Melnikov functions of arbitrary order for piecewise smooth differential systems in \(\mathbb{R}^n\) and applications. (English) Zbl 1495.34026
The authors investigate limit cycles bifurcating from periodic submanifold of \(n\)-dimensonal piecewise smooth dynamical systems with two zones separated by a hyperplane. In order to obtain the maximum number of limit cycles, they obtain arbitrary order Melnikov function by using Faá di Bruno’s Formula. As applications, they study the maximum number of crossing limit cycles bifurcating from an \(n\)-dimensional periodic submanifold caused by non-smooth centers of fold-fold type and perturbations of piecewise smooth Hamiltonian systems. Moreover, they prove that these upper bounds can be reached. Their main results extend some known results in these directions.
Reviewer: Dingheng Pi (Quanzhou)
MSC:
| 34A36 | Discontinuous ordinary differential equations |
| 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 |
| 34C29 | Averaging method for ordinary differential equations |
| 34E10 | Perturbations, asymptotics of solutions to ordinary differential equations |
Keywords:
Melnikov theory; Hilbert’s 16th problem; fold-fold singularity; limit cycle bifurcation; piecewise smooth differential systemReferences:
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