Generalized Maslov indices for non-Hamiltonian systems

TL;DR

This paper generalizes the Maslov index to non-Hamiltonian systems using Maslov-Arnold spaces, enabling topological analysis of eigenvalue problems and bifurcations like Turing instability in reaction-diffusion systems.

Contribution

It introduces Maslov-Arnold spaces to extend index theory beyond Hamiltonian systems, broadening the applicability of topological methods in dynamical systems analysis.

Findings

Defined a new class of topological spaces called Maslov-Arnold spaces.

Constructed hyperplane Maslov-Arnold spaces dense in the Grassmannian.

Applied the generalized index to interpret Turing bifurcations topologically.

Abstract

We extend the definition of the Maslov index to a broad class of non-Hamiltonian dynamical systems. To do this, we introduce a family of topological spaces--which we call Maslov-Arnold spaces--that share key topological features with the Lagrangian Grassmannian, and hence admit a similar index theory. This family contains the Lagrangian Grassmannian, and much more. We construct a family of examples, called hyperplane Maslov-Arnold spaces, that are dense in the Grassmannian, and hence are much larger than the Lagrangian Grassmannian (which is a submanifold of positive codimension). The resulting index is then used to study eigenvalue problems for non-symmetric reaction-diffusion systems. A highlight of our analysis is a topological interpretation of the Turing instability: the bifurcation that occurs as one increases the ratio of diffusion coefficients corresponds to a change in the generalized Maslov index.