Abstract

Time-periodic standing waves are demonstrated to be low-dimensional by use of the proper orthogonal decomposition (POD). Moreover, the nonlinear dynamics of the system restricted to this low-dimensional linear subspace are shown to accurately recover the spatio-temporal full PDE dynamics. A global set of modes, generated with sequential POD, are then used to produce time-periodic standing wave branches as a function of the period. This representation quantitatively reproduces the entire branch, including both large- and small amplitude solutions, using only a few POD modes. This technique offers a new direction of exploration for this challenging problem, including an efficient way to characterize the bifurcation structure and stability of these solutions.

Keywords

  1. proper orthogonal decomposition
  2. periodic orbits
  3. standing water waves

MSC codes

  1. 37C27
  2. 62H25
  3. 65M70

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Published In

cover image SIAM Journal on Applied Dynamical Systems
SIAM Journal on Applied Dynamical Systems
Pages: 1033 - 1061
ISSN (online): 1536-0040

History

Submitted: 30 August 2011
Accepted: 31 May 2012
Published online: 13 September 2012

Keywords

  1. proper orthogonal decomposition
  2. periodic orbits
  3. standing water waves

MSC codes

  1. 37C27
  2. 62H25
  3. 65M70

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