Chaotic braided solutions via rigorous numerics: chaos in the Swift-Hohenberg equation

We prove that the stationary Swift-Hohenberg equation has chaotic dynamics on a critical energy level for a large (continuous) range of parameter values. The first step of the method relies on a computer assisted, rigorous, continuation method to prove the existence of a periodic orbit with certain geometric properties. The second step is topological: we use this periodic solution as a skeleton, through which we braid other solutions, thus forcing the existence of infinitely many braided periodic orbits. A semiconjugacy to a subshift of finite type shows that the dynamics is chaotic. © 2008 Society for Industrial and Applied Mathematics.