EVENT DETAILS AND ABSTRACT

Title: Weakly unstable systems in the presence of neutrally stable modes
Speaker: Professor J.D. Crawford
Speaker Info: U. Pittsburgh
Brief Description:
Special Note:
Abstract:

Near the onset of a linear instability, the nonlinear evolution of the mode amplitudes can be described using an expansion in the amplitude of the unstable modes. Since the growth rates are very small near onset, nonlinear effects often act to saturate the instability before the amplitudes grow appreciably; for this reason such expansions have proved a powerful tool for studying the nonlinear states emerging from the bifurcation. In addition, the prototypical dissipative examples share another characteristic: except for the unstable modes, all other linear modes are exponentially damped. In particular, there are no neutrally stable modes with nonlinear couplings to the unstable modes. If such neutral modes are present, the amplitude equations for the unstable modes, i.e. normal forms on the unstable manifold, can have very different features. We discuss several illustrative examples: the normal form for a Hopf/saddle node mode interaction, the Vlasov equation from plasma physics, and a continuum model for the Kuramoto system of phase oscillators. The normal forms in these examples exhibit singularities that we interpret as evidence of new nonlinear scalings and also show no evidence of being finitely determined.