Instabilities induced by a weak breaking of a strong spatial resonance

PHYSICA D-NONLINEAR PHENOMENA

卷:191

Article

Dawes, JHP ; Postlethwaite, CM ; Proctor, MRE

关键词: pattern;    symmetry;    mode interaction;    bifurcation;    heteroclinic cycle;   

DOI  :  10.1016/j.physd.2003.11.009

来源: Elsevier

【 摘 要 】

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. (C) 2003 Elsevier B.V. All rights reserved.

【 授权许可】

Free