More limit cycles than expected in Liénard equations
By Freddy Dumortier, D. Panazzolo, Robert Roussarie. 2007
106 citations recorded by the source corpus.
The paper deals with classical polynomial Lienard equations, i.e. planar vector fields associated to scalar second order differential equations x"+ f(x)x' + x = 0 where f is a polynomial. We prove that for a well-chosen polynomial f of degree 6, the equation exhibits 4 limit cycles. It induces that for n ≥ 3 there exist polynomials f of degree 2n such that the related equations exhibit more than n limit cycles. This contradicts the conjecture of Lins, de Melo and Pugh stating that for Lienard equations as above, with f of degree 2n, the maximum number of limit cycles is n. The limit cycles that we found are relaxation oscillations which appear in slow-fast systems at the boundary of classical polynomial Lienard equations. More precisely we find our example inside a family of second order differential equations ex" + f μ (x)x' + x = 0. 0. Here, f μ is a well-chosen family of polynomials of degree 6 with parameter μ ∈ R 4 and e is a small positive parameter tending to 0. We use bifurcations from canard cycles which occur when two extrema of the critical curve of the layer equation are crossing (the layer equation corresponds to e = 0). As was proved by Dumortier and Roussarie (2005) these bifurcations are controlled by a rational integral computed along the critical curve of the layer equation, called the slow divergence integral. Our result is deduced from the study of this integral.
Subfields: Geometry and Topology, Statistical and Nonlinear Physics
View Paper: https://www.semanticscholar.org/paper/246cecc122da4fe01b4ab4c4edf94b1bc334ada6
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In Geometry and Topology
In Statistical and Nonlinear Physics
Citations in the corpus
- A Survey on the Inverse Integrating Factor (2009, background): “4 Singular perturbations As some recent research papers show, see [45], limit periodic sets containing an infinite number of critical points may have a cyclicity higher than expected.”
- Uniqueness of limit cycles for Liénard differential equations of degree four (2012, unknown): “Roussarie [3] gave a counterexample to this”