Spatio-Temporal Phase Patterns Near a Hopf Bifurcation in 2D Systems
The spontaneous nucleation of spatio-temporal patterns in systems driven far from thermal equilibrium by uniform constraints remains the subject of intensive theoretical and experimental research. Despite the complexity of the dynamics which gives rise to this phenomenon, great progress has been achieved in the understanding of pattern formation and stability near instability points where the reduction of the dynamics leads to amplitude equations for the patterns. Furthermore, since most of these structures appear via continuous symmetry breaking effects, long range fluctuations are expected to develop spontaneously in the ordered regime. The corresponding long wavelength modes which play the role of Goldstone modes in driven systems may be described by the appropriate phase dynamics. The case of translational symmetry-breaking has been widely investigated in the case of nonlinear reaction-diffusion equations, Rayleigh-Bénard, Taylor-Couette, convective or hydrodynamical instabilities in normal fluids or liquid crystals, …/1–3/. In the case of oscillations of the limit cycle type associated with a Hopf bifurcation, temporal symmetry breaking occurs and the phase dynamics leads to various kinds of spatiotemporal behaviors. Among them, concentric or spiral chemical waves and turbulent structures associated with the 1d Kuramoto- Sivashinsky equation have been widely investigated /4/.
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