On macromodeling of nonlinear systems with time periodic coefficients
type: Item
instance of: scholarly article
title: On macromodeling of nonlinear systems with time periodic coefficients
zbMATH Open document ID: 1112.34023
author: Sangram Redkar
author: Eric A. Butcher
author: S. C. Sinha
published in: Communications in Nonlinear Science and Numerical Simulation
publication date: 2006-10-31T00:00:00Z
review text: The model order reduction (MOR) is a process which consists in replacing a large dynamical system by an equivalent small-scale system that can approximate the dynamics of the initial system: this macromodel is asked to preserve the desired stability and bifurcation characteristics of the original large-scale system. This paper suggests four practical order reduction techniques: a linear projection method, a nonlinear projection method based on singular perturbation, a post-processing approach and one technique using the time periodic invariant manifold. The linear and nonlinear systems discussed in this paper have time periodic coefficients. Given a periodic \(n\times n\)-matrix \(A\), with primitive period \(T,\) the linear dynamical evolution system \[ \dot{x}(t)=A(t)x(t),\quad A(t+T)=A(t), \] where \(x\) is an \(n\) vector is transformed using the Lyapunov-Floquet (L-F) transformation such that the linear parts of the new set of equations become time-invariant. Given a \(T\) time-periodic nonlinear function \(f(x,t),\) the time periodic manifold reduction allows one to study the critical dynamics of a nonlinear system \[ \dot{x}(t)=A(t)x(t)+f(x,t),\quad A(t+T)=A(t), \] with much smaller dimension than the original. It is based on the center manifold theory. Also, the numerical error due to linear approximation analysis is provided and the authors discuss advantages and limitations of the techniques by means of a practical example consisting of a system modeling the dynamics of two inverted pendulums moving in the horizontal plane with time-dependent load acting on each pendulum.
Mathematics Subject Classification ID: 34C20
Mathematics Subject Classification ID: 34C45
Mathematics Subject Classification ID: 93A30
zbMATH DE Number: 5069042
zbMATH Keywords: order reduction
zbMATH Keywords: nonlinear time periodic systems
reviewed by: Smail Djebali
MaRDI profile type: Publication
full work available at URL: j.cnsns.2004.11.005
OpenAlex ID: W2040381116
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DOI: 10.1016/J.CNSNS.2004.11.005
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description: scientific article; zbMATH DE number 5069042