Variational problems with compactness defect and homoclinic orbits of Hamiltonian systems.

Summary In this thesis, we propose a variational approach to the problem of the existence and multiplicity of homoclinic orbits of Hamiltonian systems. The Hamiltonians considered are defined on an even dimensional real vector space with the usual symplectic form. They have a hyperbolic equilibrium, and we call homoclinic orbits, non-constant solutions which tend to this equilibrium in the two infinite directions of time. We obtain the existence of one or more of these orbits under global assumptions on the Hamiltonian. In the first part of this thesis, the Hamiltonian depends periodically on time. Under convexity and superquadraticity assumptions, we show the existence of an infinite number of homoclinic orbits, without the classical transversality condition. Our result takes the form of an alternative: either the homoclinic orbits are uncountable, or there is an infinite set of homoclinic orbits, possessing a particular structure that we call approximate Bernoulli shift. In this second case, the topological entropy of the system is strictly positive. In the second part, the Hamiltonian system is autonomous. We present the first result on the existence of homoclinic orbits under assumptions invariant by symplectic transformations.