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DCDS-B

Two-sided error estimates are derived for the strong error of convergence
of the stochastic theta method. The main result is based on two ingredients.
The first one shows how the theory of convergence can be embedded into standard
concepts of consistency, stability and convergence by an appropriate
choice of norms and function spaces. The second one is a
suitable stochastic generalization of Spijker's norm (1968) that is known to
lead to two-sided error estimates for deterministic one-step methods.
We show that the stochastic theta method is bistable with
respect to this norm and that well-known results on the optimal
$\mathcal{O}(\sqrt{h})$ order of convergence follow from this
property in a natural way.

keywords:
stochastic Spijker norm.
,
two-sided error estimate
,
stochastic theta method
,
SODE
,
consistency
,
bistability

DCDS-B

This paper is devoted to the numerical analysis of abstract semilinear parabolic
problems $u'(t) = Au(t) + f(u(t)), u(0)=u^0,$ in some general Banach space $E$. We
prove a shadowing Theorem that compares solutions of the continuous problem with those of a semidiscrete approximation (time stays continuous) in the neighborhood of a hyperbolic equilibrium. We allow rather general discretization schemes following the theory of discrete approximations developed by F. Stummel, R.D. Grigorieff and G. Vainikko. We use a compactness principle to show that the decomposition of the flow into growing and decaying solutions persists for this general type of approximation. The main assumptions of our results are naturally satisfied for operators with compact resolvents and can be verified for finite element as well as finite difference methods. In this way we obtain a unified approach to shadowing results derived e.g. in the finite element context ([19, 20, 21]).

JCD

By a classical theorem transversal homoclinic points of maps lead
to shift dynamics on a maximal invariant set, also referred to as
a homoclinic tangle. In this paper we study the fate of
homoclinic tangles in parameterized systems from the viewpoint of
numerical continuation and bifurcation theory. The new
bifurcation result shows that the maximal invariant set near a
homoclinic tangency, where
two homoclinic tangles collide, can be characterized by a system
of bifurcation equations that is indexed by a symbolic sequence.
These bifurcation equations consist of a finite or infinite set of
hilltop normal forms known from singularity theory.
For the Hénon family we determine numerically the connected components
of branches with multi-humped homoclinic orbits that pass through
several tangencies.
The homoclinic network found by numerical continuation is explained
by combining our bifurcation result with graph-theoretical arguments.

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