Two-sided error estimates for the stochastic theta method
Wolf-Jürgen Beyn Raphael Kruse
Discrete & Continuous Dynamical Systems - B 2010, 14(2): 389-407 doi: 10.3934/dcdsb.2010.14.389
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
Shadowing for discrete approximations of abstract parabolic equations
Wolf-Jürgen Beyn Sergey Piskarev
Discrete & Continuous Dynamical Systems - B 2008, 10(1): 19-42 doi: 10.3934/dcdsb.2008.10.19
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]).
keywords: hyperbolic equilibria semidiscretization abstract parabolic problems Theory of shadowing compact convergence of resolvents.
Continuation and collapse of homoclinic tangles
Wolf-Jürgen Beyn Thorsten Hüls
Journal of Computational Dynamics 2014, 1(1): 71-109 doi: 10.3934/jcd.2014.1.71
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.
keywords: bifurcation of homoclinic orbits. Homoclinic tangency symbolic dynamics numerical continuation

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