Let f be a C1 function defined over Rn and definable in a given o-minimal structure M expanding the real field. We prove here a gradient-like inequality at infinity in a neighborhood of an asymptotic critical value c. When f is C2 we use this inequality to discuss the trivialization by the gradient flow of f in a neighborhood of a regular asymptotic critical level.

We introduce connection graphs for both continuous and discrete dynamical systems. We prove the existence of connection graphs for Morse decompositions of isolated invariant sets.

This survey is an introduction to some of the methods, techniques and concepts from algebraic topology and related areas (homotopy theory, shape theory) which can be fruitfully applied to study problems concerning continuous dynamical systems. To this end two instances which exemplify the interaction between topology and dynamics are considered, namely, Conley’s index theory and the study of some properties of certain attractors.

It is proved that near a compact, invariant, proper subset of a C⁰ flow on a locally compact, connected metric space, at least one, out of twenty eight relevant dynamical phenomena, will necessarily occur. Theorem 1 shows that the connectedness of the phase space implies the existence of a considerably deeper classification of topological flow behaviour in the vicinity of compact invariant sets than that described in the classical theorems of Ura-Kimura and Bhatia. The proposed classification brings...

In this paper we will be interested in results surrounding the following basic question: what are the homotopy properties that one can extract from a gradient flow? We approach this question by decomposing it into three parts: 1. Identify what are the homotopical objects that are provided by the flow (e.g. critical points, Conley indexes). 2. Discover what are the relations that have to be satisfied by these objects (e.g. Morse inequalities, Lusternik-Schnirelmann type inequalities). 3. (The Realizability...

For an analytic function f:ℝⁿ,0 → ℝ,0 having a critical point at the origin, we describe the topological properties of the partition of the family of trajectories of the gradient equation ẋ = ∇f(x) attracted by the origin, given by characteristic exponents and asymptotic critical values.

This paper deals with some characterizations of gradient-like continuous random dynamical systems (RDS). More precisely, we establish an equivalence with the existence of random continuous section or with the existence of continuous and strict Liapunov function. However and contrary to the deterministic case, parallelizable RDS appear as a particular case of gradient-like RDS.The obtained results are generalizations of well-known analogous theorems in the framework of deterministic dynamical systems....

We deal with a subshift of finite type and an equilibrium state μ for a Hölder continuous function. Let αⁿ be the partition into cylinders of length n. We compute (in particular we show the existence of the limit) $li{m}_{n\to \infty }{n}^{-1}log{\sum }_{j=0}^{\tau ₙ\left(x\right)}\mu \left(\alpha ⁿ\left(T^j\left(x\right)\right)\right)$, where $\alpha ⁿ\left(T^j\left(x\right)\right)$ is the element of the partition containing $T^j\left(x\right)$ and τₙ(x) is the return time of the trajectory of x to the cylinder αⁿ(x).

We study the complexity of the flow in the region of attraction of an isolated invariant set. More precisely, we define the instablity depth, which is an ordinal and measures how far an isolated invariant set is from being asymptotically stable within its region of attraction. We provide upper and lower bounds of the instability depth in certain cases.

Topological spaces with generalized symmetries are defined and extensions of the Conley index of a compact isolated invariant set of the flow preserving the structures introduced are proposed. One of the two new indexes is constructed with no additional assumption on the examined set in terms of symmetry invariance.