Nonlinear Sciences – Pattern Formation and Solitons
Scientific paper
2003-11-21
Nonlinear Sciences
Pattern Formation and Solitons
43 pages. Elsevier style files used
Scientific paper
10.1016/j.physd.2003.11.009
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.
Dawes Jonathan H. P.
Postlethwaite Claire M.
Proctor Michael R. E.
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