Numerical dynamics

AAU

 

Computation of nonautonomous invariant and
inertial manifolds

(with M. Rasmussen), Numerische Mathematik 112, 2009,
449-483


We derive a numerical scheme to compute invariant manifolds for time-variant discrete dynamical systems, i.e., nonautonomous difference equations. Our universally applicable method is based on a truncated Lyapunov-Perron operator and computes invariant manifolds using a system of nonlinear algebraic equations which can be solved both locally using (nonsmooth) inexact Newton, and globally using continuation algorithms. Compared to other algorithms, our approach is quite flexible, since it captures time-dependent, nonsmooth, noninvertible or implicit equations and enables us to tackle the full hierarchy of strongly stable, stable and center-stable manifolds, as well as their unstable counterparts.

 

Our results are illustrated using a test example and are applied to a population dynamical model and the Hénon map. Finally, we discuss a linearly implicit Euler-Bubnov-Galerkin discretization of a reaction diffusion equation in order to approximate its inertial manifold.

Computation of integral manifolds for Carathéodory differential equations

(with M. Rasmussen) IMA Journal of Numerical Analysis 30(2),
2010, 401-430


We derive two numerical approximation schemes for local invariant manifolds of nonautonomous ordinary differential equations which can be measurable in time and Lipschitzian in the spatial variable. Our approach is inspired by previous work of Jolly and Rosa (2005), `Computation of non-smooth local center manifolds', IMA Journal of Numerical Analysis, 25, 698--725, on autonomous ODEs and based on truncated Lyapunov-Perron operators. Both of our methods are applicable to the full hierarchy of strongly stable, stable,     center-stable and the corresponding unstable manifolds, and exponential refinement strategies yield exponential convergence.

 

Several examples illustrate our approach.

Discrete inertial manifolds

Mathematische Nachrichten 281(6), 2008, 847-878


This work is devoted to attractive invariant manifolds for nonautonomous difference equations, occurring in the discretization theory for evolution equations. Such invariant sets provide a discrete counterpart to inertial manifolds of dissipative FDEs and evolutionary PDEs. We discuss their essential properties, like smoothness, the existence of an asymptotic phase, normal hyperbolicity and attractivity in a nonautonomous framework of pullback attraction.

 

As application we show that inertial manifolds of the Allen-Cahn and complex Ginzburg-Landau equation persist under discretization. For the Ginzburg-Landau equation we can also estimate the dimension of the inertial manifold.

Attractive invariant fiber bundles

Applicable Analysis 86(6), 2007, 687-722


This work on implicit nonautonomous difference equations (iterations) is devoted to an abstract technical result on the existence of attractive invariant manifolds. We investigate their existence, smoothness and their asymptotic phase property using invariant foliations. Such results lay the basic foundation of further investigations on discretization methods for evolutionary differential equations.

Christian Pötzsche (Mar 2023)

Integral manifolds under explicit variable time-step discretization

(with S. Keller) Journal of Difference Equations and Applications
12(3-4), 2006, 321-342


We study the behavior of the “full hierarchy“ of integral manifolds, i.e., in particular those of stable, center-stable, center-unstable and unstable type, for nonautonomous ordinary differential equations in Banach spaces under explicit one-step discretization with varying step-sizes. Or main results on Cm-closeness under such discretizations are formulated in a quantitative fashion and turn out to be an easy consequence of a general theorem on the existence of invariant fiber bundles within the “calculus on time scales“.

 

This paper is dedicated to the memory of our teacher Prof. Dr. Bernd Aulbach. As an excellent teacher he introduced us to the theory of dynamical systems, advised us throughout our studies and has always been an inspiration. As a mathematician he was full of ideas, visions and plans. And even beyond mathematics we were benefitting from his humanity.

Numerical dynamics of integrodifference equations:
Basics and discretization errors in a C0-setting
Applied Mathematics and Computation 354, 2019, 422-443


Integrodifference equations are successful and popular models in theoretical ecology to describe spatial dispersal and temporal growth of populations with nonoverlapping generations. In relevant situations, such infinite-dimensional discrete dynamical systems have a globally attractive periodic solution. We show that this property persists under sufficiently accurate spatial (semi-) discretizations of collocation- and degenerate kernel-type using linear splines. Moreover, convergence preserving the order of the method is established. This justifies theoretically that simulations capture the behavior of the original problem. Several numerical illustrations confirm our results.

Numerical dynamics of integrodifference equations:
Global attractivity in a C0-setting
SIAM Journal on Numerical Analysis 57(5), 2019, 2121-2141


Besides being interesting infinite-dimensional dynamical systems in discrete time, integrodifference equations successfully model growth and dispersal of populations with nonoverlapping generations, and are often illustrated by simulations. This paper points towards and initiates a mathematical foundation of such simulations using generic methods to numerically discretize (and solve) integral equations. We tackle basic properties of a flexible class of integrodifference equations, as well as of their collocation and degenerate kernel semi-discretizations on the state space of continuous functions over a compact domain. Moreover, various estimates for the global discretization error are provided. Numerical simulations illustrate and confirm our theoretical results.

Uniform convergence of Nyström discretizations on Hölder spaces

Journal of Integral Equations and Applications 34(2), 2022, 247-255


We establish that Nyström discretizations of linear Fredholm integral operators on Hölder spaces converge in the operator norm while preserving the consistency order of the quadrature or cubature rule. This allows to employ tools from classical perturbation theory, rather than collective compactness, when studying numerical approximations of integral operators, as well as applications in for instance the field of nonautonomous dynamical systems.

Numerical dynamics of integrodifference equations:
Hierarchies of invariant bundles in Lp(𝜴)
to appear in Numerical Functional Analysis and Optimization, 2023


We study how the ''full hierarchy" of invariant manifolds for nonautonomous integrodifference equations on the Banach spaces of $p$-integrable functions behaves under spatial discretization of Galerkin type. These manifolds include the classical stable, center-stable, center, center-unstable and unstable ones, and can be represented as graphs of $C^m$-functions. For kernels with a smoothing property, our main result establishes closeness of these graphs in the $C^{m-1}$-topology under numerical discretizations preserving the convergence order of the method. It is formulated in a quantitative fashion and technically results from a general perturbation theorem on the existence of invariant bundles (i.e.\ nonautonomous invariant manifolds).

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