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Poincaré return map near a planar center. (English) Zbl 1088.37006

Summary: We consider a germ of smooth planar vector fields having at the origin a singularity of center type. We are interested in the Poincaré return map and equivalently in the displacement function on a transversal section. The study of this topic already dates back to Lyapunov.
Here, we propose another approach than the Lyapunov one did: we construct a multi-valued normal form to find the asymptotics of the displacement function. In the context of the perturbation of a Hamiltonian vector field, we can deduce the asymptotics of each Melnikov function. This way we can cover also the case of a degenerate center. An advantage of the technique coming from our approach is that the calculations are elementary, i.e., we avoid the difficult line integrals that are encountered when using the Abelian integral approach, and the technique is directly implementable in most computer algebra packages.

MSC:

37C10 Dynamics induced by flows and semiflows
34C07 Theory of limit cycles of polynomial and analytic vector fields (existence, uniqueness, bounds, Hilbert’s 16th problem and ramifications) for ordinary differential equations
34C20 Transformation and reduction of ordinary differential equations and systems, normal forms
34C08 Ordinary differential equations and connections with real algebraic geometry (fewnomials, desingularization, zeros of abelian integrals, etc.)
37G15 Bifurcations of limit cycles and periodic orbits in dynamical systems
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