On Conley index theory for non-smooth dynamical systems. (English) Zbl 0968.37006

Kunze, M.; Küpper, T.; Li, Y.

Differ. Integral Equ. 13, No. 4-6, 479-502 (2000).

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.

Reviewer: R.Srzednicki (Kraków)

MSC:

37B30	Index theory for dynamical systems, Morse-Conley indices
34A60	Ordinary differential inclusions
70F40	Problems involving a system of particles with friction

Keywords:

isolated invariant set; Conley index; differential inclusion