Nonlinear Oscillations in One-Zone Stellar Model

Abstract

If one-zone models are available for some pulsating stars and the motion remains close to adiabaticity, the oscillation and stability criterion of the star have been analyzed by canonical perturbation theory and optimal control theory, separately under different function values on the right-hand side of Eq. (10). We found that (i) If the introduction of a given perturbation function of time b(t) is substituted into the right-hand side of Eq. (10), and energy-optimal control of a nonlinear oscillator is satisfied, the oscillation in one-zone model is unstable. (ii) If is subject to the constraint , and time-optimal control is satisfied, the oscillation in one-zone stellar model is stable. (iii) The periodic oscillation criterion is obtained by a simple method, if b is regarded as a constant of integration in Eq. (10). It shows that the adiabatic exponent is all important. When is greater than or equal to 4/3, the periodic oscillation should occur in one-zone stellar model; when is less than 4/3, and under the additional condition of , the periodic oscillation could not occur in this model.