An upper estimate for the number of limit cycles of even-degree Liénard equations in the focus case



We give an upper estimate for the number of limit cycles of the Liénard equations \( \dot{x} \) = y − F(x), \( \dot{y} \) = −x of even degree in the case where its unique singular point (0, 0) is a focus.

M. Caubergh and F. Dumortier [1] obtained an explicit linear upper estimate for the number of large-amplitude limit cycles of such equations. We estimate the number of mid-amplitude limit cycles of the Liénard equations using the growth-and-zeros theorem proved by Ilyashenko and Yakovenko in [7].

Our estimate depends on four parameters: n, C, a 1, and R. Let \( F(x) = {x^n} + \sum\limits_{i = 1}^{n - 1} {{a_i}{x^i}} \), where n = 2l is the even degree of the monic polynomial F without a constant term, |a i | < C for all i, so C is the size of a compact subset in the space of parameters, R is the size of the neighborhood of the origin, such that there are at most l limit cycles located outside of this neighborhood, |a 1 | stands for the distance from the equation linearization to the center case in the space of parameters, and 2 − |a 1 | stands for the distance from the equation linearization to the node case in the space of parameters.

Key words and phrases

Limit cycles Poincaré map Liénard equations Hilbert’s 16th problem Hilbert–Smale problem 

2000 Mathematics Subject Classification

Primary 34C07; secondary 34M10 


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Copyright information

© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  1. 1.Institute for Information Transmission Problems (Kharkevich Institute)Russian Academy of SciencesMoscowRussia

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