Shape and Morse theory of attractors

Kapitanski, Lev; Rodnianski, Igor

doi:10.1002/(sici)1097-0312(200002)53:2<218::aid-cpa2>3.0.co;2-w

Cited by 76 publications

(86 citation statements)

References 34 publications

Supporting

Mentioning

Contrasting

Unclassified

Order By: Relevance

“…We have the following fundamental result which generalizes the corresponding one for semiflows on metric spaces [9]. Theorem 6.1.…”

Section: Morse Decompositionsmentioning

confidence: 80%

Attractors of Local Semiflows on Topological Spaces

Li¹,

Wang²,

Xiong³

2017

Journal of the Korean Mathematical Society

View full text Add to dashboard Cite

Abstract. In this paper we introduce a notion of an attractor for local semiflows on topological spaces, which in some cases seems to be more suitable than the existing ones in the literature. Based on this notion we develop a basic attractor theory on topological spaces under appropriate separation axioms. First, we discuss fundamental properties of attractors such as maximality and stability and establish some existence results. Then, we give a converse Lyapunov theorem. Finally, the Morse decomposition of attractors is also addressed.

show abstract

“…We have the following fundamental result which generalizes the corresponding one for semiflows on metric spaces [9]. Theorem 6.1.…”

Section: Morse Decompositionsmentioning

confidence: 80%

Attractors of Local Semiflows on Topological Spaces

Li¹,

Wang²,

Xiong³

2017

Journal of the Korean Mathematical Society

View full text Add to dashboard Cite

show abstract

“…It is known that for a stable attractor K the inclusion K → A(K) is a shape equivalence ( [5], [16], [17], [19], [23], [32], [33]), so the polynomial p(A(K), K) is trivial. When K, more generally, has only internal explosions, the following result holds.…”

Section: Where S Is the Total Number Of Components In A(k) − K And R mentioning

confidence: 99%

Unstable attractors in manifolds

Sánchez-Gabites

2010

Trans. Amer. Math. Soc.

View full text Add to dashboard Cite

Abstract. Assume that K is a compact attractor with basin of attraction A(K) for some continuous flow ϕ in a space M . Stable attractors are very well known, but otherwise (without the stability assumption) the situation can be extremely wild. In this paper we consider the class of attractors with no external explosions, where a mild form of instability is allowed.After obtaining a simple description of the trajectories in A(K) − K we study how K sits in A(K) by performing an analysis of the Poincaré polynomial of the pair (A(K), K). In case M is a surface we obtain a nice geometric characterization of attractors with no external explosions, as well as a converse to the well known fact that the inclusion of a stable attractor in its basin of attraction is a shape equivalence. Finally, we explore the strong relations which exist between the shape (in the sense of Borsuk) of K and the shape (in the intuitive sense) of the whole phase space M , much in the spirit of the Morse-Conley theory.

show abstract

“…[21][22][23][24]). For example, for the Lorenz system for the control parameter r = 2.5 there are three local attractors: the origin and two symmetrical fixed points X 1 , 2 (±2, ∓2, 1.5) while for r = 28 there is a single local attractor which is global too, the known Lorenz strange attractor.…”

Section: Remarkmentioning

confidence: 99%

Synthesizing attractors of Hindmarsh–Rose neuronal systems

Danca

Wang

2010

Nonlinear Dyn

View full text Add to dashboard Cite

In this paper a periodic parameter-switching scheme is applied to the Hindmarsh-Rose neuronal system to synthesize certain attractors. Results show numerically, via computer graphic simulations, that the obtained synthesized attractor belongs to the class of all admissible attractors for the Hindmarsh-Rose neuronal system and matches the averaged attractor obtained with the control parameter replaced with the averaged switched parameter values. This feature allows us to imagine that living beings are able to maintain vital behavior while the control parameter switches so that their dynamical behavior is suitable for the given environment.

show abstract

Shape and Morse theory of attractors

Cited by 76 publications

References 34 publications

Attractors of Local Semiflows on Topological Spaces

Attractors of Local Semiflows on Topological Spaces

Unstable attractors in manifolds

Synthesizing attractors of Hindmarsh–Rose neuronal systems

Contact Info

Product

Resources

About