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Instabilities induced by a weak breaking of a strong spatial resonance

J. H. P. Dawes·C. M. Postlethwaite·M. R. E. Proctor
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Abstract

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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J. H. P. Dawes, C. M. Postlethwaite, M. R. E. Proctor. 2003-11-21. Instabilities induced by a weak breaking of a strong spatial resonance. https://doi.org/10.1016/j.physd.2003.11.009

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