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Archive for Rational Mechanics and Analysis

, Volume 197, Issue 3, pp 1033–1051 | Cite as

Computer-Assisted Methods for the Study of Stationary Solutions in Dissipative Systems, Applied to the Kuramoto–Sivashinski Equation

  • Gianni ArioliEmail author
  • Hans Koch
Article

Abstract

We develop some computer-assisted techniques for the analysis of stationary solutions of dissipative partial differential equations, of their stability, and of their bifurcation diagrams. As a case study, these methods are applied to the Kuramoto–Sivashinski equation. This equation has been investigated extensively, and its bifurcation diagram is well known from a numerical point of view. Here, we rigorously describe the full graph of solutions branching off the trivial branch, complete with all secondary bifurcations, for parameter values between 0 and 80. We also determine the dimension of the unstable manifold for the flow at some stationary solution in each branch.

Keywords

Bifurcation Diagram Open Neighborhood Bifurcation Point Unstable Manifold Positive Real Part 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Supplementary material

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Copyright information

© Springer-Verlag 2010

Authors and Affiliations

  1. 1.Dipartimento di Matematica and MOXPolitecnico di MilanoMilanItaly
  2. 2.Department of MathematicsUniversity of Texas at AustinAustinUSA

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