Defect-Mediated Turbulence in Spatio-Temporal Patterns
A lot of interest has been devoted, these last few decades, to systems driven far from equilibrium by an external parameter (see for instance the book edited by Wesfreid and Zaleski, 1984). The study of such problems involving many degrees of freedom and displaying nonlinear behaviours is expected to give some clues in the understanding of complex spatiotemporal phenomena, such as hydrodynamic turbulence. Many model equations, displaying spatio-temporal complexity (Kuramoto and Tsuzuki, 1976; Pomeau and Manneville, 1979; Chaté and Manneville, 1987), have been studied. We show in this paper that, in the framework of amplitude equations (Newell and Whitehead, 1969; Segel, 1969) which describe nonlinear dynamics near a bifurcation point, a new form of turbulence may occur (Coullet et al., 1989), which is associated with topological defects. First, we consider the simplest model which leads to such behaviours, namely the amplitude equation associated with a Hopf bifurcation. Then, same considerations are applied to amplitude equations corresponding to a Hopf bifurcation where space translational invariance is also broken, and which describe the occurence of wave patterns in an anisotropic medium.
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