An algorithmic approach to chain recurrence. (English) Zbl 1099.37010
Summary: We give a new definition of the chain recurrent set of a continuous map using finite spatial discretizations. This approach allows for an algorithmic construction of isolating blocks for the components of Morse decompositions which approximate the chain recurrent set arbitrarily closely as well as discrete approximations of Conley’s Lyapunov function. This is a natural framework in which to develop computational techniques for the analysis of qualitative dynamics including rigorous computer-assisted proofs.
MSC:
| 37B35 | Gradient-like behavior; isolated (locally maximal) invariant sets; attractors, repellers for topological dynamical systems |
| 37B20 | Notions of recurrence and recurrent behavior in topological dynamical systems |
| 37B25 | Stability of topological dynamical systems |
| 37B30 | Index theory for dynamical systems, Morse-Conley indices |
| 37C70 | Attractors and repellers of smooth dynamical systems and their topological structure |
| 37M99 | Approximation methods and numerical treatment of dynamical systems |
