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Dimension Reduction methods for Infinite Dimensional Systems
Author(s) -
Steindl Alois
Publication year - 2003
Publication title -
pamm
Language(s) - English
Resource type - Journals
ISSN - 1617-7061
DOI - 10.1002/pamm.200310197
Subject(s) - dynamical systems theory , reduction (mathematics) , dimension (graph theory) , nonlinear system , dimensional reduction , dynamical system (definition) , statistical physics , dimensionality reduction , inertial frame of reference , nonlinear dynamical systems , work (physics) , mathematics , point (geometry) , physics , mathematical analysis , classical mechanics , computer science , geometry , pure mathematics , mathematical physics , quantum mechanics , artificial intelligence , thermodynamics
From experiments and also from computer simulation of dynamical systems it is well known that for many dynamical phenomena in physics or engineering, which are modelled by infinite dimensional dynamical systems, the asymptotic behavior can be accurately described by replacing the original infinite dimensional system by a low dimensional system represented by so‐called essential variables. Such a dimension reduction of a dynamical system turns out to be central, both for a qualitative and quantitative understanding of its behaviour. In [1] Approximate Inertial Manifolds are presented, which perform extremely well for nonlinear evolution equations, but don't work as expected for the dynamics of a fluid conveying tube. By comparing the results for different internal damping values it can be seen that the larger gaps and the location of the cluster point in the spectrum for the weaker damping improve the approximation quality considerably.