A generalized index theory for non-Hamiltonian system

TL;DR

This paper introduces a new topological invariant called the degree-index to extend Morse index theory to non-Hamiltonian systems, establishing an equality with the Morse index and applying it to reaction-diffusion models.

Contribution

It develops the degree-index as a substitute for the Maslov index in non-Hamiltonian systems and proves an index equality in this broader context.

Findings

Degree-index is defined via Brouwer degree of a determinant map.

Proved the Morse index equals the degree-index in non-selfadjoint systems.

Applied the theory to 1D reaction-diffusion systems.

Abstract

Morse index theory provides an elegant and useful tool for describing several aspects of a Lagrangian system in terms of its variational properties. In the classical framework it provides an equality between the spectral properties of a second order selfadjoint differential operator arising by the second variation of the Lagrangian action functional and the oscillation properties of the space of solutions of the associated boundary value problem. These oscillation properties are described by means of the Maslov index, which is a symplectic invariant intimately related to the Hamiltonian nature of the problem. Driven by the desire to get an analogous result in the non-Hamiltonian context for the investigation of the dynamical properties of dissipative systems, in this paper we start to introduce a new topological invariant, the so-called degree-index, defined in terms of the Brouwer degree of a suitable determinant map of a boundary matrix, which provides one possible substitute of the Maslov index in this non-Hamiltonian framework and finally we prove the equality between the Morse index and the degree-index in this non-selfadjoint setting through a new abstract trace formula. In the last part of the paper we apply our theoretical results to some 1D reaction-diffusion systems.