Zhang B, Chan P Y, Dong X, Sun F, Chan H B
Department of Physics, The Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong, China.
William Mong Institute of Nano Science and Technology, The Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong, China.
Sci Rep. 2025 May 30;15(1):18965. doi: 10.1038/s41598-025-98844-w.
Critical slowing down of the dynamics of a system near bifurcation points leads to long recovery times towards stable states in response to perturbations. Analogously, for systems initially in an unstable state, the relaxation also becomes slow near bifurcation points. Here we explore the onset of self-sustained oscillations in a sideband-driven electromechanical resonator, when the zero-amplitude state changes from stable to unstable. As the system moves away from the unstable zero-amplitude state due to thermal fluctuations, the vibration amplitude increases exponentially with time until nonlinear effects limit the growth and the system settles into stable self-sustained vibrations. We show that the first passage time for the amplitude to reach a threshold value is random and follows a non-Gaussian distribution. On the other hand, the rate of exponential buildup remains constant for different build-up events. As the system approaches a bifurcation point, the build-up of vibrations slows down drastically. The mean and the standard deviation of the first passage time as well as the inverse rate of exponential rise exhibit power law scaling with the distance to either the supercritical or subcritical Hopf bifurcation point with exponent of - 1.
系统在分岔点附近动力学的临界减慢会导致系统在受到扰动后向稳定状态恢复的时间变长。类似地，对于初始处于不稳定状态的系统，在分岔点附近弛豫也会变慢。在此，我们探讨当零振幅状态从稳定变为不稳定时，边带驱动的机电谐振器中自激振荡的起始情况。当系统由于热涨落而远离不稳定的零振幅状态时，振动幅度随时间呈指数增长，直到非线性效应限制了增长，系统进入稳定的自激振动状态。我们表明，振幅达到阈值的首次通过时间是随机的，且遵循非高斯分布。另一方面，不同增长事件的指数增长速率保持恒定。当系统接近分岔点时，振动的增长会急剧减慢。首次通过时间的均值和标准差以及指数上升的倒数速率与到超临界或亚临界霍普夫分岔点的距离呈现幂律缩放关系，指数为 -1。
