On Conley index theory for non-smooth dynamical systems (Q5933773)

The paper presents a version of the Conley index for differential inclusions of the form \(\dot x\in F(x)\) a.e., where \(F\) is an upper semicontinuous (in some strong sense) multi-valued vector-field on \(\mathbb R^n\) with closed convex values satisfying some boundedness condition. The index of a suitably defined isolated invariant set is equal to the usual Conley index of the corresponding invariant set of the equation \(\dot x=f(x)\), where \(f\) is a smooth ``almost section'' sufficiently close to \(F\). The index is well-defined and satisfies standard properties. Some theorems on existence of bifurcation points are proved as an application of the index.