On the stability of limit cycles for planar differential systems

JOURNAL OF DIFFERENTIAL EQUATIONS

卷:213

Article

Giacomini, H ; Grau, A

关键词: limit cycle;    planar differential system;    stability;    hyperbolicity;    polynomial vector field;   

DOI  :  10.1016/j.jde.2005.02.010

来源: Elsevier

【 摘 要 】

We consider a planar differential system (x) over dot = P(x, y), (y) over dot = Q(x, y), where P and Q are l(1) functions in some open set U subset of R-2, and () over dot = (d)/(dt). T. Let y be a periodic orbit of the system in U. Let f (x, y) : U subset of R-2 -> R be a l(1) function such that P (x, y) (partial derivative f)/(partial derivative x) (x, y) + Q(x, y) (partial derivative f)/(partial derivative y)(x, y) = k(x, y) f(x, y), where k(x, y) is a l(l) function in U and gamma subset of {(x, y) | f (x, y) = 0}. We assume that if p is an element of U is such that f(p) = 0 and del f (p) = 0, then p is a singular point. We prove that integral(0)(T) ((partial derivative P)/(partial derivative x) + (partial derivative Q)/(partial derivative y)) (gamma(t))dt =integral(0)(T) k(gamma(t)) dt, where T > 0 is the period of gamma. As an application, we take profit from this equality to show the hyperbolicity of the known algebraic limit cycles of quadratic systems. (c) 2005 Elsevier Inc. All rights reserved.

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