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

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## Abstract

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* = 2*l* 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## References

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