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Conley index theory is a very powerful tool in the study of dynamical systems, differential equations and bifurcation theory. In this paper, we make an attempt to generalize the Conley index to discrete random dynamical systems. And we mainly follow the Conley index for maps given by Franks and Richeson in [6]. Furthermore, we simply discuss the relations of isolated invariant sets between time-continuous random dynamical systems and the corresponding time-$h$ maps. For applications we give several examples to illustrate our results.
In this paper, we introduce concepts of pathwise random almost periodic and almost automorphic solutions for dynamical systems generated by non-autonomous stochastic equations. These solutions are pathwise stochastic analogues of deterministic dynami
Multivector fields provide an avenue for studying continuous dynamical systems in a combinatorial framework. There are currently two approaches in the literature which use persistent homology to capture changes in combinatorial dynamical systems. The
The topological method for the reconstruction of dynamics from time series [K. Mischaikow, M. Mrozek, J. Reiss, A. Szymczak. Construction of Symbolic Dynamics from Experimental Time Series, Physical Review Letters, 82 (1999), 1144-1147] is reshaped t
In this note we prove that a fractional stochastic delay differential equation which satisfies natural regularity conditions generates a continuous random dynamical system on a subspace of a Holder space which is separable.