The chain recurrent set, attractors, and explosions
Louis Block,John E. Franke +1 more
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In this paper, it was shown that for any continuous map f of a compact space to itself, if the non-wandering set equals the chain recurrent set then f does not permit Ω-explosions.Abstract:
Charles Conley has shown that for a flow on a compact metric space, a point x is chain recurrent if and only if any attractor which contains the & ω-limit set of x also contains x. In this paper we show that the same statement holds for a continuous map of a compact metric space to itself, and additional equivalent conditions can be given. A stronger result is obtained if the space is locally connected.It follows, as a special case, that if a map of the circle to itself has no periodic points then every point is chain recurrent. Also, for any homeomorphism of the circle to itself, the chain recurrent set is either the set of periodic points or the entire circle. Finally, we use the equivalent conditions mentioned above to show that for any continuous map f of a compact space to itself, if the non-wandering set equals the chain recurrent set then f does not permit Ω-explosions. The converse holds on manifolds.read more
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Journal ArticleDOI
Chain Transitivity, Attractivity, and Strong Repellors for Semidynamical Systems
TL;DR: Some properties of internally chain transitive sets for continuous maps in metric spaces are presented in this paper, where applications are made to attractivity, convergence, strong repellors, uniform persistence, and permanence.
Journal ArticleDOI
Uniform persistence and repellors for maps
Josef Hofbauer,Joseph W.-H. So +1 more
TL;DR: In this article, conditions for an isolated invariant set M of a map to be a repellor were established in terms of the stable set of M and refined in two ways by considering (i) a Morse decomposition for M, and (ii) the invariantly connected components of the chain recurrent set.
The fundamental theorem of dynamical systems
TL;DR: The Fundamental Theorem of Dynamical Systems (FTHS) as discussed by the authors is a theorem of Charles Conley concerning the decomposition of spaces on which dynamical systems can be defined.
Journal ArticleDOI
Dynamics of a Periodically Pulsed Bio-reactor Model
Hal L. Smith,Xiao-Qiang Zhao +1 more
TL;DR: In this paper, global attractivity and uniform persistence are established for both single species growth and two species competition in a periodically pulsed bio-reactor model in terms of principal eigenvalues of the periodic-parabolic eigenvalue problem by appealing to the theories of monotone discrete dynamical systems.
Journal ArticleDOI
Global attractions in competitive systems
John E. Franke,Abdul-Aziz Yakubu +1 more
TL;DR: In this article, the authors established the ecological principle of mutual exclusion as a mathematical theorem in a discrete system modeling the interactions of pioneer species, and proved that mutual exclusion can be expressed as a deterministic function.
References
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MonographDOI
Isolated Invariant Sets and the Morse Index
TL;DR: On stable properties of the solution set of an ordinary differential equation, see as mentioned in this paper and the Morse index continuuation bibliography for a complete survey of the literature on flow stability and flow properties.
Book
Differentiable dynamics;: An introduction to the orbit structure of diffeomorphisms
TL;DR: The subject of differentiable dynamical systems in the form recently developed by the group of mathematicians associated with S. Smale and M. Peixoto in the United States and with Ja.
Book ChapterDOI
Lectures on Dynamical Systems
TL;DR: A basic question in the theory of dynamical systems is to study the asymptotic behaviour of orbits as mentioned in this paper, which has led to the development of many different subjects in mathematics, including ergodic theory, hamiltonian mechanics, and qualitative theory of differential equations.