Middle Point Reduction of the Chain-recurrent Set

Describing dynamical systems requires capability to isolate periodic behaviour. In Lyapunov's theory, the qualitative behaviour of a dynamical system given by a differential equation can be described by a scalar function that decreases along solutions: the Complete Lyapunov Function. The chain-recurrent set will produce constant values of an associated complete Lyapunov function and zero values of its orbital derivative. Recently, we have managed to isolate the chain-recurrent set of different dynamical systems in 2- and 3- dimensions. An overestimation, however, is always obtained. In this paper, we present a method to reduce such overestimation based on geometrical middle points for 2-dimensional systems.