diagram between possible pulsational states to be mapped out. In this picture, the boundaries of the instability strip where pulsation sets in during the star's evolution correspond to a Hopf bifurcation. The existence of a center manifold eliminates the possibility of chaotic (i.e. irregular) pulsations on the time scale of the period. Although resonant amplitude equations are sufficiently complex to also allow for chaotic solutions, this is a very different chaos because it is in the temporal variation of the amplitudes and occurs on a long time scale. While long term irregular behavior in the temporal variations of the pulsation amplitudes is possible when amplitude equations apply, this is not the general situation. Indeed, for the majority of the observations and modeling, the pulsations of these stars occur with constant Fourier amplitudes, leading to regular pulsations that can be periodic or multi-periodic (quasi-periodic in the mathematical literature). Irregular pulsations The light curves of intrinsic variable stars with large amplitudes have been known for centuries to exhibit behavior that goes from extreme regularity, as for the classical Cepheids and the RR Lyrae stars, to extreme irregularity, as for the so-called Irregular variables. In the Population II stars this irregularity gradually increases from the low period W Virginis variables through the RV Tauri variables into the regime of the semiregular variables. Low-dimensional chaos in stellar pulsations is the current interpretation of this established phenomenon. Regular behavior of the Cepheids The regular behavior of the Cepheids has been successfully modeled with numerical hydrodynamics since the 1960s, and from a theoretical point of view it is easily understood as due to the presence of center manifold which arises because of the weakly dissipative nature of the dynamical system. This, and the fact that the pulsations are weakly nonlinear, allows a description of the system in terms of amplitude equations and a construction of the bifurcation diagram (see also bifurcation theory) of the possible types of pulsation (or limit cycles), such fundamental mode pulsation, first or second overtone pulsation, or more complicated, double-mode pulsations in which several modes are excited with constant amplitudes. The boundaries of the instability strip where pulsation sets in during the star's evolution correspond to a Hopf bifurcation. Irregularity of Population II stars In contrast, the irregularity of the large amplitude Population II stars is more challenging to explain. The variation of the pulsation amplitude over one period implies large dissipation,