## Journals

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### Open Access Journals

DCDS

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.

DCDS

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.

DCDS-B

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.

CPAA

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.

CPAA

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.

DCDS

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.

DCDS-S

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.

DCDS

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.

CPAA

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.

CPAA

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.## Year of publication

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