×

Periodic solutions of perturbed Hamiltonian systems in the plane by the use of the Poincaré-Birkhoff theorem. (English) Zbl 1277.34046

This paper is dedicated to the \(100\)th anniversary of Poincaré’s last geometric theorem. This theorem, also called Poincaré-Birkhoff theorem, deals with the number of fixed points of an area-preserving homeomorphism of an annular region of the plane with a twisting condition. The paper starts with a description of the history of this well-known theorem together with the most important references; see also, for instance, the articles by P. Le Calvez and J. Wang [Proc. Am. Math. Soc. 138, No. 2, 703–715 (2010; Zbl 1194.37069)] and C. Rebelo [Nonlinear Anal., Theory Methods Appl. 29, No. 3, 291–311 (1997; Zbl 0906.34029)] for such a description. In this last reference, a modern version of the Poincaré-Birkhoff theorem is provided, and this is the one used in the paper. The statement is that given an annular region of \(\mathbb{R}^2\) around the origin, bounded by two star-shaped curves with respect to the origin and an area preserving homeomorphism of the annular region to an annular region around the origin and under a twist condition, it can be proved that the homeomorphism has two fixed points. The assumption of star-shapedness is a delicate one and recent developments point out that it cannot be eliminated. This modern version has eliminated an invariance assumption for the boundaries of the annular region.
The authors apply this result to \(T\)-periodic perturbations of autonomous Hamiltonian systems of the form \[ J \dot{u} = \nabla \mathcal{H}(u). \] The main result, together with its corollaries, deals with the number of periodic solutions which are not broken by the perturbation. The authors provide several applications for systems for which the orbits bounding the considered annular region are not star-shaped and, thus, some preliminary changes to “good” coordinates needs to be done. The results of the authors need few regularity conditions to be applied.

MSC:

34C25 Periodic solutions to ordinary differential equations
47H25 Nonlinear ergodic theorems
34C23 Bifurcation theory for ordinary differential equations
37E30 Dynamical systems involving homeomorphisms and diffeomorphisms of planes and surfaces
37E40 Dynamical aspects of twist maps
PDFBibTeX XMLCite