Isolating Segments, Fixed Point Index, and Symbolic Dynamics

Abstract

An extension of the recently introduced Srzednicki Wo jcik method for detecting chaotic dynamics in periodically forced ordinary differential equations is presented. As an application of the method we construct a topological model for the planar equationand we show by a continuation argument that the symbolic dynamics on three symbols for the topological model continues to Eq. (1) for 0<} 0.495. 2000Academic Press

Covering relations for multidimensional dynamical systems

“…These new type of relations turn out to be very useful in studying systems exhibiting time-reversal symmetry (see [1,17]). Even in the case of direct coverings, which is considered in [15] (in a different, more abstract language), we have the advantage of providing a quite elegant geometric approach, which allows one to obtain deeper and stronger results (see [16]), when compared to the one based on the Lefschetz Fixed Point Theorem.…”

Rigorous Numerics for Partial Differential Equations: The Kuramoto—Sivashinsky Equation

“…First, finding the appropriate isolating neighborhoods is more complicated in these cases and our goal here is to emphasize the fundamental ideas associated with the methods. The second, and more important point, is that a straightforward application of the earlier numerical methods would lead to large computations-which we believe can be avoided by alternative methods (see, e.g., [25]). This latter point is currently being investigated.…”

Topological entropy for local processes

“…Theorem 11. Let ϕ be a local process on a metric space (X, d), let K be a compact subset of X and let s, t ∈ R, s t. In that case the following inequalities hold (12) and furthermore…”

“…This is the main difficulty, which makes direct adaptation of the definition of topological entropy stated previously in the setting of continuous dynamical systems [11] rather problematic. Previously, entropy of process (or more generally, complexity of the dynamics) was viewed in terms of Poincaré maps [12] or Poincaré sections [13]. This approach was quite natural, since assumption that complicated dynamics on sections should reflect complicated dynamics of the local process is intuitively reasonable.…”