Peter E. Kloeden

Pullback and Forward Attractors for a Damped Wave Equation with Delays

The existence of a pullback (and also a uniform forward) attractor is proved for a damped wave equation containing a delay forcing term which, in particular, covers the models of sine–Gordon type. The result follows from the existence of a compact set which is uniformly attracting for the two-parameter semigroup associated to the model.

A Peano theorem for fuzzy differential equations with evolving membership grade

Abstract The Peano theorem on the existence without possible uniqueness of solutions has been a perplexing problem in the theory of fuzzy differential equations. The difficulty appears to be due to the standard use of the supremum metric ∞ defined by the supremum over the Hausdorff metric between the level sets of the fuzzy sets. Another may have been the classical formulation of fuzzy differential equations in terms of the Hukuhara derivative of the level sets. Here a Peano theorem is established for fuzzy differential equations formulated in a recent paper by the authors by combining Hullermeiers suggestion of defining fuzzy differential equations at each level set via differential inclusions with Aubins morphological equations, which allow non-local set evolution. A major difference from previous publications is the use of the endograph metric end , essentially the Hausdorff metric between the endographs in R n × [ 0 , 1 ] of fuzzy sets, instead of the supremum metric ∞ . Another is that the membership grades of the fuzzy sets are also allowed to evolve under the fuzzy differential equations. The result applies for a very general class of fuzzy sets without additional assumptions of fuzzy convexity, compact supports or even normality.

Forward attraction in nonautonomous difference equations

Two types of attractors consisting of families of sets that are mapped into each other under the dynamics have been defined for nonautonomous difference equations, one using pullback convergence with information about the system in the past and the other using forward convergence with information about the system in the future. In both cases, the component sets are constructed using a pullback argument within a positively invariant family of sets. The forward attractor so constructed also uses information about the past, which is very restrictive and not essential for determining future behaviour. Here an alternative is investigated, essentially the omega-limit set of the system, which Chepyzhov and Vishik called the uniform attractor. It is shown here that this set is asymptotically positively invariant, thus providing it with an hitherto missing form of invariance, if in somewhat weaker than usual, that one expects an attractor to possess. As a consequence this set provides useful information about the behaviour in current time during the approach to the limit.

Multi-step methods for random ODEs driven by Itô diffusions

Linear multi-step methods are derived for random ordinary differential equations (RODEs) driven by the solutions of Ito stochastic differential equations (SODEs) via strong Ito-Taylor schemes for SODEs. Due to the special structure of the RODE-SODE pair it is not necessary to restrict the intensity of the noise. Pathwise convergence is established as well as the B-stability of implicit multi-step methods. Numerical comparisons are provided for explicit schemes applied to a low dimensional RODE and implicit schemes applied to a high dimensional RODE obtained with the method of lines by spatially discretizing a random partial differential equation with finite difference quotients.

Pullback and Forward Attractors of Nonautonomous Difference Equations

In 1998 at the ICDEA Poznan the first author talked about pullback attractors of nonautonomous difference equations. That talk was published as [7] in the Journal of Difference Equations & Applications in 2000. Since then the theory of nonautonomous dynamical systems has been the topic of many papers and there are some new developments, in particular concerning the construction of forward nonautonomous attractors, that will be discussed here.

Dynamics of Nonautonomous Chemostat Models

Chemostat models have a long history in the biological sciences as well as in biomathematics. Hitherto most investigations have focused on autonomous systems, that is, with constant parameters, inputs, and outputs. In many realistic situations these quantities can vary in time, either deterministically (e.g., periodically) or randomly. They are then nonautonomous dynamical systems for which the usual concepts of autonomous systems do not apply or are too restrictive. The newly developing theory of nonautonomous dynamical systems provides the necessary concepts, in particular that of a nonautonomous pullback attractor. These will be used here to analyze the dynamical behavior of nonautonomous chemostat models with or without wall growth, time-dependent delays, variable inputs and outputs. The possibility of overyielding in nonautonomous chemostats will also be discussed.

Asymptotic separation between solutions of Caputo fractional stochastic differential equations

ABSTRACT Using a temporally weighted norm, we first establish a result on the global existence and uniqueness of solutions for Caputo fractional stochastic differential equations of order whose coefficients satisfy a standard Lipschitz condition. For this class of systems, we then show that the asymptotic distance between two distinct solutions is greater than as t → ∞ for any ϵ > 0. As a consequence, the mean square Lyapunov exponent of an arbitrary non-trivial solution of a bounded linear Caputo fractional stochastic differential equation is always non-negative.

Numerical schemes for random ODEs with affine noise

Numerical schemes for random ordinary differential equations, abbreviated RODEs, with an affine structure can be derived in a similar way as for affine control systems using Taylor expansions that resemble stochastic Taylor expansions for Stratonovich stochastic differential equations. The driving noise processes can be quite general, such as Wiener processes or fractional Brownian motions with continuous sample paths or compound Poisson processes with piecewise constant sample paths, and even more general noises. Such affine-Taylor schemes of arbitrarily high order are constructed here. It is shown how their structure simplifies when the noise terms are additive or commutative. A derivative free counterpart is given and multi-step schemes are derived too. Numerical comparisons are provided for various explicit one-step and multi-step schemes in the context of a toggle switch model from systems biology.

Weak solution of stochastic differential equations with fractional diffusion coefficient

ABSTRACT In stochastic financial and biological models, the diffusion coefficients often involve the term , or more general |x|r, r ∈ (0, 1). These coefficients do not satisfy the local Lipschitz condition, which implies that the existence and uniqueness of the solution cannot be obtained by the standard conditions. This article establishes the existence of the weak solution for this class of stochastic differential equations by using the martingale representation and weak convergence methods.

Dissipative Systems with Steady States

The preservation or stability of the zero solution to Euler schemes for dissipative systems is established using Lyapunov functions.