Bifurcations for Hamiltonian systems

TL;DR

This paper develops new bifurcation results for Hamiltonian boundary value problems using advanced variational and saddle point reduction techniques, extending Rabinowitz's bifurcation theorem.

Contribution

It introduces generalized bifurcation theorems applicable to Hamiltonian systems, expanding the theoretical framework for analyzing solution bifurcations.

Findings

Proved new bifurcation results for four types of Hamiltonian boundary value problems.

Extended Rabinowitz's bifurcation theorem to more general Hamiltonian systems.

Identified conditions under which bifurcations occur in Hamiltonian boundary value problems.

Abstract

With the dual variational principle and the saddle point reduction we use the abstract bifurcation theory recently developed by author in previous work to prove many new bifurcation results for solutions of four types of Hamiltonian boundary value problems nonlinearly depending on parameters. The most interesting and important among them are those alternative results which can only be proved with our generalized versions of the famous Rabinowitz's alternative bifurcation theorem.