center manifold, or more precisely a near center manifold. In addition, it has been found that the stellar pulsations are only weakly nonlinear in the sense that their description can be limited powers of the pulsation amplitudes. These two properties are very general and occur for oscillatory systems in many other fields such as population dynamics, oceanography, plasma physics, etc. The weak nonlinearity and the long time scale of the amplitude variation allows the temporal description of the pulsating system to be simplified to that of only the pulsation amplitudes, thus eliminating motion on the short time scale of the period. The result is a description of the system in terms of amplitude equations that are truncated to low powers of the amplitudes. Such amplitude equations have been derived by a variety of techniques, e.g. the Poincaré–Lindstedt method of elimination of secular terms, or the multi-time asymptotic perturbation method, and more generally, normal form theory. For example, in the case of two non-resonant modes, a situation generally encountered in RR Lyrae variables, the temporal evolution of the amplitudes A1 and A2 of the two normal modes 1 and 2 is governed by the following set of ordinary differential equations where the Qij are the nonresonant coupling coefficients. These amplitude equations have been limited to the lowest order nontrivial nonlinearities. The solutions of interest in stellar pulsation theory are the asymptotic solutions (as time tends towards infinity) because the time scale for the amplitude variations is generally very short compared to the evolution time scale of the star which is the nuclear burning time scale. The equations above have fixed point solutions with constant amplitudes, corresponding to single-mode (A1 0, A2 = 0) or (A1 = 0, A2 0) and double-mode (A1 0, A20) solutions. These correspond to singly periodic and doubly periodic pulsations of the star. It is important to emphasize that no other asymptotic solution of the above equations exists for physical (i.e., negative) coupling coefficients. For resonant modes the appropriate amplitude equations have additional terms that describe the resonant coupling among the modes. The Hertzsprung progression in the light curve morphology of classical (singly periodic) Cepheids is the result of a well-known 2:1 resonance among the fundamental pulsation mode and the second overtone mode. The amplitude equation can be further extended to nonradial stellar pulsations. In the overall analysis of pulsating stars, the amplitude equations allow the bifurcation