Krechetnikov, R. and Marsden, J. E. (2009) On the origin and nature of finite-amplitude instabilities in physical systems. Journal of Physics A: Mathematical and Theoretical, 42 (41). Art. No. 412004. ISSN 1751-8113. http://resolver.caltech.edu/CaltechAUTHORS:20091022-143705462
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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.
|Additional Information:||Copyright © Institute of Physics and IOP Publishing Limited 2009. Received 26 May 2009, in final form 10 September 2009. Published 29 September 2009. Print publication: Issue 41 (16 October 2009). RK is grateful to Edgar Knobloch for a fruitful discussion on the subject. The authors also acknowledge the constructive input of anonymous referees.|
|Classification Code:||PACS numbers: 47.15.Fe, 47.27.Cn|
|Official Citation:||On the origin and nature of finite-amplitude instabilities in physical systems R Krechetnikov and J E Marsden 2009 J. Phys. A: Math. Theor. 42 412004 (18pp) doi: 10.1088/1751-8113/42/41/412004|
|Usage Policy:||No commercial reproduction, distribution, display or performance rights in this work are provided.|
|Deposited By:||George Porter|
|Deposited On:||22 Oct 2009 21:55|
|Last Modified:||26 Dec 2012 11:30|
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