Resonances and period doubling in the pulsations of stellar models

Abstract

The nonlinear pulsational behavior of several sequences of state-of-the-art Cepheid models is computed with a numerical hydrodynamics code. These sequences exhibit period doubling as the control parameter, the effective temperature, is changed. By following the evolution of the Floquet stability coefficients of the periodic pulsations, this period doubling is identified with the destabilization of a vibrational overtone mode through a resonance of the type (2n + 1) omega (0) equal to about 2 omega (k) (n integer). In the weakly dissipative Population I Cepheids, only a single period doubling and subsequent undoubling is observed, whereas in the case of the strongly dissipative Population II Cepheids, a cascade of period doublings and chaos can occur. The basic properties of the period doubling bifurcation are examined within the amplitude equation formalism, leaving little doubt about the resonance origin of the phenomenon. A simple model system to two coupled nonlinear oscillators which mimics the behavior of the complicated stellar models is also analyzed.