Dynamical Systems at Queen Mary

Stability analysis of nonlinear systems is usually based on Lyapunov functions (nonlocal methods) or first order approximation (local methods). Control design which is based on linearization around a desired position is far simpler in general than nonlocal design. However, there are some efficient qualitative global tools based on linearization around solutions of dynamical systems. One of these tools is the classical Bendixson criterion (or divergence test) which gives sufficient conditions for the nonexistence of periodic orbits.

A classical proof of this statement is based on the divergence theorem and cannot be generalized to the higher dimensional case. The main purpose of this talk is to present one of the possible generalizations of the Bendixson result for the case of arbitrary dimension.

In this talk we investigate this question by a method which allows to estimate the Hausdorff dimension of invariant compact sets. The conditions presented in the talk are formulated in terms of inequalities involving two eigenvalues of some matrix pencil.