@article{16710, abstract = {{In this work we present a set-oriented path following method for the computation of relative global attractors of parameter-dependent dynamical systems. We start with an initial approximation of the relative global attractor for a fixed parameter λ0 computed by a set-oriented subdivision method. By using previously obtained approximations of the parameter-dependent relative global attractor we can track it with respect to a one-dimensional parameter λ > λ0 without restarting the whole subdivision procedure. We illustrate the feasibility of the set-oriented path following method by exploring the dynamics in low-dimensional models for shear flows during the transition to turbulence and of large-scale atmospheric regime changes . }}, author = {{Gerlach, Raphael and Ziessler, Adrian and Eckhardt, Bruno and Dellnitz, Michael}}, issn = {{1536-0040}}, journal = {{SIAM Journal on Applied Dynamical Systems}}, pages = {{705--723}}, title = {{{A Set-Oriented Path Following Method for the Approximation of Parameter Dependent Attractors}}}, doi = {{10.1137/19m1247139}}, year = {{2020}}, } @article{33263, abstract = {{Dynamical systems often admit geometric properties that must be taken into account when studying their behavior. We show that many such properties can be encoded by means of quiver representations. These properties include classical symmetry, hidden symmetry, and feedforward structure, as well as subnetwork and quotient relations in network dynamical systems. A quiver equivariant dynamical system consists of a collection of dynamical systems with maps between them that send solutions to solutions. We prove that such quiver structures are preserved under Lyapunov--Schmidt reduction, center manifold reduction, and normal form reduction.}}, author = {{Nijholt, Eddie and Rink, Bob W. and Schwenker, Sören}}, issn = {{1536-0040}}, journal = {{SIAM Journal on Applied Dynamical Systems}}, keywords = {{Modeling and Simulation, Analysis}}, number = {{4}}, pages = {{2428--2468}}, publisher = {{Society for Industrial & Applied Mathematics (SIAM)}}, title = {{{Quiver Representations and Dimension Reduction in Dynamical Systems}}}, doi = {{10.1137/20m1345670}}, volume = {{19}}, year = {{2020}}, } @article{16708, abstract = {{ In this work we extend the novel framework developed by Dellnitz, Hessel-von Molo, and Ziessler to the computation of finite dimensional unstable manifolds of infinite dimensional dynamical systems. To this end, we adapt a set-oriented continuation technique developed by Dellnitz and Hohmann for the computation of such objects of finite dimensional systems with the results obtained in the work of Dellnitz, Hessel-von Molo, and Ziessler. We show how to implement this approach for the analysis of partial differential equations and illustrate its feasibility by computing unstable manifolds of the one-dimensional Kuramoto--Sivashinsky equation as well as for the Mackey--Glass delay differential equation. }}, author = {{Ziessler, Adrian and Dellnitz, Michael and Gerlach, Raphael}}, issn = {{1536-0040}}, journal = {{SIAM Journal on Applied Dynamical Systems}}, number = {{3}}, pages = {{1265--1292}}, title = {{{The Numerical Computation of Unstable Manifolds for Infinite Dimensional Dynamical Systems by Embedding Techniques}}}, doi = {{10.1137/18m1204395}}, volume = {{18}}, year = {{2019}}, } @article{16581, author = {{Dellnitz, Michael and Klus, Stefan and Ziessler, Adrian}}, issn = {{1536-0040}}, journal = {{SIAM Journal on Applied Dynamical Systems}}, pages = {{120--138}}, title = {{{A Set-Oriented Numerical Approach for Dynamical Systems with Parameter Uncertainty}}}, doi = {{10.1137/16m1072735}}, year = {{2017}}, } @article{16527, author = {{Day, S. and Junge, O. and Mischaikow, K.}}, issn = {{1536-0040}}, journal = {{SIAM Journal on Applied Dynamical Systems}}, pages = {{117--160}}, title = {{{A Rigorous Numerical Method for the Global Analysis of Infinite-Dimensional Discrete Dynamical Systems}}}, doi = {{10.1137/030600210}}, year = {{2004}}, }