Topological Methods for ODE's: Symplectic Differential Systems

Abstract

Taking inspiration from the geometrical ideas behind the classical Sturmian theory for ordinary differential equations in â„, in this paper we review some recent topological techniques to study some properties of systems of ODE's in higher dimension. More specifically, we will discuss the notion of Moslov index for symplectic differential systems, i.e., those systems of differential equations in â„ⁿ âŠ• â„^n* whose flow preserves the canonical symplectic form. Such systems appear naturally in association with the Jacobi equation along a semi-Riemannian geodesic, or, more generally, with solutions of possibly time-dependent Hamiltonians on symplectic manifolds. In this paper we review some recent results in the theory of symplectic differential systems, with special emphasis on those systems arising form semi-Riemannian geometry.