Fixed and moving limit cycles for Liénard equations
- 28 Downloads
We consider a family of planar vector fields that writes as a Liénard system in suitable coordinates. It has a fixed closed invariant curve that often contains periodic orbits of the system. We prove a general result that gives the hyperbolicity of these periodic orbits, and we also study the coexistence of them with other periodic orbits. Our family contains the celebrated Wilson polynomial Liénard equation, as well as all polynomial Liénard systems having hyperelliptic limit cycles. As an illustrative example, we study in more detail a natural 1-parametric extension of Wilson example. It has at least two limit cycles, one of them fixed and algebraic and the other one moving with the parameter, presents a transcritical bifurcation of limit cycles and for a given parameter has a non-hyperbolic double algebraic limit cycle. In order to prove that for some values of the parameter the system has exactly two hyperbolic limit cycles, we use several suitable Dulac functions.
KeywordsLiénard equation Limit cycle Bifurcations Invariant algebraic curve
Mathematics Subject ClassificationPrimary 34C07 Secondary 37C23 34C25 37C27
The first author is partially supported by the MINECO/FEDER MTM2016-77278-P and AGAUR 2017-SGR-1617 Grants. The second author is partially supported by GNAMPA, Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni.
- 24.Sotomayor, J.: Curvas definidas por Equaçoes Diferenciais no Plano. Instituto de Matemática Pura e Aplicada, Rio de Janeiro (1981)Google Scholar
- 26.Sturmfels, B.: Solving Systems of Polynomial Equations, CBMS Regional Conference Series Mathematics, vol. 97. American Mathematical Society, Providence, RI (2002) (Published for the Conference Board of the Mathematical Sciences, Washington, DC)Google Scholar
- 29.Yan-Qian, Y., et al.: Theory of Limit Cycles. Translations of Mathematical Monographs, vol. 66. American Mathematical Society, Providence, RI (1986)Google Scholar
- 30.Zhang, Z.F., et al.: Qualitative theory of differential equations. Translations of Mathematical Monographs, vol. 101. American Mathematical Society, Providence, RI (1992)Google Scholar