__Instabilities
induced by a weak breaking of a strong__

__spatial
resonance__

** 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.