Probing a subcritical instability with an amplitude expansion: An exploration of how far one can get

Paul Becherer, Alexander N. Morozov, Wim van Saarloos

Research output: Contribution to journalArticlepeer-review

Abstract

We explore methods to locate subcritical branches of spatially periodic solutions in pattern forming systems with a nonlinear finite-wavelength instability. We do so by means of a direct expansion in the amplitude of the linearly least stable mode about the appropriate reference state which one considers. This is motivated by the observation that for some equations fully nonlinear chaotic dynamics has been found to be organized around periodic solutions that do not simply bifurcate from the basic (laminar) state. We apply the method to two model equations, a subcritical generalization of the Swift-Hohenberg equation and a novel extension of the Kuramoto-Sivashinsky equation that we introduce to illustrate the abovementioned scenario in which weakly chaotic subcritical dynamics is organized around periodic states that bifurcate "from infinity" and that can nevertheless be probed perturbatively. We explore the reliability and robustness of such an expansion, with a particular focus on the use of these methods for determining the existence and approximate properties of finite-amplitude stationary solutions. Such methods obviously are to be used with caution: the expansions are often only asymptotic approximations, and if they converge their radius of convergence may be small. Nevertheless, expansions to higher order in the amplitude can be a useful tool to obtain qualitatively reliable results. (C) 2009 Published by Elsevier B.V.

Original languageEnglish
Pages (from-to)1827-1840
Number of pages14
JournalPhysica D: Nonlinear Phenomena
Volume238
Issue number18
DOIs
Publication statusPublished - Sep 2009

Keywords

  • Subcritical instabilities
  • Amplitude expansion
  • KURAMOTO-SIVASHINSKY EQUATION
  • PLANE POISEUILLE FLOW
  • EXACT COHERENT STRUCTURES
  • SWIFT-HOHENBERG EQUATION
  • UNSTABLE PARALLEL FLOWS
  • NON-LINEAR MECHANICS
  • SHEAR FLOWS
  • SPATIOTEMPORAL INTERMITTENCY
  • PATTERN SELECTION
  • WAVE DISTURBANCES

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