Symmetry Breaking Homoclinic Chaos

Abstract

The essentially spatio-temporal dynamics of extended physical systems is still poorly understood, Indeed, in spite of the success of dynamical systems theory to explain the deterministic chaos occurring in confined situations, this approach is unable to handle the complex behaviors appearing when the spatial structure is not frozen. From this point of view, turbulence may be seen as the ultimate complex system, and it seems interesting to study models of extended systems somewhat “simpler” than the Navier-Stokes equations but nevertheless retaining some of the essential physics of the problem. In such models (one-dimensional P.D.E.’s, cellular automata), a transition to turbulence via spatio-temporal intermittency is often observed [1,2,3]. Such regimes are characterized by the coexistence of clusters of laminar cells and groups of coherent cells within a sea of turbulent patches. Now, a most intriguing problem in the theory of hydrodynamic turbulence is the formation of large-scale, coherent structures (C.S.) in a flow performing random turbulent motion of small scales. The C.S. are important in transport of heat, mass and momentum, and they are responsible for the intermittent nature of turbulence in many shear and boundary layer flows. Is it possible that the dynamics of C.S. are related to spatio-temporal intermittences observed in simpler, lower-dimensional systems?