On the zeros of the Abelian integrals for a class of Liénard systems. (English) Zbl 1142.34327
Summary: A planar polynomial differential system has a finite number of limit cycles. However, finding the upper bound of the number of limit cycles is an open problem for the general nonlinear dynamical systems. In this Letter, we investigated a class of Liénard systems of the form \(\dot x, y, \dot y = f(x) + yg (x)\) with deg \(f=5\) and deg \(g=4\). We proved that the related elliptic integrals of the Liénard systems have at most three zeros including multiple zeros, which implies that the number of limit cycles bifurcated from the periodic orbits of the unperturbed system is less than or equal to 3.
MSC:
| 34C08 | Ordinary differential equations and connections with real algebraic geometry (fewnomials, desingularization, zeros of abelian integrals, etc.) |
| 34C07 | Theory of limit cycles of polynomial and analytic vector fields (existence, uniqueness, bounds, Hilbert’s 16th problem and ramifications) for ordinary differential equations |
| 37C10 | Dynamics induced by flows and semiflows |
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