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Bifurcation d’orbites périodiques d’un système Hamiltonien au voisinage d’une position d’équilibre. (French) Zbl 0598.70016
In this thesis a detailed analysis is made of some period doubling bifurcations of periodic orbits for a class of Hamiltonian systems. The Poincaré map P corresponding to a periodic orbit is first discussed in some generality and then specialized to the case of a Hamiltonian vector field, where it is shown to be a symplectic map of a codimension-2 submanifold.
For the main part of the thesis, the basic assumptions are the following. Consider a family \(X_{\nu}\) of Hamiltonian vector fields, which has a non-degenerate periodic orbit for \(\nu =\nu_ 0\). Suppose that the tangent map of \(P_{\nu_ 0}\) (at the intersection of the periodic orbit with the transversal submanifold) has an eigenvalue -1, while the other eigenvalues lie outside the unit circle. Suppose further that for \(\nu\) near \(\nu_ 0\) there is at most one quadruple of eigenvalues approaching -1. Then, among the various different possibilities, two cases are investigated, in which application of the centre manifold theorem allows one to reduce the study of the behaviour of \(P_{\nu}\) near the passage through -1 to a 2-dimensional problem.
Each of these cases is then modeled by a certain family of maps in the plane, for which the bifurcations are studied in great detail. As an application, periodic orbits for a family of 3-dimensional cubic potentials are discussed.
Reviewer: W.Sarlet
70H05 Hamilton’s equations
37G99 Local and nonlocal bifurcation theory for dynamical systems
37J99 Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems