Chain Recurrent Set Research Articles

Diffeomorphisms with the s-limit shadowing property

Let Diff(M) be the space of diffeomorphisms of a closed C ∞ manifold M endowed with the C 1-topology, and denote by ℛ(f) the chain recurrent set of f ∈ Diff(M). In this article, we show that C 1-generically f |ℛ(f) has the s-limit shadowing property if and only if f satisfies both Axiom A and no-cycle condition.

Perturbations of Set-Valued Dynamical Systems, with Applications to Game Theory

We present upper-semicontinuity results for attractors and the chain-recurrent set of differential inclusions, in particular w.r.t. discretizations, and applications to game dynamics.

Topological stability of chain recurrent sets

It is proved that for any homeomorphism on a compact metric space, if it has the shadowing property and it is expansible on chain recurrent set, then it is chain topologically stable.

Partial hyperbolicity far from homoclinic bifurcations

We prove that any diffeomorphism of a compact manifold can be C1-approximated by a diffeomorphism which exhibits a homoclinic bifurcation (a homoclinic tangency or a heterodimensional cycle) or by a diffeomorphism which is partially hyperbolic (its chain-recurrent set splits into partially hyperbolic pieces whose centre bundles have dimensions less or equal to two). We also study in a more systematic way the central models introduced in Crovisier (in press) [10].

CHARACTERIZATIONS ON CHAIN RECURRENCES

It is well known that there is a residual subset J of the space of <TEX>$C^1$</TEX>-diffeomorphisms on a compact Riemannian manifold M such that the maps f <TEX>$\mapsto$</TEX> chain recurrent set of f and f <TEX>$\mapsto$</TEX> number of chain components of f are continuous on J. In this paper we get the flow version of the above results on diffeomorphisms.

SOME PROPERTIES OF THE STRONG CHAIN RECURRENT SET

The article is devoted to exhibit some general properties of strong chain recurrent set and strong chain transitive components for a continuous map f on a compact metric space X. We investigate the relation between the weak shadowing property and strong chain transitivity. It is shown that a continuous map f from a compact metric space X onto itself with the average shadowing property is strong chain transitive.

The size of the chain recurrent set for generic maps on an n-dimensional locally (n-1)-connected compact space

The size of the chain recurrent set for generic maps on an n-dimensional locally (n-1)-connected compact space

Adaptive Set-Oriented Computation of Topological Horseshoe Factors in Area and Volume Preserving Maps

We describe an automatic chaos verification scheme, based on set-oriented numerical methods, which is especially well suited to the study of area- and volume preserving diffeomorphisms. The novel feature of the scheme is an iterative algorithm for approximating connecting orbits between collections of hyperbolic fixed and periodic points with greater and greater accuracy. The algorithm is geometric rather than graph theoretic in nature and, in contrast to existing methods, does not require the computation of chain recurrent sets. We provide several example computations in dimensions two and three.

Random chain recurrent sets for random dynamical systems

It is known by the Conley's theorem that the chain recurrent set CR(ϕ) of a deterministic flow ϕ on a compact metric space is the complement of the union of sets B(A) − A, where A varies over the collection of attractors and B(A) is the basin of attraction of A. It has recently been shown that a similar decomposition result holds for random dynamical systems (RDSs) on non-compact separable complete metric spaces, but under a so-called absorbing condition. In the present article, the authors introduce a notion of random chain recurrent sets for RDSs, and then prove the random Conley's theorem on non-compact separable complete metric spaces without the absorbing condition.

An attracting singular-hyperbolic set containing a non trivial hyperbolic repeller

The chain recurrent set of an Anosov three-dimensional vector field (e.g. [3]) is an example of a non-transitive singular-hyperbolic attracting set containing hyperbolic repellers. Observe that this example has no singularities. In this paper we construct a C ∞ three-dimensional vector field that exhibits a singular-hyperbolic attracting set containing a singularity which is not Lorenz-like and a non trivial repeller hyperbolic set.

Modeling of path-choice behavior based on the regret minimization criteria givena full travel information

This article formulates the drivers' behaviors of path-choice when travel information for a set of paths is given to each driver after finishing his or her trip.While an underpinning theory of the behavioral model stems from a regret matching theory, the basic framework is adjusted to capture the characteristic of congested networks. The notable feature of the model is that each driver is treatedas an individual decision maker as in game theory. In addition, uncertainty thata driver faces is endogenously generated as the results of mutually dependent path-choice behavior. Drivers are modeled to achieve no-regret criteria and to learn uncertain environment with stochastic approximation.Dynamic of updating the propensity of path-choice is characterized by the ODE approach, and is shownto converge to an internally chain recurrent set. However, this is not the case for the congested network with monotone cost functions. If we admit monotone cost function, then the regret matching approach with stochastic approximation has very nice feature that allows a generic travel cost function such as asymmetric Jacobian, piece-wise linear and so on.

CHAIN RECURRENCE AND ATTRACTORS IN GENERAL DYNAMICAL SYSTEMS

We introduce here notions of chain recurrent sets, attractors and its basins for general dynamical systems and prove important properties including (i) the chain recurrent set CR(f) of f on a metric space (X, d) is the complement of the union of sets B(A) -A as A varies over the collection of attractors and (ii) genericity of general dynamical systems.

Asymptotic average shadowing property on compact metric spaces

We prove that if for a continuous map f on a compact metric space X , the chain recurrent set, R ( f ) has more than one chain component, then f does not satisfy the asymptotic average shadowing property. We also show that if a continuous map f on a compact metric space X has the asymptotic average shadowing property and if A is an attractor for f , then A is the single attractor for f and we have A = R ( f ) . We also study diffeomorphisms with asymptotic average shadowing property and prove that if M is a compact manifold which is not finite with dim M = 2 , then the C 1 interior of the set of all C 1 diffeomorphisms with the asymptotic average shadowing property is characterized by the set of Ω -stable diffeomorphisms.

Cohomological estimates for cat(X,ξ)

This paper studies the homotopy invariant $\cat(X,\xi)$ introduced in \cite{farbe2}. Given a finite cell-complex $X$, we study the function $\xi\mapsto \cat(X,\xi)$ where $\xi$ varies in the cohomology space $H^1(X;\R)$. Note that $\cat(X,\xi)$ turns into the classical Lusternik - Schnirelmann category $\cat(X)$ in the case $\xi=0$. Interest in $\cat(X,\xi)$ is based on its applications in dynamics where it enters estimates of complexity of the chain recurrent set of a flow admitting Lyapunov closed 1-forms.

Generic properties of [formula omitted]-actions on the interval

We prove that generic Z 2 -actions ( f , g ) on the interval [ 0 , 1 ] have the property that both f and g have the periodic shadowing and strong tolerance stability properties, as well as their chain recurrent sets are the Cantor sets (Theorem 1).

On Strong Chain Classes of Homeomorphisms of Compact Metric Spaces

The strong chain recurrent points and strong chain transitive sets of a homeomorphism f on a compact metric space X, first introduced by Easton [4]. He note that, if chain recurrent set of f is all of X, then strong chain recurrent set and strong chain transitive set of f are useful. Here, we obtain some results about strong chain recurrency. In particular, we show that an isolated strong chain class S of generic homeomorphism f has a generic continuation Sg , where g close to f, in C ○-topology. Moreover, we have the generic persistence of Lipschitz ergodicity at f near S.

The random case of Conley's theorem: II. The complete Lyapunov function

Conley (1978 Isolated Invariant Sets and the Morse Index (Conf. Board Math. Sci. vol 38) (Providence, RI: American Mathematical Society)) constructed a complete Lyapunov function for a flow on compact metric space which is constant on orbits in the chain recurrent set and strictly decreasing on orbits outside the chain recurrent set. This indicates that the dynamical complexity focuses on the chain recurrent set and the dynamical behaviour outside the chain recurrent set is quite simple. In this paper, a similar result is obtained for random dynamical systems (RDS) under the assumption that the base space is a separable metric space endowed with a probability measure. By constructing a complete Lyapunov function, which is constant on orbits in the random chain recurrent set and strictly decreasing on orbits outside it, the random case of Conley's fundamental theorem of dynamical systems is obtained. Furthermore, this result for RDS is generalized to noncompact state spaces.

Polygonal approximation of flows

The analysis of the qualitative behavior of flows generated by ordinary differential equations often requires quantitative information beyond numerical simulation which can be difficult to obtain analytically. In this paper we present a computational scheme designed to capture qualitative information using ideas from the Conley index theory. Specifically we design an combinatorial multivalued approximation from a simplicial decomposition of the phase space, which can be used to extract isolating blocks for isolated invariant sets. These isolating blocks can be computed rigorously to provide computer-assisted proofs. We also obtain local conditions on the underlying simplicial approximation that guarantees that the chain recurrent set can be well-approximated.

Morse Decompositions for General Dynamical Systems and Differential Inclusions with Applications to Control Systems

The Morse decomposition theory for single-valued dynamical systems is extended to general dynamical systems and differential inclusions. Relations between chain recurrent sets, chain recurrent components (chain control sets), and Morse decompositions for open-loop systems are discussed. The robustness of feedback laws of closed-loop systems with respect to small time delays and sample-hold controls is also addressed.

The chain recurrent set for maps of compacta

The chain recurrent set for maps of compacta