Instabilities induced by a weak breaking of a strong
Jonathan Dawes, Claire Postlethwaite and Mike Proctor.
DAMTP, University of Cambridge
Using 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.
The ODEs for exact resonance have been studied by many previous authors and may contain traveling waves, heteroclinic cycles and complex dynamics. Analytic progress on the PDE version is a substantial challenge but it is possible to discuss in detail the stability of the steady 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-three bifurcation point. The new instabilities we find can be described in terms of reduced `normal form' PDEs near secondary codimension-two points. Of particular interest is the interaction of two distinct phase instabilities.
Analysis of the spatially periodic solutions leads naturally to stability of the robust heteroclinic cycle in the ODEs. This becomes destabilized by long-wavelength perturbations and a stable periodic orbit is generated close to the cycle.