Existence and Uniqueness of Limit Cycle in Discontinuous Planar Differential Systems

详细信息    查看全文

• 作者：Fangfang Jiang ; Jitao Sun
• 关键词：Discontinuous system ; Sliding limit cycle ; Existence ; Uniqueness ; Stability
• 刊名：Qualitative Theory of Dynamical Systems
• 出版年：2016
• 出版时间：April 2016
• 年：2016
• 卷：15
• 期：1
• 页码：67-80
• 全文大小：589 KB
• 参考文献：1.Filippov, A.: Differential Equations with Discontinuous Righthand Sides: Control Systems. Kluwer Academic, Dordrecht (1988)CrossRef
  2.Luo, C.: Discontinuous Dynamical Systems. Higher Education Press, Springer-Verlag, Beijing and Berlin Heidelberg (2012)CrossRef MATH
  3.Zhang, Z., Ding, T., Huang, W., Dong, Z.: Qualitative Theory of Differential Equations. Science Press, Beijing (2006)
  4.Luo, X., Guo, S.: Existence and uniqueness of solutions for fractional boundary value problem with fractional boundary value conditions. Qual. Theory Dyn. Syst. 13, 1–17 (2014)MathSciNet CrossRef MATH
  5.Li, C., Llibre, J.: Uniqueness of limit cycles for Liénard differential equations of degree four. J. Differ. Equ. 252, 3142–3162 (2012)MathSciNet CrossRef MATH
  6.Llibre, J., Ordóñez, M., Ponce, E.: On the existence and uniqueness of limit cycles in planar continuous piecewise linear systems without symmetry. Nonlinear Anal. RWA 14, 2002–2012 (2013)MathSciNet CrossRef MATH
  7.Chang, G., Zhang, T., Han, M.: On the number of limit cycles of a class of polynomial systems of Liénard type. J. Math. Anal. Appl. 408, 775–780 (2013)MathSciNet CrossRef MATH
  8.Llibre, J., Lopes, B., De Moraes, J.: Limit cycles for a class of continuous and discontinuous cubic polynomial differential systems. Qual. Theory Dyn. Syst. 13, 129–148 (2014)MathSciNet CrossRef MATH
  9.Sun, J.: Necessary and sufficient condition for oscillating solutions to the system \(x^{\prime }=h(y)-F(x), y^{\prime }=-g(x)\) . Nanjing Daxue Xuebao Shuxue Banniankan 9, 113–119 (1992). ChineseMATH
  10.Freire, E., Ponce, E., Torres, F.: Canonical discontinuous planar piecewise linear systems. SIAM J. Appl. Dyn. Syst. 11, 181–211 (2012)MathSciNet CrossRef MATH
  11.Huan, S., Yang, X.: Existence of limit cycles in general planar piecewise linear systems of saddle-saddle dynamics. Nonlinear Anal. TMA 92, 82–95 (2013)MathSciNet CrossRef MATH
  12.Llibre, J., Ponce, E., Torres, F.: On the existence and uniqueness of limit cycles in Liénard differential equations allowing discontinuities. Nonlinearity 21, 2121–2142 (2008)MathSciNet CrossRef MATH
  13.Jiang, F., Sun, J., On the uniqueness of limit cycles in discontinuous Liénard-type systems. Electron. J. Qual. Theory Differ. Equ. 71, 1–12 (2014)
  14.Carmona, V., Fernndez Garca, S., Freire, E., Torres, F.: Melnikov theory for a class of planar hybrid systems. Phys. D 248, 44–54 (2013)MathSciNet CrossRef MATH
  15.Cardin, P., Carvalho, T., Llibre, J.: Limit cycles of discontinuous piecewise linear differential systems. Internat. J. Bifur. Chaos 11, 3181–3194 (2011)MathSciNet CrossRef MATH
  16.Artés, J., Llibre, J., Medrado, J.: Piecewise linear differential systems with two real saddles. Math. Comput. Simul. 95, 13–22 (2014)MathSciNet CrossRef
  
• 作者单位：Fangfang Jiang (1)
  Jitao Sun (1)
  
  1. Department of Mathematics, Tongji University, Shanghai, 200092, China
  
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Mathematics
  Dynamical Systems and Ergodic Theory
  Difference and Functional Equations
  
• 出版者：Birkh盲user Basel
• ISSN：1662-3592

文摘

In this paper, we investigate the existence and uniqueness of limit cycle for a class of discontinuous planar piecewise smooth differential systems with four regions separated by three vertical parallel lines. We first show the continuity and qualitative properties of half Poincaré mappings, and then by some analysis techniques we provide several criteria concerning the existence, uniqueness and stability of limit cycle for the discontinuous planar differential system. Finally, an example possessing a unique stable sliding limit cycle is presented to illustrate the obtained result.