The Kneser property of the weak solutions of the three dimensional Navier-Stokes equations
Peter E. Kloeden José Valero
The Kneser theorem for ordinary differential equations without uniqueness says that the attainability set is compact and connected at each instant of time. We establish corresponding results for the attainability set of weak solutions for the 3D Navier-Stokes equations satisfying an energy inequality. First, we present a simplified proof of our earlier result with respect to the weak topology in the space $H$. Then we prove that this result also holds with respect to the strong topology on $H$ provided that the weak solutions satisfying the weak version of the energy inequality are continuous. Finally, using these results, we show the connectedness of the global attractor of a family of setvalued semiflows generated by the weak solutions of the NSE satisfying suitable properties.
keywords: weak connectedness Attainability set Navier-Stokes equations global weak attractor. globally modified Navier-Stokes Equations weak compactness Kneser property
Lyapunov's second method for nonautonomous differential equations
Lars Grüne Peter E. Kloeden Stefan Siegmund Fabian R. Wirth
Converse Lyapunov theorems are presented for nonautonomous systems modelled as skew product flows. These characterize various types of stability of invariant sets and pullback, forward and uniform attractors in such nonautonomous systems.
keywords: nonautonomous attractor. nonautonomous differential equation Lyapunov's second method nonautonomous dynamical system Lyapunov function stability
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
Asymptotic invariance and the discretisation of nonautonomous forward attracting sets
Peter E. Kloeden
The $\omega$-limit set $\omega_B$ of a nonautonomous dynamical system generated by a nonautonomous ODE with a positive invariant compact absorbing set $B$ is shown to be asymptotic positive invariant in general and asymptotic negative invariant if, in addition, the vector field is uniformly continuous in time on the absorbing set. This set has been called the forward attracting set of the nonautonomous dynamical system and is related to Vishik's concept of a uniform attractor. If $\omega_B$ is also assumed to be uniformly attracting, then its upper semi continuity in a parameter and the upper semi continuous convergence of its counterparts under discretisation by the implicit Euler scheme are established.
keywords: asymptotic positive invariance asymptotic negative invariance implicit Euler scheme forward attracting set upper semi continuous convergence. uniform attractor Nonautonomous dynamical system upper semi continuous dependence on parameters omega limit points
Peter E. Kloeden Alexander M. Krasnosel'skii Pavel Krejčí Dmitrii I. Rachinskii
Alexei Vadimovich Pokrovskii was an outstanding mathematician, a scientist with very broad mathematical interests, and a pioneer in the mathematical theory of systems with hysteresis. He died unexpectedly on September 1, 2010 at the age 62. For the previous nine years he had been Professor and Head of Applied Mathematics at University College Cork in Ireland.

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Pullback attractors of reaction-diffusion inclusions with space-dependent delay
Peter E. Kloeden Thomas Lorenz

Inspired by biological phenomena with effects of switching off (maybe just for a while), we investigate non-autonomous reaction-diffusion inclusions whose multi-valued reaction term may depend on the essential supremum over a time interval in the recent past (but) pointwise in space. The focus is on sufficient conditions for the existence of pullback attractors. If the multi-valued reaction term satisfies a form of inclusion principle standard tools for non-autonomous dynamical systems in metric spaces can be applied and provide new results (even) for infinite time intervals of delay. More challenging is the case without assuming such a monotonicity assumption. Then we consider the parabolic differential inclusion with the time interval of delay depending on space and extend the approaches of norm-to-weak semigroups to a purely metric setting. This provides completely new tools for proving pullback attractors of non-autonomous dynamical systems in metric spaces.

keywords: Reaction-diffusion equation parabolic differential inclusion with space-dependent delay dead core retarded functional evolution inclusion existence of solutions non-autonomous pullback attractors set-valued dynamical system with norm-to-weak closed graph
The uniform attractor of a multi-valued process generated by reaction-diffusion delay equations on an unbounded domain
Yejuan Wang Peter E. Kloeden
The existence of a uniform attractor in a space of higher regularity is proved for the multi-valued process associated with the nonautonomous reaction-diffusion equation on an unbounded domain with delays for which the uniqueness of solutions need not hold. A new method for checking the asymptotical upper-semicompactness of the solutions is used.
keywords: uniform attractor Multi-valued process asymptotical upper-semicompactness reaction-diffusion equation with delay.
Mean-square random attractors of stochastic delay differential equations with random delay
Fuke Wu Peter E. Kloeden
The existence of a random attactor is established for a mean-square random dynamical system (MS-RDS) generated by a stochastic delay equation (SDDE) with random delay for which the drift term is dominated by a nondelay component satisfying a one-sided dissipative Lipschitz condition. It is shown by Razumikhin-type techniques that the solution of this SDDE is ultimately bounded in the mean-square sense and that solutions for different initial values converge exponentially together as time increases in the mean-square sense. Consequently, similar boundedness and convergence properties hold for the MS-RDS and imply the existence of a mean-square random attractor for the MS-RDS that consists of a single stochastic process.
keywords: Razumikhin-type theorem random delay ultimate boundedness. mean-square random dynamical system (MS-RDS) Mean-square random attractor
Tomás Caraballo Amadeu Delshams Àngel Jorba Peter E. Kloeden Rafael Obaya
The theory of dynamical systems has undergone some spectacular and fascinating developments in the past century, as the readers of this journal are well aware, with the focus predominately on autonomous systems. There are many ways in which one could classify the work that has been done, but one that stands clearly in the forefront is the distinction between dissipative systems with their attractors and conservative systems, in particular Hamiltonian systems.
    Another classification is between autonomous and nonautonomous systems. Of course, the latter subsumes the former as special case, but with the former having special structural features, i.e., the semigroup evolution property, which has allowed an extensive and seemingly complete theory to be developed. Although not as extensive, there have also been significant developments in the past half century on nonautonomous dynamical systems, in particular the skew-product formalism involving a cocycle evolution property which generalizes the semigroup property of autonomous systems. This has been enriched in recent years by advances on random dynamical systems, which are roughly said a measure theoretic version of a skew-product flow. In particular, new concepts of random and nonautonomous attractors have been introduced and investigated.

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Asymptotic behaviour of random tridiagonal Markov chains in biological applications
Peter E. Kloeden Victor Kozyakin
Discrete-time discrete-state random Markov chains with a tridiagonal generator are shown to have a random attractor consisting of singleton subsets, essentially a random path, in the simplex of probability vectors. The proof uses the Hilbert projection metric and the fact that the linear cocycle generated by the Markov chain is a uniformly contractive mapping of the positive cone into itself. The proof does not involve probabilistic properties of the sample path $\omega$ and is thus equally valid in the nonautonomous deterministic context of Markov chains with, say, periodically varying transitions probabilities, in which case the attractor is a periodic path.
keywords: linear cocycles positive cones Hilbert metric random attractors. Random Markov chain uniformly contracting cocycles

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