Efficient energy transfer often occurs between oscillation modes in a resonator when they are tuned to internal resonance. We design the eigenfrequencies of two vibrational modes of an electromechanical resonator to be close to a ratio of $3:1$ and demonstrate that the energy supplied to the upper mode can be controllably transferred to the lower mode. With the lower mode vibrating with a period tripled that of the upper mode, the discrete time-translation symmetry imposed by the periodic drive is broken. The lower mode settles into one of three stable period-tripled states with different phases. This channel for energy transfer from the upper mode can be turned on or off without changing system parameters. When the upper mode itself becomes multistable under strong resonant or parametric drive, additional sets of coexisting period-tripled states emerge in the lower mode. In the latter case, we measure a total of six coexisting vibration states with identical amplitude but phases differing by $\pi /3$. Excitation of coexisting states with three different phases could open new opportunities in designing mechanical memory based on ternary logic. Coupled resonators with period-tripled states can also be used to model complex interacting systems with spin equal to one.

中文翻译：

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内部共振耦合机电模式中的能量转移到三倍周期状态

当谐振器调谐到内部谐振时，通常在谐振器中的振荡模式之间发生有效的能量转移。我们将机电谐振器的两种振动模式的特征频率设计为接近$3:1$并证明提供给上模的能量可以可控地转移到下模。随着下模的振动周期是上模的三倍，周期性驱动施加的离散时间平移对称性被打破。较低的模式进入具有不同相位的三个稳定的三倍周期状态之一。无需更改系统参数即可打开或关闭此用于从上层模式传输能量的通道。当上模本身在强共振或参数驱动下变为多稳态时，在下模中会出现额外的共存三倍周期状态集。在后一种情况下，我们总共测量了六个振幅相同但相位不同的共存振动状态$\pi /3$. 激发具有三个不同阶段的共存状态可以为基于三元逻辑的机械存储器设计开辟新的机会。具有三倍周期状态的耦合谐振器也可用于模拟自旋等于 1 的复杂相互作用系统。