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Limit set

About: Limit set is a research topic. Over the lifetime, 840 publications have been published within this topic receiving 14500 citations.


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743 citations

Journal ArticleDOI
TL;DR: A survey of time-reversal symmetry in dynamical systems can be found in this paper, where the relation of time reversible dynamical sytems to equivariant and Hamiltonian systems is discussed.

483 citations

Journal ArticleDOI
TL;DR: In this article, it was shown that limit sets of such systems cannot be more complicated than invariant sets of systems of one lower dimension, and orthogonal projection along any positive direction maps a limit set homeomorphically and equivariantly onto an invariant set of a Lipschitz vector field in a hyperplane.
Abstract: A vector field in n-space determines a competitive (or cooperative) system of differential equations provided all the off-diagonal terms of its Jacobian matrix are nonpositive (or nonnegative). The principal result is that limit sets of such systems cannot be more complicated than invariant sets of systems of one lower dimension. In fact orthogonal projection along any positive direction maps a limit set homeomorphically and equivariantly onto an invariant set of a Lipschitz vector field in a hyperplane. Limit sets are nowhere dense, unknotted and unlinked. In dimension 2 every trajectory is eventually monotone. In dimension 3 a compact limit set which does not contain an equilibrium is a closed orbit or a cylinder of closed orbits. Introduction. One of the most interesting questions to ask about a dynamical system is: what is the long-run behavior of its trajectories? In many systems it is natural to expect, or at least hope, that almost all trajectories either converge to an equilibrium or asymptotically approach a closed orbit (= periodic trajectory). Unfortu- nately there are many systems that not only lack this convenient property, but cannot even be approximated by systems that have it. Such systems are often said to be "chaotic" or to possess "strange attractors". To make matters worse, it is very hard to discover the long-run behavior of any but the simplest systems. Research on this problem has bifurcated into two quite different methodologies. A great deal of recent work has gone toward exploring the consequences of various assumptions about the large scale structure of the system, e.g., hyperbolicity of the nonwandering set, structural stability, ergodicity, and so forth. The basic examples come from geometry and physics; the mathematical tech- niques tend to be topological. For a recent overview of this work see Smale (15, Chapt. I).

389 citations

Journal ArticleDOI
TL;DR: In this paper, smooth dynamical systems having contracting and expanding invariant foliations (of not necessarily complementary dimensions) are investigated, and the K-property is established for such systems under additional assumptions.
Abstract: Smooth dynamical systems having contracting and expanding invariant foliations (of not necessarily complementary dimensions) are investigated. Ergodicity and the K-property are established for such dynamical systems under additional assumptions.

381 citations

Journal ArticleDOI
TL;DR: A theoretical analysis of the mathematical mechanisms underlying this complexity from the viewpoint of modern dynamical systems theory on a density-dependent Leslie model with two age classes and the existence of a "strange attractor" is demonstrated.
Abstract: The dynamics of density-dependent population models can be extraordinarily complex as numerous authors have displayed in numerical simulations. Here we commence a theoretical analysis of the mathematical mechanisms underlying this complexity from the viewpoint of modern dynamical systems theory. After discussing the chaotic behavior of one-dimensional difference equations we proceed to illustrate the general theory on a density-dependent Leslie model with two age classes. The pattern of bofurcations away from the equilibrium point is investigated and the existence of a "strange attractor" is demonstrated--i.e. an attracting limit set which is neither an equilibrium nor a limit cycle. Near the strange attractor the system exhibits essentially random behavior. An approach to the statical analysis of the dynamics in the chaotic regime is suggested. We then generalize our conclusions to higher dimensions and continuous models (e.g. the nonlinear von Foerster equation).

376 citations


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Performance
Metrics
No. of papers in the topic in previous years
YearPapers
20236
202230
202122
202015
201919
201822