Bifurcations of Periodic Solutions Satisfying the Zero-Hamiltonian Constraint in Reversible Differential Equations

Abstract

This is a study of the existence of bifurcation branches for the problem of finding even, periodic solutions in fourth-order, reversible Hamiltonian systems such that the Hamiltonian evaluates to zero along each solution on the branch. The class considered here is a generalization of both the Swift--Hohenberg and extended Fisher--Kolmogorov equations that have been studied in several recent papers. We obtain the existence of local bifurcations from a trivial solution under mild restrictions on the nonlinearity and obtain existence and disjointness results regarding the global nature of the resulting bifurcating continua for the case where the Hamiltonian has a single-well potential.The local results rest on two abstract bifurcation theorems which also have applications to sixth-order problems and which show that the curves of zero-Hamiltonian solutions are contained within two-dimensional manifolds of solutions of both negative and positive Hamiltonian.