Numerical dynamics of integrodifference equations: Forward dynamics and pullback attractors

Huy Huynh; Peter E. Kloeden; Christian Pötzsche

doi:10.3934/jcd.2023003

Article Contents

2023, Volume 10, Issue 2: 345-370. Doi: 10.3934/jcd.2023003

This issue Previous Article Approximated exponential integrators for the stochastic Manakov equation Next Article

Numerical dynamics of integrodifference equations: Forward dynamics and pullback attractors

1.
Institut für Mathematik, Universität Klagenfurt, 9020 Klagenfurt, Austria
2.
Mathematisches Institut, Universität Tübingen, 72076 Tübingen, Germany

Received: January 2022

Revised: February 2023

Early access: April 2023

Published: April 2023

The work of Huy Huynh has been supported by the Austrian Science Fund (FWF) under grant number P 30874-N35.
Peter E. Kloeden thanks the Universität Klagenfurt for support and hospitality

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Abstract

In order to determine the dynamics of nonautonomous equations both their forward and pullback behavior need to be understood. For this reason we provide sufficient criteria for the existence of such attracting invariant sets in a general setting of nonautonomous difference equations in metric spaces. In addition it is shown that both forward and pullback attractors, as well as forward limit sets persist and that the latter two notions even converge under perturbation.

As concrete application, we study integrodifference equations over the continuous functions under spatial discretization of collocation type. Integrodifference equation and Pullback attractor and Forward attractor and Urysohn operator

Keywords:

Mathematics Subject Classification: 37C70, 37G35, 45G15, 65R20, 65P40.

Citation:

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Figure 1. Shape of generalized Beverton-Holt functions $ g_\alpha(z): = \tfrac{z}{1+z^\alpha} $ for $ \alpha = \tfrac{1}{2} $ (dashed), $ \alpha = 1 $ (solid) and $ \alpha = 2 $ (dash dotted)

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Figure 2. Sequences of sets containing functions in the fibers $ {\mathcal A}_n^\ast(t) $, $ 0\leq t\leq 10 $, for $ \alpha \in\left\{\frac{1}{2}, 1\right\} $, respectively from left to right

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Figure 3. Development of the averaged error $ \|\xi^n-\xi^{2n}\| $ for $ n \in\left\{2^4, \ldots, 2^{10}\right\} $ nodes and parameters $ \alpha \in\left\{\frac{1}{2}, 1,2\right\} $

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Table 1. Values of the approximate convergence rates $ c(n) $ for $ n\in\left\{{2^4, \ldots, 2^{10}}\right\} $ nodes and parameters $ \alpha\in\left\{{\tfrac{1}{2}, 1, 2}\right\} $

$ n $	$ \alpha=\tfrac{1}{2} $	$ \alpha=1 $
16	2.112614126300029	2.100856100109834
32	2.055209004601208	2.051123510423984
64	2.026777868073563	2.025681916479720
128	2.013096435137189	2.012865860858589
256	2.006536458546063	2.006433295172438
512	2.003256451919377	2.003220576642864
1024	2.001624177537549	2.001610772707442

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Table 2. Values of the approximate convergence rates $ c(n) $ for $ n\in\left\{{2^4, \ldots, 2^{10}}\right\} $ nodes and final times $ T\in\left\{{10, 50}\right\} $

$ n $	$ T=10 $	$ T=50 $
16	2.2837654186745091	2.0849355987819758
32	2.1026388978422323	2.0460569864690159
64	2.0383208155788273	2.0248832042126796
128	2.0182226164652075	2.0155618789522607
256	2.0137461227397369	1.9944650101138011
512	2.0168071129306111	2.0415755409000083
1024	2.1561021773053772	2.6711767140843996

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