A family index theorem for periodic Hamiltonian systems and bifurcation

TL;DR

This paper establishes an index theorem for families of linear periodic Hamiltonian systems, linking it to classical results and applying it to analyze bifurcations of periodic solutions in nonlinear systems.

Contribution

It introduces a new index theorem for families of Hamiltonian systems, extending classical results and enabling bifurcation analysis of nonlinear systems.

Findings

Proves an index theorem analogous to Atiyah-Singer for Hamiltonian systems

Compares the new theorem with Salamon-Zehnder result for one-parameter families

Applies the theorem to study bifurcation of periodic solutions

Abstract

We prove an index theorem for families of linear periodic Hamiltonian systems, which is reminiscent of the Atiyah-Singer index theorem for selfadjoint elliptic operators. For the special case of one-parameter families, we compare our theorem with a classical result of Salamon and Zehnder. Finally, we use the index theorem to study bifurcation of branches of periodic solutions for families of nonlinear Hamiltonian systems.