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Instabilities induced by a weak breaking of a strong spatial resonance

  • Dawes, J. H. P.
  • Postlethwaite, C. M.
  • Proctor, M. R. E.
Publication Date
Nov 21, 2003
Submission Date
Nov 21, 2003
DOI: 10.1016/j.physd.2003.11.009
arXiv ID: nlin/0311045
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Through multiple-scales and symmetry arguments we derive a model set of amplitude equations describing the interaction of two steady-state pattern-forming instabilities, in the case that the wavelengths of the instabilities are nearly in the ratio 1:2. In the case of exact 1:2 resonance the amplitude equations are ODEs; here they are PDEs. We discuss the stability of spatially-periodic solutions to long-wavelength disturbances. By including these modulational effects we are able to explore the relevance of the exact 1:2 results to spatially-extended physical systems for parameter values near to this codimension-two bifurcation point. These new instabilities can be described in terms of reduced `normal form' PDEs near various secondary codimension-two points. The robust heteroclinic cycle in the ODEs is destabilised by long-wavelength perturbations and a stable periodic orbit is generated that lies close to the cycle. An analytic expression giving the approximate period of this orbit is derived.

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