A Computational Approach to Conley’s Decomposition Theorem
نویسندگان
چکیده
The discrete dynamics generated by a continuous map can be represented combinatorially by an appropriate multivalued map on a discretization of the phase space such as a cubical grid or triangulation. In this paper we provide explicit algorithms and computational complexity bounds for computing dynamical structures for the resulting combinatorial multivalued maps. Specifically we focus on the computation attractor-repeller pairs and Lyapunov functions for Morse decompositions. These discrete Lyapunov functions are weak Lyapunov functions and well-approximate a continuous Lyapunov function for the underlying map.
منابع مشابه
Random Chain Recurrent Sets for Random Dynamical Systems ∗
It is known by the Conley’s theorem that the chain recurrent set CR(φ) of a deterministic flow φ on a compact metric space is the complement of the union of sets B(A) − A, where A varies over the collection of attractors and B(A) is the basin of attraction of A. It has recently been shown that a similar decomposition result holds for random dynamical systems on noncompact separable complete met...
متن کاملThe random case of Conley’s theorem: III. Random semiflow case and Morse decomposition
In the first part of this paper, we generalize the results of the author [25, 26] from the random flow case to the random semiflow case, i.e. we obtain Conley decomposition theorem for infinite dimensional random dynamical systems. In the second part, by introducing the backward orbit for random semiflow, we are able to decompose invariant random compact set (e.g. global random attractor) into ...
متن کاملThe random case of Conley’s theorem
The well-known Conley’s theorem states that the complement of chain recurrent set equals the union of all connecting orbits of the flow φ on the compact metric space X , i.e. X − CR(φ) = ⋃ [B(A) − A], where CR(φ) denotes the chain recurrent set of φ, A stands for an attractor and B(A) is the basin determined by A. In this paper we show that by appropriately selecting the definition of random at...
متن کاملAn Algorithmic Approach to Chain Recurrence
In this paper 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 i...
متن کاملScenario-based modeling for multiple allocation hub location problem under disruption risk: multiple cuts Benders decomposition approach
The hub location problem arises in a variety of domains such as transportation and telecommunication systems. In many real-world situations, hub facilities are subject to disruption. This paper deals with the multiple allocation hub location problem in the presence of facilities failure. To model the problem, a two-stage stochastic formulation is developed. In the proposed model, the number of ...
متن کامل