Robustness of asymptotic stability to small time delays
Desheng Li P.E. Kloeden
The robustness of asymptotic stability properties of ordinary differential equations with respect to small constant time delays is investigated. First, a local robustness result is established for compact asymptotically stable sets of systems with nonlinearities which need be only continuous, so the solutions may even be non-unique. The proof is based on the total stability of the differential inclusion obtained by inflating the original system. Using this first result, it is shown that an exponentially asymptotically stable equilibrium of a nonlinear equation which is Lipschitz in a neighborhood of the equilibrium remains exponentially asymptotically stable under small time delays. Then a global result regarding robustness of exponential dissipativity to small time delays is established with the help of a Lyapunov function for nonlinear systems which satisfy a global Lipschitz condition. The extension of these results to variable time delays is indicated. Finally, conditions ensuring the continuous convergence of the delay system attractors to the attractor of the system without delays are presented.
keywords: Lyapunov functions Setvalued dynamical systems global attractors exponential dissipativity delay differential equations differential inclusions continuous convergence. asymptotic stability
The perturbation of attractors of skew-product flows with a shadowing driving system
P.E. Kloeden Victor S. Kozyakin
The influence of the driving system on a skew-product flow generated by a triangular system of differential equations can be perturbed in two ways, directly by perturbing the vector field of the driving system component itself or indirectly by perturbing its input variable in the vector field of the coupled component. The effect of such perturbations on a nonautonomous attractor of the driven component is investigated here. In particular, it is shown that a perturbed nonautonomous attractor with nearby components exists in the indirect case if the driven system has an inflated nonautonomous attractor and that the direct case can be reduced to this case if the driving system is shadowing.
keywords: perturbations attractors Skew product flow shadowing.
Equi-Attraction and the continuous dependence of attractors on time delays
P.E. Kloeden Pedro Marín-Rubio
Under appropriate regularity conditions it is shown that the continuous dependence of the global attractors $\mathcal{A}_\tau$ of semi dynamical systems $S^{(\tau)}(t)$ in $C([-\tau,0];Z)$ with $Z$ a Banach space and time delay $\tau \in [T_*,T^$*$]$, where $T_* > 0$, is equivalent to the equi-attraction of the attractors. Examples and counter examples posed in this right framework are provided.
keywords: extended semiflows and attractors parametric attractors continuity of attractors and equi-attraction. Semiflows for delay differential equations
Pitchfork and transcritical bifurcations in systems with homogeneous nonlinearities and an almost periodic time coefficient
P.E. Kloeden
The zero solution of a vector valued differential equation with an autonomous linear part and a homogeneous nonlinearity multiplied by an almost periodic function is shown to undergo pitchfork or transcritical bifurcations to small nontrivial almost periodic soutions as a leading simple real eigenvalue of the linear part crosses the imaginary axis.
keywords: Nonautonomous differential equation almost periodic solutions transcritical bifurcation. pitchfork bifurcation
Asymptotic behaviour of the nonautonomous SIR equations with diffusion
María Anguiano P.E. Kloeden
The existence and uniqueness of positive solutions of a nonautonomous system of SIR equations with diffusion are established as well as the continuous dependence of such solutions on initial data. The proofs are facilitated by the fact that the nonlinear coefficients satisfy a global Lipschitz property due to their special structure. An explicit disease-free nonautonomous equilibrium solution is determined and its stability investigated. Uniform weak disease persistence is also shown. The main aim of the paper is to establish the existence of a nonautonomous pullback attractor is established for the nonautonomous process generated by the equations on the positive cone of an appropriate function space. For this an energy method is used to determine a pullback absorbing set and then the flattening property is verified, thus giving the required asymptotic compactness of the process.
keywords: asymptotic stability pullback attractors flattening property. asymptotic compactness nonautonomous equilibria nonautonomous dynamical systems SIR epidemic model with diffusion
Uniform nonautonomous attractors under discretization
P.E. Kloeden Victor S. Kozyakin
A nonautonomous or cocycle dynamical system that is driven by an autonomous dynamical system acting on a compact metric space is assumed to have a uniform pullback attractor. It is shown that discretization by a one-step numerical scheme gives rise to a discrete time cocycle dynamical system with a uniform pullback attractor, the component subsets of which converge upper semi continuously to their continuous time counterparts as the maximum time step decreases to zero. The proof involves a Lyapunov function characterizing the uniform pullback attractor of the original system.
keywords: perturbations discretization. Cocycle dynamical systems attractors
Non-autonomous attractors for integro-differential evolution equations
Tomás Caraballo P.E. Kloeden
We show that infinite-dimensional integro-differential equations which involve an integral of the solution over the time interval since starting can be formulated as non-autonomous delay differential equations with an infinite delay. Moreover, when conditions guaranteeing uniqueness of solutions do not hold, they generate a non-autonomous (possibly) multi-valued dynamical system (MNDS). The pullback attractors here are defined with respect to a universe of subsets of the state space with sub-exponetial growth, rather than restricted to bounded sets. The theory of non-autonomous pullback attractors is extended to such MNDS in a general setting and then applied to the original integro-differential equations. Examples based on the logistic equations with and without a diffusion term are considered.
keywords: Integro-differential equation set-valued process differential equation with infinite delay pullback attractor. set-valued non-autonomous dynamical system
Numerical and finite delay approximations of attractors for logistic differential-integral equations with infinite delay
Tomás Caraballo P.E. Kloeden Pedro Marín-Rubio
The upper semi-continuous convergence of approximate attractors for an infinite delay differential equation of logistic type is proved, first for the associated truncated delay equation with finite delay and then for a numerical scheme applied to the truncated equation.
keywords: Attractors for delay differential equations numerical and theoretical approximations of solutions.
Equivalence of invariant measures and stationary statistical solutions for the autonomous globally modified Navier-Stokes equations
P.E. Kloeden Pedro Marín-Rubio José Real
A new proof of existence of solutions for the three dimensional system of globally modified Navier-Stokes equations introduced in [3] by Caraballo, Kloeden and Real is obtained using a smoother Galerkin scheme. This is then used to investigate the relationship between invariant measures and statistical solutions of this system in the case of temporally independent forcing term. Indeed, we are able to prove that a stationary statistical solution is also an invariant probability measure under suitable assumptions.
keywords: Modified Navier-Stokes model statistical solutions invariant measures.
Pullback V-attractors of the 3-dimensional globally modified Navier-Stokes equations
P.E. Kloeden José A. Langa José Real
The existence and finite fractal dimension of a pullback attractor in the space $V$ for a three dimensional system of the nonautonomous Globally Modified Navier-Stokes Equations on a bounded domain is established under appropriate properties on the time dependent forcing term. These equations were proposed recently by Caraballo et al and are obtained from the Navier- Stokes Equations by a global modification of the nonlinear advection term. The existence of the attractor is obtained via the flattening property, which is verified.
keywords: weak solutions flattening property existence and uniqueness of strong solutions 3-dimensional Navier-Stokes equations nonautonomous and pullback attractors.

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