Coupled skinny baker's maps and the Kaplan-Yorke conjecture
| Autor: | Maik Gröger, Brian R. Hunt |
|---|---|
| Jazyk: | angličtina |
| Rok vydání: | 2013 |
| Předmět: |
Lyapunov function
Class (set theory) Pure mathematics Conjecture Dynamical systems theory Applied Mathematics Dimension (graph theory) General Physics and Astronomy Statistical and Nonlinear Physics Dynamical Systems (math.DS) Coupling (probability) Kaplan–Yorke conjecture Measure (mathematics) symbols.namesake FOS: Mathematics symbols 37C45 (Primary) 37C40 37D45 37C20 28C20 (Secondary) Mathematics - Dynamical Systems Mathematical Physics Mathematics |
| Popis: | The Kaplan-Yorke conjecture states that for "typical" dynamical systems with a physical measure, the information dimension and the Lyapunov dimension coincide. We explore this conjecture in a neighborhood of a system for which the two dimensions do not coincide because the system consists of two uncoupled subsystems. We are interested in whether coupling "typically" restores the equality of the dimensions. The particular subsystems we consider are skinny baker's maps, and we consider uni-directional coupling. For coupling in one of the possible directions, we prove that the dimensions coincide for a prevalent set of coupling functions, but for coupling in the other direction we show that the dimensions remain unequal for all coupling functions. We conjecture that the dimensions prevalently coincide for bi-directional coupling. On the other hand, we conjecture that the phenomenon we observe for a particular class of systems with uni-directional coupling, where the information and Lyapunov dimensions differ robustly, occurs more generally for many classes of uni-directionally coupled systems (also called skew-product systems) in higher dimensions. 33 pages, 3 figures |
| Databáze: | OpenAIRE |
| Externí odkaz: |
|