Semi-invariants and their use for stability analysis of planar systems

Abstract

Semi-invariants and relative characteristic functions extend to nonlinear systems the concept of eigenvectoreigenvalue pair for linear systems, and are, therefore, very useful to depict the behaviour of the system. In this article, semi-invariants are used to construct explicitly Lyapunov functions useful for studying the stability of the origin, for continuous-time systems. Moreover, using well-known tools from differential geometry such as (orbital) symmetries, it is shown how semi-invariants can be found for several classes of systems. Important connections with centre manifold theory are pointed out. By using the proposed general techniques, a new proof of a result claimed to Bendixson (Bendixson, I. (1901), 'Sur les courbes de´finies par des e´quations diffe´rentielles', Acta Mathematics, 24, 1-88) is given.