A note on higher order Melnikov functions
Copyright policy )A note on higher order Melnikov functions
The paper comments some facts related to the weakened 16th Hilbert's problem about limit cycles. The authors deal with small polynomial perturbations of Hamiltonian systems in the plane \(dH-\varepsilon \omega=0\) such that the first displacement map near a periodic orbit of the unperturbed system is of the form \(\Delta H=\varepsilon^kM_k(h)+O(\varepsilon^{k+1})\), with \(k>1\). They discuss the form of \(M_k(h)\) and present several examples (some of them new) where \(M_k\) is an Abelian integral or not. The second case may occur e.g. in the presence of symmetry. One can find a further result in this field in the very recent preprint of \textit{C. Christopher} and \textit{P. Mardešić} [Monodromy and tangential center problems, \url{arXiv:0803.2110}].
- Polish Academy of Sciences Poland
- University of Burgundy France
- Institut de Mathématiques de Bourgogne France
- University of Warsaw Poland
- Institute of Mathematics Poland
Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests, Ordinary differential equations and connections with real algebraic geometry (fewnomials, desingularization, zeros of abelian integrals, etc.), Planar polynomial system, Abelian integral, Topological structure of integral curves, singular points, limit cycles of ordinary differential equations, Theory of limit cycles of polynomial and analytic vector fields (existence, uniqueness, bounds, Hilbert's 16th problem and ramifications) for ordinary differential equations, higher-order Melnikov integral, weakened Hilbert's 16th problem, limit cycle, Dynamics induced by flows and semiflows, displacement map
Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests, Ordinary differential equations and connections with real algebraic geometry (fewnomials, desingularization, zeros of abelian integrals, etc.), Planar polynomial system, Abelian integral, Topological structure of integral curves, singular points, limit cycles of ordinary differential equations, Theory of limit cycles of polynomial and analytic vector fields (existence, uniqueness, bounds, Hilbert's 16th problem and ramifications) for ordinary differential equations, higher-order Melnikov integral, weakened Hilbert's 16th problem, limit cycle, Dynamics induced by flows and semiflows, displacement map
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