Abstract
We describe the basic lattice structures of attractors and repellers in dynamical systems.The structure of distributive lattices allows for an algebraic treatment of gradient-like dynamics in general dynamical systems, both invertible and noninvertible. We separate those properties which rely solely on algebraic structures from those that require some topological arguments, in order to lay a foundation forthe development of algorithms to manipulate these structures computationally.
Highlights
As is made clear by Conley [6], attractors are central to our theoretical understanding of global nonlinear dynamics in that they form the basis for robust decompositions of gradient-like structures
It is not surprising that attractors appear as standard topics in nonlinear dynamics [15], and that their structure has been studied in a wide variety of settings
We make no attempt to provide even a cursory list of references on the subject, but we do remark that the theory developed in [1] applies in a general setting in which the dynamics is generated by relations
Summary
As is made clear by Conley [6], attractors are central to our theoretical understanding of global nonlinear dynamics in that they form the basis for robust decompositions of gradient-like structures. Our goal is to develop a computational theory, which raises the following fundamental question: Given a finite sublattice of attractors, does there exists a finite sublattice of attracting neighborhoods such that Inv(·, φ) produces a lattice isomorphism? This section, which consists of pure point set topological arguments, is followed by Section 4, which contains a demonstration that attractors and repellers have the algebraic structure of bounded, distributive lattices. We present a proof of this thorem by first proving a general result, Theorem 5.13, concerning the lifting of lattice homomorphisms over bounded, distributive lattices We apply this theorem to the lifting of a finite sublattice of attractors (or repellers) to a lattice of attracting (or repelling) neighborhoods.
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