On the origin and nature of finite-amplitude instabilities in physical systems

Abstract

Finite-amplitude instabilities are ubiquitous, but their theory and precise definitions require clarification. In this work, we discuss the interrelation of various notions connected with finite-amplitude instabilities and offer a precise context for these phenomena. Then we establish a connection between non-normality of linear operators, energy conservation by nonlinear operators and the existence of finite-amplitude instabilities in finite- and infinite-dimensional dynamical systems, both in the conservative and dissipative cases. Such a connection may at first appear counter-intuitive since it relates intrinsically linear and nonlinear phenomena, but it follows naturally from the properties of linear and nonlinear operators when they appear together in a dynamical system. In particular, the main theorem of this communication proves that non-normality is a necessary condition for a finite-amplitude instability. It is demonstrated that this phenomenon is relevant to a wide class of physical systems with energy-conserving nonlinearities.