Related papers: Conley index for random dynamical systems
The analysis of global dynamics, particularly the identification and characterization of attractors and their regions of attraction, is essential for complex nonlinear and hybrid systems. Combinatorial methods based on Conley's index theory…
A theorem is established where the computation of the discrete Conley index for zero dimensional basic sets is given with respect to the dynamical information contained in the associated structure matrices. A classification of the reduced…
The aim of this paper is to explore the possibilities of Conley index techniques in the study of heteroclinic connections between finite and infinite invariant sets. For this, we remind the reader of the Poincar\'e compactification: this…
In this paper, we use Conley index theory to examine the Poincare index of an isolated invariant set. We obtain some limiting conditions on a critical point of a planar vector field to be an isolated invariant set. As a result we show the…
In Conley index theory one may study an invariant set $S$ by decomposing it into an attractor $A$, a repeller $R$, and the orbits connecting the two. The Conley indices of $S$, $A$ and $R$ fit into an exact sequence where a certain…
In the first part of this paper, we generalize the results of the author \cite{Liu,Liu2} from the random flow case to the random semiflow case, i.e. we obtain Conley decomposition theorem for infinite dimensional random dynamical systems.…
In this paper we study the Lorenz equations using the perspective of the Conley index theory. More specifically, we examine the evolution of the strange set that these equations posses throughout the different values of the parameter. We…
We study the behaviour of discrete dynamical systems generated by a continuous map $f$ of a compact real interval into itself where at randomly chosen times a function different from $f$ - so called impulse function is applied. We show that…
It is known by the Conley's theorem that the chain recurrent set $CR(\phi)$ of a deterministic flow $\phi$ 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…
In a previous work, the author established a nonautonomous Conley index based on the interplay between a nonautonomous evolution operator and its skew-product formulation. This index is refined to obtain a Conley index for families of…
We prove that every combinatorial dynamical system in the sense of Forman, defined on a family of simplices of a simplicial complex, gives rise to a multivalued dynamical system F on the geometric realization of the simplicial complex.…
Conley in \cite{Con} constructed a complete Lyapunov function for a flow on compact metric space which is constant on orbits in the chain recurrent set and is strictly decreasing on orbits outside the chain recurrent set. This indicates…
We propose a new framework for Conley index theory. The main feature of our approach is that we do not use the notion of index pairs. We introduce, instead, the notions of compactifiable subsets and index neighbourhoods, and formulate and…
We present new methods of automating the construction of index pairs, essential ingredients of discrete Conley index theory. These new algorithms are further steps in the direction of automating computer-assisted proofs of semi-conjugacies…
Consider a sequence of i.i.d. random variables $X_n$ where each random variable is refreshed independently according to a Poisson clock. At any fixed time $t$ the law of the sequence is the same as for the sequence at time 0 but at random…
We study a random dynamical system such that one transformation is randomly selected from a family of transformations and then applied on each iteration. For such random dynamical systems, we consider estimates of absolutely continuous…
We investigate the (linearized) Morse index of solutions to Hamiltonan systems, with a focus on convex Hamiltonians functions and sign-changing radial solutions. For strongly coupled systems, we describe the profile of the radial solutions…
The dynamics by iteration of a function on a compact metric space, sometimes called a cascade, can be extended to the dynamics of a closed relation on such a space. Here we apply this relation dynamics to study semiflows (and their relation…
We investigate the convergence towards periodic orbits in discrete dynamical systems. We examine the probability that a randomly chosen point converges to a particular neighborhood of a periodic orbit in a fixed number of iterations, and we…
The connection matrix is a powerful algebraic topological tool from Conley index theory that captures relationships between isolated invariant sets. Conley index theory is a topological generalization of Morse theory in which the connection…