limit cycles from a two-dimensional center inside Rn, Nonlinear Analysis, 72(3–4), 2010, 1387–1392.

  19. [19]

    Llibre, J., Wu, H. and Yu, J., Linear estimate for the number of limit cycles of a perturbed cubic polynomial differential system, Nonlinear Analysis, 70(1), 2009, 419–432.

  20. [20]

    Han, M., On Hopf cyclicity of planar systems, J. Math. Anal. Appl., 245(2), 2000, 404–422.

  21. [21]

    Han, M., Chen, G. and Sun, C., On the number of limit cycles in near-Hamiltonian polynomial systems, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 17(6), 2007, 2033–2047.

  22. [22]

    Wu, Y., Cao, Y. and Han, M., Bifurcations of limit cycles in a z3-equivariant quartic planar vector field, Chaos, Solitons & Fractals, 38(4), 2008, 1177–1186.

  23. [23]

    Yang, J. and Han, M., Limit cycle bifurcations of some Liénard systems with a cuspidal loop and a homoclinic loop, Chaos, Solitons & Fractals, 44(4–5), 2011, 269–289.

  24. [24]

    Han, M. and Chen, J., On the number of limit cycles in double homoclinic bifurcations, Sci. China, Ser. A, 43(9), 2000, 914–928.

  25. [25]

    Han, M., Cyclicity of planar homoclinic loops and quadratic integrable systems, Sci. China, Ser. A, 40(12), 1997, 1247–1258.

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Author information

Correspondence to Feng Liang.

Additional information

This work was supported by the National Natural Science Foundation of China (No. 11271261), the Natural Science Foundation of Anhui Province (No. 1308085MA08), and the Doctoral Program Foundation (2012) of Anhui Normal University.

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Liang, F., Han, M. On the number of limit cycles in small perturbations of a piecewise linear Hamiltonian system with a heteroclinic loop. Chin. Ann. Math. Ser. B 37, 267–280 (2016). https://doi.org/10.1007/s11401-016-0946-8

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Keywords

  • Limit cycle
  • Heteroclinic loop
  • Melnikov function
  • Chebyshev system
  • Bifurcation
  • Piecewise smooth system

2000 MR Subject Classification

  • 34C05
  • 34C07
  • 37G15