Time-Dependent Ginzburg-Landau Equations for Hard-Mode Instabilities

An equation for self-oscillations with small amplitudes near hard-mode instabilities is derived by means of a reductive perturbation approach. The present equation can be used to discuss both normally and invertedly bifurcating cases, bacause the present equation is equivalent to each set of equations resulting from reductive perturbations up to the fifth order, in spite of the difference of expansion parameters for reductive perturbations according as the types of bifurcations. In this paper, an explicit expression of the present equation is given for the Fitz Hugh-Nagumo model, while the present method can be used generally for hard-mode instabilities. Some typical behavior of hard-mode instabilities is also discussed with the aid of the present equation.