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

Semi-invariants and relative characteristic functions extend to nonlinear systems the concept of eigenvector–eigenvalue 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 Bendixson, I. 1901. Sur les courbes définies par des équations différentielles. Acta Mathematica, 24: 1–88.  [Google Scholar]), ‘Sur les courbes définies par des équations différentielles’, Acta Mathematics, 24, 1–88) is given.