Abstract
The application of bifurcation theory to nonlinear physical problems is reviewed. It is shown that the topological singularities and bifurcation processes deriving from the concept of structural stability determine the most significant phenomena observed in both structure formation and structure recognition. From this emerges a unifying geometrical framework for the description of nonlinear physical systems which, when passing through instabilities, exhibit analogous critical behavior both at the microscopic and macroscopic levels. After a survey on the basic concepts of singularity and bifurcation theory some new developments are outlined. These include nonlinear conservation laws in various physical fields, the relation between analytical and topological singularities in the inverse scattering problem and in phonon focusing, interacting Hopf and steady-state bifurcations in nonlinear evolution equations and applications to optical bistability and neuronal activity.
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