Criteria for strong and weak random attractors

The theory of random attractors has different notions of attraction, amongst them pullback attraction and weak attraction. We investigate necessary and sufficient conditions for the existence of pullback attractors as well as of weak attractors

Measure Attractors and Markov Attractors

The actions induced by a random dynamical system on spaces of probability measures on the state space are investigated, and generalisations of the notion of an attractor are discussed and compared. For the particular case of a random dynamical system generated by a stochastic differential equation the notion of an attractor for the associated Markov semigroup had previously been discussed in several instances in the literature. It is re-discovered here as a special case of a more general notion of an attractor in the space of measures

Correction to: Criteria for Strong and Weak Random Attractors

In the article 'Criteria for Strong and Weak Random Attractors' necessary and sufficient conditions for strong attractors and weak attractors are studied. In this note we correct two of its theorems on strong attractors.Comment: 4 page

Stabilization of linear systems by rotation

We introduce the concept of `stabilization by rotation' for deterministic linear systems with negative trace. This concept encompasses the well known concept of "vibrational stabilization" introduced by Meerkov in the 1970s and is a deterministic version of 'stabilization by noise' for stochastic systems as introduced by Arnold and coworkers in the 1980s. It is shown that a linear system with negative trace can be stabilized by adding a skew-symmetric matrix, multiplied by a suitable scalar so-called 'gain function' (possibly a constant) which is suffciently large. To overcome the problem of what is "suffciently large", we also present a servo mechanism which which tunes the gain function by learning from the trajectory until finally the trajectory tends to zero. This approach allows to show that one of Meerkov's assumptions for vibrational stabilization is superfluous. Moreover, while Meerkov as well as Arnold and coworkers assume that a stabilizing periodic function or the noise has suffciently large frequency and amplitude, we also provide a servo mechanism to determine this function dynamically in a deterministic setup

Stabilization of linear systems by dynamic high-gain rotation

We discuss stabilisation of linear systems by dynamic high-gainrotation. The existence of stabilizing rotations is established forsystems with negative trace, and an adaptive method to choose thecontroller gain is presented. The stabilization is robust with respectto arbitrary (possibly time-varying) skew-symmetric perturbations,which is also illustrated by a numerical example

Towards a Morse theory for random dynamical systems

A generalization of the concepts of deterministic Morse theory to random dynamical systems is presented. Using the notions of attraction and repulsion in probability, the main building blocks of Morse theory such as attractor-repeller sets, Morse sets, and the Morse decomposition are obtained for random dynamical systems

The effect of noise on the Chaffee-Infante Equation: a nonlinear case study

We investigate the effect of perturbing the Chafee-Infante scalar reaction diffusion equation, ut - [Delta]u = [beta]u - u3, by noise. While a single multiplicative Itô noise of sufficient intensity will stabilise the origin, its Stratonovich counterpart leaves the dimension of the attractor essentially unchanged. We then show that a collection of multiplicative Stratonovich terms can make the origin exponentially stable, while an additive noise of sufficient richness reduces the random attractor to a single point

Non-Markovian invariant measures are hyperbolic

AbstractSuppose μ is an invariant measure for a smooth random dynamical system on a d-dimensional Riemannian manifold. We prove that αμ⩽dEμ(max{0,−λμd}), where αμ is the relative entropy of μ, λμd is thesmallest Lyapunov exponent associated with μ, and Eμ denotes integration with respect to μ

Non-Markovian invariant measures are hyperbolic

Suppose [mu] is an invariant measure for a smooth random dynamical system on a d-dimensional Riemannian manifold. We prove that [alpha][mu][less-than-or-equals, slant]dE[mu](max{0,-[lambda][mu]d}), where [alpha][mu] is the relative entropy of [mu], [lambda][mu]d is thesmallest Lyapunov exponent associated with [mu], and E[mu] denotes integration with respect to [mu].