Bifurcations of Periodic Solutions Satisfying the Zero-hamiltonian Constraint in Fourth-order Differential Equations

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 generalisation 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, with further restrictions, obtain results regarding the global nature of the resulting bifurcating continua.