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

详细信息    查看全文

• 作者：Feng Liang ; Maoan Han
• 关键词：Limit cycle ; Heteroclinic loop ; Melnikov function ; Chebyshev system ; Bifurcation ; Piecewise smooth system ; 34C05 ; 34C07 ; 37G15
• 刊名：Chinese Annals of Mathematics - Series B
• 出版年：2016
• 出版时间：March 2016
• 年：2016
• 卷：37
• 期：2
• 页码：267-280
• 全文大小：199 KB
• 参考文献：[1]di Bernardo, M., Budd, C. J., Champneys, A. R. and Kowalczyk, P., Piecewise-Smooth Dynamical Systems, Theory and Applications, Springer-Verlag, London, 2008.MATH
  [2]Andronov, A. A., Khaikin, S. E. and Vitt, A. A., Theory of Oscillators, Pergamon Press, Oxford, 1965.
  [3]Kunze, M., Non-smooth Dynamical Systems, Springer-Verlag, Berlin, 2000.CrossRef MATH
  [4]Filippov, A. F., Differential Equations with Discontinuous Righthand Sides, Kluwer Academic, Netherlands, 1988.CrossRef
  [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.CrossRef MathSciNet MATH
  [6]Budd, C. J., Non-smooth dynamical systems and the grazing bifurcation, Nonlinear Mathematics and Its Applications, Cambridge Univ. Press, Cambridge, 1996.
  [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.CrossRef MathSciNet MATH
  [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.CrossRef MathSciNet MATH
  [9]Han, M. and Zhang, W., On Hopf bifurcation in nonsmooth planar systems, J. Differential Equations, 248, 2010, 2399–2416.CrossRef MathSciNet MATH
  [10]Coll, B., Gasull, A. and Prohens R., Degenerate Hopf bifurcations in discontinuous planar systems, J. Math. Anal. Appl., 253(2), 2001, 671–690.CrossRef MathSciNet MATH
  [11]Gasull, A. and Torregrosa, J., Center-focus problem for discontinuous planar differential equations, Int. J. Bifur. Chaos, 13(7), 2003, 1755–1765.CrossRef MathSciNet MATH
  [12]Chen, X. and Du, Z., Limit cycles bifurcate from centers of discontinuous quadratic systems, Comput. Math. Appl., 59(12), 2010, 3836–3848.CrossRef MathSciNet MATH
  [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.CrossRef MathSciNet
  [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]Karlin, S. and Studden, W., Tchebycheff Systems: With Applications in Analysis and Statistics, Interscience Publishers, New York, 1966.MATH
  [16]Du, Z. and Zhang, W., Melnikov method for homoclinic bifurcation in nonlinear impact oscillators, Comput. Math. Appl., 50(3–4), 2005, 445–458.CrossRef MathSciNet MATH
  [17]Battelli, F. and Feckan, M., Bifurcation and chaos near sliding homoclinics, J. Differential Equations, 248(9), 2010, 2227–2262.CrossRef MathSciNet MATH
  [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.CrossRef MathSciNet MATH
  [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.CrossRef MathSciNet MATH
  [20]Han, M., On Hopf cyclicity of planar systems, J. Math. Anal. Appl., 245(2), 2000, 404–422.CrossRef MathSciNet MATH
  [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.CrossRef MathSciNet MATH
  [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.CrossRef MathSciNet MATH
  [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.CrossRef MathSciNet MATH
  [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.CrossRef MathSciNet MATH
  [25]Han, M., Cyclicity of planar homoclinic loops and quadratic integrable systems, Sci. China, Ser. A, 40(12), 1997, 1247–1258.CrossRef MathSciNet MATH
  
• 作者单位：Feng Liang (1)
  Maoan Han (2)
  
  1. Institute of Mathematics, Anhui Normal University, Wuhu, Anhui, 241000, China
  2. Institute of Mathematics, Shanghai Normal University, Shanghai, 200234, China
  
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Mathematics
  Applications of Mathematics
  Chinese Library of Science
  
• 出版者：Springer Berlin / Heidelberg
• ISSN：1860-6261

文摘

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. Keywords Limit cycle Heteroclinic loop Melnikov function Chebyshev system Bifurcation Piecewise smooth system