An extended method of Lyapunov for the analysis of dynamical systems with weakly attracting equilibria

Abstract

Several types of stability definitions and criteria have been developed to describe the behaviour of dynamical systems, of which Lyapunov stability is perhaps the most impactful concept and method. It enables us to determine asymptotic properties of solutions without solving the underlying differential equations. The approach, historically and methodologically, is highly germane to cases when the motions of interest are attracting to some invariant sets, orbits, or equilibria. Such motions are often compatible with the notion of Lyapunov stability, but many relevant dynamical systems exhibit motions that are attracting or approaching some relevant domains of the system's state space, while such motions and corresponding domains do not conform with the classical notion of Lyapunov stability.

An example is an equilibrium which is both an attractor and a repeller, whose attractor basin has a positive Lebesgue measure. Such equilibria are termed weak attractors in a Milnor sense. Small perturbations to these may lead to long transients or ghost attractors -- a property which is important for the understanding and modelling of processes in ecology, evolution, and chemical kinetics. Unfortunately, since these motions are not stable in the sense of Lyapunov, the conventional analysis machinery based on the method of Lyapunov for the identification and characterization of these motions does not apply.

The aim of the thesis is to develop Lyapunov-like criteria of weak attractivity of unstable equilibria in systems of non-linear autonomous coupled ordinary differential equations. We showed that one can determine the weak attractivity of an equilibrium by examining Jacobians and Hessians of the vector fields that can be derived from the original equations. The conditions are based on previous studies (Gorban, Tyukin, Steur $ \& $ Nijmeijer, SIAM Journal on Control and Optimization, 2013; Gorban, Tyukin, Nijmeier, IEEE CDC, 2014). Applications include mathematical modelling of metastable processes in ecology, evolution, and kinetics, as well as inverse problems and parameter estimation/fitting of systems of non-linear ODEs with nonlinear dependence on parameters.