The displacement map associated to polynomial unfoldings of planar Hamiltonian vector fields

By L. Gavrilov, Iliya D. Iliev. 2003

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We study the displacement map associated to small one-parameter polynomial unfoldings of polynomial Hamiltonian vector fields on the plane. Its leading term, the generating function M ( t ), has an analytic continuation in the complex plane and the real zeroes of M ( t ) correspond to the limit cycles bifurcating from the periodic orbits of the Hamiltonian flow. We give a geometric description of the monodromy group of M ( t ) and use it to formulate sufficient conditions for M ( t ) to satisfy a differential equation of Fuchs or Picard-Fuchs type. As examples, we consider in more detail the Hamiltonian vector fields [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="01i" /] and [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="02i" /], possessing a rotational symmetry of order two and three, respectively. In both cases M ( t ) satisfies a Fuchs-type equation but in the first example M ( t ) is always an Abelian integral (that is to say, the corresponding equation is of Picard-Fuchs type) while in the second one this is not necessarily true. We derive an explicit formula of M ( t ) and estimate the number of its real zeroes.

Subfields: Applied Mathematics, Geometry and Topology, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/94f82da8018610c2cf9be4cf953408ca4ddb601c

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In Geometry and Topology

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