Uniform persistence and chain recurrence

Garay, Barnabás M.

doi:10.1016/0022-247x(89)90114-5

Cited by 52 publications

(50 citation statements)

References 4 publications

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“…Theorem 4.3 generalizes Theorem 3,1 of , Theorem 4.2 of Hofbauer and So (1989), Theorem 2 of Garay (1989), and Theorem 3 of Teng and Duan (1990).…”

Section: (I) 3 R Is Uniformly Persistent (Ii) ~ Is Weakly Persistentmentioning

confidence: 99%

“…With such a generalization, a uniform persistence theorem has been established for certain dynamical systems which are not necessarily dissipative. Garay (1989) generalized the main theorem of by using Conley's (1978) theory of invariant sets and a theorem obtained by Ura and Kimura (1960) and Bhatia (1969). Briefly speaking, the UraKimur-Bhatia theorem means that in a local compact space, either an isolated compact invar/ant set is asymptotically stable (positively or negatively) or there exist two points not in the compact set, whose omega or alpha limit sets belong to the compact set, respectively.…”

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confidence: 99%

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Uniform persistence and flows near a closed positively invariant set

1994

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In this paper, the behavior of a continuous flow in the vicinity of a closed positively .invariant subset in a metric space is investigated. The main theorem in this part in some sense generalizes previous results concerning classification of the flow near a compact invariant set in a locally compact metric space which was described by Ura-Kimura (1960) and Bhatia (1969). By applying the obtained main theorem, we are able to prove two persistence theorems. In the first one, several equivalent statements are established, which unify and generalize earlier results based on Liapunov-like functions and those about the equiyalence of weak uniform persistence and uniform persistence. The second theorem generalizes the classical uniform persistence theorems based on analysis of the flow on the boundary by relaxing point dissipativity and invariance of the boundary. Several examples are given which show that our theorems will apply to a wider rarity of ecological models.

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“…Theorem 4.3 generalizes Theorem 3,1 of , Theorem 4.2 of Hofbauer and So (1989), Theorem 2 of Garay (1989), and Theorem 3 of Teng and Duan (1990).…”

Section: (I) 3 R Is Uniformly Persistent (Ii) ~ Is Weakly Persistentmentioning

confidence: 99%

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confidence: 99%

Uniform persistence and flows near a closed positively invariant set

1994

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“…Part of the motivation for considering the connection between chain recurrence and attractors came from several recent papers dealing with dynamics on noncompact spaces [3,5,6, 10].…”

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confidence: 99%

Noncompact chain recurrence and attraction

Hurley¹

1992

Proc. Amer. Math. Soc.

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Abstract.Both this paper and Chain recurrence and attraction in noncompact spaces, [Ergodic Theory Dynamical Systems (to appear)] are concerned with the question of extending certain results obtained by C. Conley for dynamical systems on compact spaces to systems on arbitrary metric spaces. The basic result is the analogue of Conley's theorem that characterizes the chain recurrent set of / in terms of the attractors of / and their basins of attraction. The point of view taken in the above-mentioned paper was that the given metric was of primary importance rather than the topology that it generated. The purpose of this note is to give results that depend on the topology induced by a metric rather than on the particular choice of the metric.The goal of this paper is to extend a theorem of C. Conley from the setting of dynamical systems on compact metric spaces to metric spaces that are only locally compact. Conley's result connects the chain recurrent set of / with the collection of attractors and basins of attraction of /, as follows:Theorem (Conley). If X is a compact metric space and f: X -> X is continuous, then the chain recurrent set of f is the complement of the union of the sets B(A)-A, as A varies over the collection of attractors of f ; here B(A) denotes the basin of attraction of A (the set of points whose omega-limit sets lie in A).Definitions are given in the next section. An earlier paper [7] described one extension of Conley's theorem to the noncompact case. In [7] it was assumed that the distances defined by the given metric on X were themselves important; as a consequence some of the dynamical structures (the analogues of the chain recurrent set and of attractors and their basins) could change if the metric was changed-even if the new metric induced the same topology as the old. There are circumstances where this point of view is appropriate (see [8]), but in general it is preferable to have a theory in which the dynamical structures are invariant under changes of metric (or equivalently, under topological conjugacies). Describing such a theory is the main goal of this paper. There are two main results. The first is the counterpart of Conley's theorem, and the second is the existence of global Lyapunov functions in the case that X is second countable (which is also a generalization of a theorem of Conley).The approach taken in this paper was suggested by John Franks and the author benefitted from conversations with him. Part of the motivation for

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“…In this section we answer the question for "reasonable" vector fields when n = 3. To prove this theorem we use Garay's characterization of uniform persistence in terms of the stable sets of the connected components of R(f, ∂R 3 + ) [8]. Theorem 3 (Garay 1989)…”

Section: Can We Approximate F By a Vector Field With Robust Communities?mentioning

confidence: 99%

Successional stability of vector fields in dimension three

Schreiber

1999

Proc. Amer. Math. Soc.

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Abstract. A topological invariant, the community transition graph, is introduced for dissipative vector fields that preserve the skeleton of the positive orthant. A vector field is defined to be successionally stable if it lies in an open set of vector fields with the same community transition graph. In dimension three, it is shown that vector fields for which the origin is a connected component of the chain recurrent set can be approximated in the C 1 Whitney topology by a successionally stable vector field.

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Uniform persistence and chain recurrence

Cited by 52 publications

References 4 publications

Uniform persistence and flows near a closed positively invariant set

Uniform persistence and flows near a closed positively invariant set

Noncompact chain recurrence and attraction

Successional stability of vector fields in dimension three

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