Dissipation-induced instabilities in finite dimensions

Krechetnikov, R. and J. E. Marsden

Rev. of Mod. Phys., 79, (2007), 519-553

Abstract:

The goal of this work is to introduce a coherent theory of the counterintuitive phenomena of dynamical destabilization under the action of dissipation. While the existence of one class of dissipation-induced instabilities was known to Sir Thomson Lord Kelvin , it was not realized until recently that there is another major type of these phenomena hinted at by one of Merkin’s theorems; in fact, these two cases exhaust all the generic possibilities. The theory grounded on the Thomson-Tait-Chetayev and Merkin theorems and on the geometric understanding introduced in this paper leads to the conclusion that ubiquitous dissipation is one of the paramount mechanisms by which instabilities develop in nature. Along with a historical review, the main theoretical achievements are put in a general context, thus unifying the current knowledge in this area and the multitude of relevant physical problems scattered over a vast literature. This general view also highlights the striking connection to various areas of mathematics. To appeal to the reader’s intuition and experience, a large number of motivating examples are provided. The paper contains some new unpublished results and insights, and, finally, open questions are formulated to provide an impetus for future studies. While this review focuses on the finite-dimensional case, where the theory is relatively complete, a brief discussion of the current state of knowledge in the infinite-dimensional case, typified by partial differential equations, is also given.