Attractors for nonautonomous and random dynamical systems perturbed by impulses
Björn Schmalfuss
Nonautonomous and random dynamical systems perturbed by impulses are considered. The impulses form a flow. Over this flow the perturbed system also has the structure of a new nonautonomous/random dynamical system. The long time behavior of this system is considered. In particular the existence of an attractor is proven. The result can be applied to a large class of dissipative systems given by partial or ordinary differential equations. As an example of this class of problems the Lorenz system is studied. For another problem given by a one-dimensional affine differential equation and perturbed by affine impulses, the attractor can be calculated explicitly.
keywords: attractors. nonautonomous systems Impulse equations random dynamical systems
Lyapunov functions and attractors under variable time-step discretization
Peter E. Kloeden Björn Schmalfuss
A one-step numerical scheme with variable time--steps is applied to an autonomous differential equation with a uniformly asymptotically stable set, which is compact but otherwise of arbitrary geometric shape. A Lyapunov function characterizing this set is used to show that the resulting nonautonomous difference equation generated by the numerical scheme has an absorbing set. The existence of a cocycle attractor consisting of a family of equivariant sets for the associated discrete time cocycle is then established and shown to be close in the Hausdorff separation to the original stable set for sufficiently small maximal time-steps.
keywords: cocycle attractor. uniform asymptotic stability cocycle numerical scheme Lyapunov function
Unstable invariant manifolds for a nonautonomous differential equation with nonautonomous unbounded delay
Arne Ogrowsky Björn Schmalfuss
In this paper we deal with a nonautonomous differential equation with a nonautonomous delay. The aim is to establish the existence of an unstable invariant manifold to this differential equation for which we use the Lyapunov-Perron transformation. However, the delay is assumed to be unbounded which makes it necessary to use nonclassical methods.
keywords: nonautonomous delay cocycles. Unstable invariant manifolds nonautonomous differential equations
Asymptotic behaviour of a stochastic semilinear dissipative functional equation without uniqueness of solutions
Tomás Caraballo María J. Garrido–Atienza Björn Schmalfuss José Valero
In this work we present the existence and uniqueness of pullback and random attractors for stochastic evolution equations with infinite delays when the uniqueness of solutions for these equations is not required. Our results are obtained by means of the theory of set-valued random dynamical systems and their conjugation properties.
keywords: Multivalued non-autonomous and random dynamical systems functional stochastic equations conjugacy method. pullback and random attractors
Attractors for a random evolution equation with infinite memory: Theoretical results
Tomás Caraballo María J. Garrido-Atienza Björn Schmalfuss José Valero

The long-time behavior of solutions (more precisely, the existence of random pullback attractors) for an integro-differential parabolic equation of diffusion type with memory terms, more particularly with terms containing both finite and infinite delays, as well as some kind of randomness, is analyzed in this paper. We impose general assumptions not ensuring uniqueness of solutions, which implies that the theory of multivalued dynamical system has to be used. Furthermore, the emphasis is put on the existence of random pullback attractors by exploiting the techniques of the theory of multivalued nonautonomous/random dynamical systems.

keywords: Pullback and random attractor random dynamical system random delay equation infinite delay
Qualitative behavior of a class of stochastic parabolic PDEs with dynamical boundary conditions
Igor Chueshov Björn Schmalfuss
We consider non-linear parabolic stochastic partial differential equations with dynamical boundary conditions and with a noise which acts in the domain but also on the boundary and is presented by the temporal generalized derivative of an infinite dimensional Wiener process. We prove that solutions to this stochastic partial differential equation generate a random dynamical system. Under additional conditions we show that this system is monotone. Our main result states the existence of a compact global (pullback) attractor.
keywords: stochastic PDE monotonicity. random dynamical systems dynamical boundary conditions pullback attractor
Pathwise solutions of SPDEs driven by Hölder-continuous integrators with exponent larger than $1/2$ and random dynamical systems
Yong Chen Hongjun Gao María J. Garrido–Atienza Björn Schmalfuss
This article is devoted to the existence and uniqueness of pathwise solutions to stochastic evolution equations, driven by a Hölder continuous function with Hölder exponent in $(1/2,1)$, and with nontrivial multiplicative noise. As a particular situation, we shall consider the case where the equation is driven by a fractional Brownian motion $B^H$ with Hurst parameter $H>1/2$. In contrast to the article by Maslowski and Nualart [17], we present here an existence and uniqueness result in the space of Hölder continuous functions with values in a Hilbert space $V$. If the initial condition is in the latter space this forces us to consider solutions in a different space, which is a generalization of the Hölder continuous functions. That space of functions is appropriate to introduce a non-autonomous dynamical system generated by the corresponding solution to the equation. In fact, when choosing $B^H$ as the driving process, we shall prove that the dynamical system will turn out to be a random dynamical system, defined over the ergodic metric dynamical system generated by the infinite dimensional fractional Brownian motion.
keywords: Stochastic PDEs fractional Brownian motion pathwise solutions random dynamical systems.
Inertial manifolds for stochastic pde with dynamical boundary conditions
Peter Brune Björn Schmalfuss
In this article we investigate the dynamics of stochastic partial differential equations with dynamical boundary conditions. We prove that such a problem with Lipschitz continuous non--linearity generates a random dynamical system. The main result is to show that this random dynamical system has an inertial manifold. Under additional assumptions on the non--linearity this manifold is differentiable.
keywords: dynamical boundary conditions. inertial manifolds stochastic partial differential equations Random dynamical systems
Pathwise solutions and attractors for retarded SPDEs with time smooth diffusion coefficients
Hakima Bessaih María J. Garrido–Atienza Björn Schmalfuss
In this paper we study the long--time dynamics of mild solutions to retarded stochastic evolution systems driven by a Hilbert-valued Brownian motion. For this purpose, we begin by showing the existence and uniqueness of a cocycle solution of such an equation. We do not assume that the noise is given in additive form or that it is a very simple multiplicative noise. However, we need some smoothing property for the coefficient in front of the noise. The main idea of this paper consists of expressing the stochastic integral in terms of non-stochastic integrals and the noisy path by using an integration by parts. This latter term causes that at first, only a local mild solution can be obtained, since in order to apply the Banach fixed point theorem it is crucial to have the Hölder norm of the noisy path to be sufficiently small. Subsequently, by using appropriate stopping times, we shall derive the existence and uniqueness of a global mild solution. Furthermore, the asymptotic behavior is investigated by using the Random Dynamical Systems theory. In particular, we shall show that the global mild solution generates a random dynamical system that, under an appropriate smallness condition for the time lag, has an associated random attractor.
keywords: Hilbert-valued Brownian motion random attractors. Stochastic PDEs pathwise solutions
Random dynamical systems for stochastic partial differential equations driven by a fractional Brownian motion
María J. Garrido–Atienza Kening Lu Björn Schmalfuss
In this paper we study nonlinear stochastic partial differential equations (SPDEs) driven by a fractional Brownian motion (fBm) with the Hurst parameter bigger than $1/2$. We show that these SPDEs generate random dynamical systems (or stochastic flows) by using the stochastic calculus for an fBm where the stochastic integrals are defined by integrands given by fractional derivatives. In particular, we emphasize that the coefficients in front of the fractional noise are non-trivial.
keywords: random dynamical systems. Stochastic PDEs fractional Brownian motion

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