We find a bound on the parameter range for which the Henon map exhibits a complete binary horseshoe as well as a subshift of finite type. We use this limit to assign global symbols to orbits and use continuation from the limit to study their bifurcations. © 2008 Society for Industrial and Applied Mathematics. Abstract: Chaotic dynamics can be effectively studied by continuation from an anti-integrable limit. Using this structure theorem on innite-to-one factor. A semiconjugacy to a subshift of finite type shows that the dynamics is chaotic. subshift of nite type is a composition of a nite-to-one factor code and a class degree one factor code. 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. 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. In the next section the basic definitions of CAs and the classification of Wolfram are given. In fact, we have shown that simple subshift rules are chaotic on the set 2:0, and that alternating and double alternating subshift rules are chaotic on 2:1 or 2:2. (iii) Any sensitive subshift of finite type is cofinitely sensitive. The most interesting property of these rules is their chaotic nature. © 2008 Society for Industrial and Applied Mathematics.ĪB - We prove that the stationary Swift-Hohenberg equation has chaotic dynamics on a critical energy level for a large (continuous) range of parameter values. 4 that sensitivity is redundant in Devaneys definition of chaos) Theorem 1. A semiconjugacy to a subshift of finite type shows that the dynamics is chaotic. (2)When F f11g, X(F) is a shift of nite type called the golden mean shift. Some examples (and non-examples): (1)The full shift is a shift of nite type, corresponding to F. N2 - We prove that the stationary Swift-Hohenberg equation has chaotic dynamics on a critical energy level for a large (continuous) range of parameter values. Jenkinson J3 established the analogous fact for functions of summable variation defined on finite alphabet subshifts of finite type, and Bousch B2 further. A subshift X Q n2Z Ais called a subshift of nite type (or usually just a shift of nite type) if there exists a nite set of words Fsuch that X X(F). map g: X X can be analyzed by using certain naturally related subshifts of finite type. T1 - Chaotic braided solutions via rigorous numerics: chaos in the Swift-Hohenberg equation Introduction to Applied Nonlinear Dynamical Systems and Chaos.
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