Magnus' expansion for time-periodic systems: Parameter-dependent approximations

Eric Butcher, Ma'en Sari, Ed Bueler, Tim Carlson

Research output: Contribution to journalArticle

12 Citations (Scopus)

Abstract

Magnus' expansion solves the nonlinear Hausdorff equation associated with a linear time-varying system of ordinary differential equations by forming the matrix exponential of a series of integrated commutators of the matrix-valued coefficient. Instead of expanding the fundamental solution itself, that is, the logarithm is expanded. Within some finite interval in the time variable, such an expansion converges faster than direct methods like Picard iteration and it preserves symmetries of the ODE system, if present. For time-periodic systems, Magnus expansion, in some cases, allows one to symbolically approximate the logarithm of the Floquet transition matrix (monodromy matrix) in terms of parameters. Although it has been successfully used as a numerical tool, this use of the Magnus expansion is new. Here we use a version of Magnus' expansion due to Iserles [Iserles A. Expansions that grow on trees. Not Am Math Soc 2002;49:430-40], who reordered the terms of Magnus' expansion for more efficient computation. Though much about the convergence of the Magnus expansion is not known, we explore the convergence of the expansion and apply known convergence estimates. We discuss the possible benefits to using it for time-periodic systems, and we demonstrate the expansion on several examples of periodic systems through the use of a computer algebra system, showing how the convergence depends on parameters.

Original languageEnglish (US)
Pages (from-to)4226-4245
Number of pages20
JournalCommunications in Nonlinear Science and Numerical Simulation
Volume14
Issue number12
DOIs
StatePublished - Dec 2009
Externally publishedYes

Fingerprint

Magnus Expansion
Time varying systems
Periodic Systems
Dependent
Approximation
Logarithm
Picard Iteration
Linear Time-varying Systems
Matrix Exponential
Convergence Estimates
Computer algebra system
Transition Matrix
Monodromy
Fundamental Solution
Commutator
System of Ordinary Differential Equations
Direct Method
Electric commutators
Converge
Symmetry

Keywords

  • Chebyshev polynomials
  • Magnus expansion
  • Time-periodic systems

ASJC Scopus subject areas

  • Modeling and Simulation
  • Numerical Analysis
  • Applied Mathematics

Cite this

Magnus' expansion for time-periodic systems : Parameter-dependent approximations. / Butcher, Eric; Sari, Ma'en; Bueler, Ed; Carlson, Tim.

In: Communications in Nonlinear Science and Numerical Simulation, Vol. 14, No. 12, 12.2009, p. 4226-4245.

Research output: Contribution to journalArticle

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