Amplitude equations

Amplitude equations are reduced dynamical equations that describe the slow evolution of pattern amplitudes near a bifurcation point, where a homogeneous state loses stability and patterned states emerge. They are universal: the same amplitude equation governs convection rolls, Turing stripes, and Faraday waves, because near onset the system's behavior is determined by symmetry rather than by microscopic details. The derivation of amplitude equations from the full governing equations — via multiple-scale analysis or center manifold reduction — is one of the triumphs of applied bifurcation theory, allowing physicists to predict pattern selection without solving the full nonlinear problem.

The canonical example is the Swift-Hohenberg equation, which describes the amplitude of stripe-forming instabilities in systems with reflection symmetry. The Swift-Hohenberg equation captures not only the growth of the pattern but also the slow modulations of its phase and the competition between different orientations.