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Classical Liénard equations of degree \(n\geqslant 6\) can have \([\frac{n-1}{2}]+2\) limit cycles. (English) Zbl 1215.37038

Consider the Lienard equation
\[ \dot{x}=y-F(x), \dot{y}=-x, \]
where \(F\) is a polynomial of degree \(n\). Recall that the Smale-Hilbert problem is to find, in terms of \(n\), the maximum number of limit cycles. In 1976, A. Lins, W. de Melo and C. C. Pugh [Lect. Notes Math. 597, 335–357 (1977; Zbl 0362.34022)] conjectured that the Lienard equation has at most \([\frac{n-1}{2}]\) limit cycles (where \([\frac{n-1}{2}]\) denotes the largest integer less than or equal to \(\frac{n-1}{2}\)). Many research papers published in the last 35 years support the conjecture. In this paper, the authors give a counterexample to this conjecture. They prove the existence of the Lienard equation of degree \(6\) having \(4\) limit cycles. The theory of geometric singular perturbations is used to prove the result.

MSC:

37G15 Bifurcations of limit cycles and periodic orbits in dynamical systems
34E17 Canard solutions to ordinary differential equations
34C05 Topological structure of integral curves, singular points, limit cycles of ordinary differential equations
34C26 Relaxation oscillations for ordinary differential equations

Citations:

Zbl 0362.34022

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References:

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