Bifurcation of critical points for continuous families of C^2 functionals of Fredholm type

TL;DR

This paper investigates how the spectral flow of Hessians in continuous families of C^2 functionals of Fredholm type indicates bifurcation points, providing estimates for their number and applications to Hamiltonian systems.

Contribution

It establishes a link between spectral flow non-vanishing and bifurcation, offering methods to estimate bifurcation points and their count in parameterized families.

Findings

Spectral flow non-vanishing implies bifurcation of critical points.

Provides bounds on the number of bifurcation points.

Applies results to periodic orbits in Hamiltonian systems.

Abstract

Given a continuous family of C^2 functionals of Fredholm type, we show that the non-vanishing of the spectral flow for the family of Hessians along a known (trivial) branch of critical points not only entails bifurcation of nontrivial critical points but also allows to estimate the number of bifurcation points along the branch. We use this result for several parameter bifurcation, estimating the number of connected components of the complement of the set of bifurcation points and apply our results to bifurcation of periodic orbits of Hamiltonian systems. By means of a comparison principle for the spectral flow, we obtain lower bounds for the number of bifurcation points of periodic orbits on a given interval in terms of the coefficients of the linearization.