Flattening, squeezing and the existence of random attractors

By Peter E. Kloeden, José A. Langa. Proceedings of the Royal Society A, 2007

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The study of qualitative properties of random and stochastic differential equations is now one of the most active fields in the modern theory of dynamical systems. In the deterministic case, the properties of flattening and squeezing in infinite-dimensional autonomous dynamical systems require the existence of a bounded absorbing set and imply the existence of a global attractor. The flattening property involves the behaviour of individual trajectories while the squeezing property involves the difference of trajectories. It is shown here that the flattening property is implied by the squeezing property and is in fact weaker, since the attractor in a system with the flattening property can be infinite-dimensional, whereas it is always finite-dimensional in a system with the squeezing property. The flattening property is then generalized to random dynamical systems, for which it is called the pullback flattening property. It is shown to be weaker than the random squeezing property, but equivalent to pullback asymptotic compactness and pullback limit-set compactness, and thus implies the existence of a random attractor. The results are also valid for deterministic non-autonomous dynamical systems formulated as skew-product flows.

Subfields: Computational Theory and Mathematics, Computer Networks and Communications, Control and Systems Engineering

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Citations in the corpus

Sufficient and Necessary Criteria for Existence of Pullback Attractors for Non-compact Random Dynamical Systems (2012, background): “he equation. The concept of pullback attractors for random dynamical systems was introduced by the authors in [16, 18, 24], and the existence of such attractors for compact systems was established in [1, 7, 8, 14, 15, 16, 17, 18, 21, 22, 24] and the references therein. By a compact random dynamical ”
Existence, Stability and Bifurcation of Random Complete and Periodic Solutions of Stochastic Parabolic Equations (2013, background): “ous stochastic one. The concept of random attractor for autonomous systems was introduced in [18, 25, 44], and the existence of such attractors have been established for various equations, see, e.g., [1, 4, 5, 6, 7, 8, 9, 10, 16, 17, 18, 19, 25, 26, 27, 32, 33, 44, 49, 50]. For the existence of inva”