On Transients, Lyapunov Functions and Turing Instabilities

Abstract

Motivated by the papers [84, 85], this thesis considers the concepts of reactivity, Lyapunov stability and Turing patterns. We introduce the notion of P-reactivity, a new measure for transient dynamics. We extend a result by Shorten and Narendra [108] regarding joint dissipativity for second order systems. We derive an easy verifiable formula that determines systems P-reactivity with respect to a norm induced by the positive definite matrix P. An optimization problem aiming to determine the positive definite P with respect to which a stable system is most reactive is posed and solved numerically for second order systems. The stability radius is adopted as a measure of robustness of joint disspaptivity. We characterise the stability radius of joint dissipativity when the underlying systems are subject to certain specific perturbation structures. A detailed robustness analysis of the Shorten and Narendra conditions is also presented. Using the notion of common Lyapunov function we show that the necessary condition in [85] is a special case of a more powerful (i.e tighter) necessary condition. Specifically, we show that if the linearised reaction matrix and the diffusion matrix share a common Lyapunov function, then Turing instability is not possible. The existence of common Lyapunov functions is readily checked using semi-definite programming. We also further extend this to include more complicated movement mechanisms such as chemotaxis. Unlike the traditional techniques, this new necessary condition can be used to check Turing instability for systems with any dimension and any number of parameters. We apply our new conditions to various models in literature.