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 (A 1 ≠ ≠ 0, A 2 = 0) or (A 1 = 0, A 2 ≠ ≠ 0) and double-mode (A 1 ≠ ≠ 0, A 2 ≠ ≠ 0) solutions. These correspond to singly periodic and doubly periodic pulsations of the star. No other asymptotic solution of the above equations exists for physical (i.e., negative) coupling coefficients.