The local cyclicity problem: Melnikov method using Lyapunov constants. (English) Zbl 1504.34082
C. Chicone and M. Jacobs [J. Differ. Equations 91, No. 2, 268–326 (1991; Zbl 0733.34045)] proved the equivalence between Lyapunov constants and the first Melnikov function for determining the number of limit cycles bifurcating from the quadratic centers. In the paper, the authors generalize the equivalence for polynomial systems of any degree and use the equivalence to update the local cyclicity of monodromic equilibrium points for polynomial systems of degree \(6\), i.e., \(\mathcal{M}(6)\geq 44\). In addition, the authors use averaging theory to prove that the local cyclicity for piecewise polynomial vector fields of degree \(4\) and \(5\) satisfies \(\mathcal{M}^{c}_{p}(4)\geq 43 \) and \(\mathcal{M}^{c}_{p}(5)\geq 65 \), respectively.
Reviewer: Kuilin Wu (Guiyang)
MSC:
| 34C07 | Theory of limit cycles of polynomial and analytic vector fields (existence, uniqueness, bounds, Hilbert’s 16th problem and ramifications) for ordinary differential equations |
| 34C05 | Topological structure of integral curves, singular points, limit cycles of ordinary differential equations |
| 34C23 | Bifurcation theory for ordinary differential equations |
| 34E10 | Perturbations, asymptotics of solutions to ordinary differential equations |
Citations:
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