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Global random attractors and random point attractors for random dynamical systems have been studied for several decades. Here we introduce two intermediate concepts: $Delta$-attractors are characterized by attracting all deterministic compact sets of Hausdorff dimension at most $Delta$, where $Delta$ is a non-negative number, while cc-attractors attract all countable compact sets. We provide two examples showing that a given random dynamical system may have various different $Delta$-attractors for different values of $Delta$. It seems that both concepts are new even in the context of deterministic dynamical systems.
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 the secon
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
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
We show that the twisted planar random walk - which results by summing up stationary increments rotated by multiples of a fixed angle - is recurrent under diverse assumptions on the increment process. For example, if the increment process is alpha-mi
We study the number and distribution of the limit cycles of a planar vector field whose component functions are random polynomials. We prove a lower bound on the average number of limit cycles when the random polynomials are sampled from the Kostlan-