Bifurcations for Hamiltonian systems
Guangcun Lu
TL;DR
The paper advances bifurcation theory for parameter-dependent Hamiltonian systems by combining dual variational principles, saddle-point reduction, and Maslov-type index analysis. It proves myriad new bifurcation results for non-autonomous boundary value problems, generalized periodic orbits, brake orbits, and Hamiltonian paths, including Rabinowitz- and Fadell–Rabinowitz-type alternatives. Central to the approach are index crossing criteria derived from the fundamental solution $\gamma_\lambda$, the Maslov-type indices $(i_{\tau,M}, \nu_{\tau,M})$, and, in several cases, the Conley–Zehnder index; these yield existence and multiplicity results through finite-dimensional reductions and equivariant variational methods. The work thus provides a comprehensive, index-theoretic framework for bifurcations in a broad class of Hamiltonian boundary value problems with rich symmetry structures, with potential applications to rotating periodic orbits, brake orbits, and orbit bifurcations in autonomous and non-autonomous settings.
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.