scholarly journals The generalized recurrent set and strong chain recurrence

2016 ◽  
Vol 38 (2) ◽  
pp. 788-800 ◽  
Author(s):  
JIM WISEMAN
Keyword(s):  
Chain Recurrence ◽  
Chain Recurrent Set ◽  
Recurrent Set

Fathi and Pageault have recently shown a connection between Auslander’s generalized recurrent set$\text{GR}(f)$and Easton’s strong chain recurrent set. We study$\text{GR}(f)$by examining that connection in more detail, as well as connections with other notions of recurrence. We give equivalent definitions that do not refer to a metric. In particular, we show that$\text{GR}(f^{k})=\text{GR}(f)$for any$k>0$, and give a characterization of maps for which the generalized recurrent set is different from the ordinary chain recurrent set.

SIMPLE AND COMPLEX DYNAMICS FOR ONE-DIMENSIONAL MANIFOLD MAPS

1995 ◽  
Vol 05 (05) ◽  
pp. 1351-1355
Author(s):  
VLADIMIR FEDORENKO
Keyword(s):  
Topological Entropy ◽  
Complex Dynamics ◽  
Periodic Points ◽  
Dimensional Manifold ◽  
One Dimensional ◽  
Circle Maps ◽  
Interval Maps ◽  
Chain Recurrent Set ◽  
Recurrent Set

We give a characterization of complex and simple interval maps and circle maps (in the sense of positive or zero topological entropy respectively), formulated in terms of the description of the dynamics of the map on its chain recurrent set. We also describe the behavior of complex maps on their periodic points.

Chain recurrence and attraction in non-compact spaces

1991 ◽  
Vol 11 (4) ◽  
pp. 709-729 ◽  
Author(s):  
Mike Hurley
Keyword(s):  
Metric Space ◽  
Basins Of Attraction ◽  
The Other ◽  
Compact Metric Space ◽  
Chain Recurrence ◽  
Fundamental Interest ◽  
Compact Spaces ◽  
Continuous Map ◽  
Chain Recurrent Set ◽  
Recurrent Set

AbstractIn the study of a dynamical systemf:X→Xgenerated by a continuous mapfon a compact metric spaceX, thechain recurrent setis an object of fundamental interest. This set was defined by C. Conley, who showed that it has two rather different looking, but equivalent, definitions: one given in terms of ‘approximate orbits’ through individual points (pseudo-orbits, or ε-chains), and the other given in terms of the global structure of the class of ‘attractors’ and ‘basins of attraction’ off. The first of these definitions generalizes directly to dynamical systems on any metric space, compact or not. The main purpose of this paper is to extend the second definition to non-compact spaces in such a way that it remains equivalent to the first.

scholarly journals Chain recurrence and discretisation

1997 ◽  
Vol 55 (1) ◽  
pp. 63-71 ◽  
Author(s):  
Barnabas M. Garay ◽  
Josef Hofbauer
Keyword(s):  
Lower Semicontinuity ◽  
Variable Stepsize ◽  
Chain Recurrence ◽  
Limit Sets ◽  
Chain Recurrent Set ◽  
Numerical Dynamics ◽  
Upper And Lower Semicontinuity ◽  
Recurrent Set

Upper and lower semicontinuity results for the chain recurrent set are shown to remain valid in numerical dynamics with constant stepsizes. It is also pointed out that the chain recurrent set contains numerical ω–limit sets for discretisations with a variable stepsize sequence approaching zero.

Shadowing, thick sets and the Ramsey property

2015 ◽  
Vol 36 (5) ◽  
pp. 1582-1595 ◽  
Author(s):  
PIOTR OPROCHA
Keyword(s):  
Arbitrary Sequence ◽  
Ramsey Property ◽  
Shadowing Property ◽  
Full Characterization ◽  
Equivalent Properties ◽  
Chain Recurrent Set ◽  
Recurrent Set

We provide a full characterization of relations between the shadowing property and the thick shadowing property. We prove that they are equivalent properties for non-wandering systems, the thick shadowing property is always a consequence of the shadowing property, and the thick shadowing property on the chain-recurrent set and the thick shadowing property are the same properties. We also provide a full characterization of the cases when for any family ${\mathcal{F}}$ with the Ramsey property an arbitrary sequence of points can be ${\it\varepsilon}$-traced over a set from ${\mathcal{F}}$.

scholarly journals Bifurcation and chain recurrence

1983 ◽  
Vol 3 (2) ◽  
pp. 231-240 ◽  
Author(s):  
M. Hurley
Keyword(s):  
Compact Manifold ◽  
Invariant Set ◽  
Residual Subset ◽  
Chain Recurrence ◽  
Chain Component ◽  
Residual Set ◽  
Image Position ◽  
Chain Recurrent Set ◽  
Recurrent Set

AbstractWe show that there is a residual subset of the set of C1 diffeomorphisms on any compact manifold at which the mapis continuous. As this number is apt to be infinite, we prove a localized version, which allows one to conclude that if f is in this residual set and X is an isolated chain component for f, then(i) there is a neighbourhood U of X which isolates it from the rest of the chain recurrent set of f, and(ii) all g sufficiently C1 close to f have precisely one chain component in U, and these chain components approach X as g approaches f.(ii) is interpreted as a generic non-bifurcation result for this type of invariant set.

scholarly journals Vector fields with transverse foliations, II

1986 ◽  
Vol 6 (2) ◽  
pp. 193-203 ◽  
Author(s):  
Sue Goodman
Keyword(s):  
Large Class ◽  
Periodic Orbits ◽  
Vector Fields ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Complete Answer ◽  
Necessary And Sufficient ◽  
Singular Flow ◽  
Chain Recurrent Set ◽  
Recurrent Set

AbstractWhen does a non-singular flow on a 3-manifold have a 2-dimensional foliation everywhere transverse to it? A complete answer is given for a large class of flows, those with 1-dimensional hyperbolic chain recurrent set. We find a simple necessary and sufficient condition on the linking of periodic orbits of the flow.

INVESTIGATION OF DYNAMICAL SYSTEMS USING SYMBOLIC IMAGES: EFFICIENT IMPLEMENTATION AND APPLICATIONS

2006 ◽  
Vol 16 (12) ◽  
pp. 3451-3496 ◽  
Author(s):  
V. AVRUTIN ◽  
P. LEVI ◽  
M. SCHANZ ◽  
D. FUNDINGER ◽  
G. OSIPENKO
Keyword(s):  
Dynamical Systems ◽  
Unified Framework ◽  
Connecting Orbits ◽  
Investigation Methods ◽  
System Flow ◽  
As Graph ◽  
Symbolic Images ◽  
Chain Recurrent Set ◽  
Adequate Data ◽  
Recurrent Set

Symbolic images represent a unified framework to apply several methods for the investigation of dynamical systems both discrete and continuous in time. By transforming the system flow into a graph, they allow it to formulate investigation methods as graph algorithms. Several kinds of stable and unstable return trajectories can be localized on this graph as well as attractors, their basins and connecting orbits. Extensions of the framework allow, e.g. the calculation of the Morse spectrum and verification of hyperbolicity. In this work, efficient algorithms and adequate data structures will be presented for the construction of symbolic images and some basic operations on them, like the localization of the chain recurrent set and periodic orbits. The performance of these algorithms will be analyzed and we show their application in practice. The focus is not only put on several standard systems, like Lorenz and Ikeda, but also on some less well-known ones. Additionally, some tuning techniques are presented for an efficient usage of the method.

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