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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

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 consider discrete time systems $x_{k+1}=U(x_{k};\lambda)$,
$x\in\R^{N}$, with a complex parameter $\lambda$.
The map $U(\cdot;\lambda)$ at infinity contains a principal linear
term, a bounded positively homogeneous nonlinearity, and a smaller
part. We describe the sets of parameter values for which the
large-amplitude $n$-periodic trajectories exist for a fixed $n$.
In the related problems on small periodic orbits near zero,
similarly defined parameter sets, known as Arnold tongues, are more narrow.

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