The displacement map associated to polynomial unfoldings of planar Hamiltonian vector fields
| dc.creator | Gavrilov, Lubomir | |
| dc.creator | Iliev, Iliya D. | |
| dc.date | 2003-05-21 | |
| dc.date | 2004-12-16 | |
| dc.date.accessioned | 2026-07-07T09:41:42Z | |
| dc.date.available | 2026-07-07T09:41:42Z | |
| dc.description | 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 $\dot{z}=i\bar{z}-i(z+\bar{z})^3$ and $\dot{z}=iz+\bar{z}^2$, 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.} | |
| dc.description | This is a revised verion containing 36 pages and 8 figures | |
| dc.identifier | https://arxiv.org/abs/math/0305301 | |
| dc.identifier | http://arxiv.org/abs/math/0305301 | |
| dc.identifier | American J. of Math., 127 (2005) 1153-1190 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/161918 | |
| dc.subject | Dynamical Systems | |
| dc.subject | 34C07, 34C08 | |
| dc.title | The displacement map associated to polynomial unfoldings of planar Hamiltonian vector fields | |
| dc.type | text |