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arXiv · 2006.14517

Generalized Maslov indices for non-Hamiltonian systems

Thomas John Baird·Paul Cornwell·Graham Cox·Christopher Jones·Robert Marangell
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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.

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Thomas John Baird, Paul Cornwell, Graham Cox, Christopher Jones, Robert Marangell. 2020-06-25. Generalized Maslov indices for non-Hamiltonian systems. https://arxiv.org/abs/2006.14517

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