Amplitude Equations for Systems with Competing Instabilities

We describe how to derive ordinary differential equations to predict the time dependence of a system governed by partial differential equations when that system is near to the polycritical condition for the onset of several instabilities. The method is illustrated for the general case of two competing instabilities and then a specific physical example of this case is worked out in detail. The basic idea is to extend the Krylov–Bogolyubov–Mitropolsky method of nonlinear mechanics to bifurcation problems.