Robustness anylysis of periodic oscillations in continuous-time crystals

<sec>Continuous time crystals represent a novel state in many-body systems that can self-organize into timeperiodic oscillations without external periodic driving. Recent experiments have achieved such systems in dissipative solid-state materials, where persistent temporal order is autonomously sustained. A decisive characteristic of time crystals is their robustness, meaning that despite various disturbances, including fluctuations in internal parameters and external noise, they can still maintain rhythmic behavior, which has scientific value and echnological application potential. Although previous studies have shown that specific experimental parameters have robustness, thare is a lack of a systematic framework for quantifying and predicting their resilience to disturbances, and the underlying physics of this robustnessis still not fully understood. The key unresolved problems include how nonlinear interactions and feedback mechanisms contribute to stability, and what the critical thresholds are for parameter variations beyond which temporal order collapses.</sec><sec>This work addresses these gaps by systematically analyzing how internal parameters and external influences affect the oscillation period and overall stability. Internally, the dynamics are determined by dipole-dipole interactions and atomic transition strengths, which define the temporal symmetry breaking that occurs in the system. Externally, the response of the system is controlled by the strength of the optical driving field and the energy dissipation rate. A key finding is the determination of an intrinsic feedback mechanism for a dynamically stabile time crystal. This mechanism plays a role in restoring force, correcting deviations caused by minor disturbances, and maintaining the coherence of oscillatory phase.</sec><sec>Moreover, the system displays nonlinear dynamical behavior, characterized by two different states: one is stable oscillation continuing under moderate disturbance, and the other is stronger disturbance causing dynamical phase transition, resulting in switching between disordered or dynamically unstable state and stable state. These results provide a comprehensive understanding of the various behaviors observed in continuous time crystals and lay an important theoretical foundation for utilizing their unique properties in advanced applications such as quantum information processing and precision metrology.</sec>