Phase Synchronization of two anharmonic NEMS oscillators

I. Theoretical derivation of synchronization equations for two anharmonic oscillators We start with the slow time equation[1] for two oscillators with two feedback[2, 3] terms: one which is common to both oscillators (fc), and one that affects the corresponding oscillator only (fi, i=1,2). These are í µí±‘í µí°´̃ 1 í µí±‘í µí±‡ − í µí±– (í µí»¿ 1 2 + í µí¼† 11 |í µí°´̃ 1 | 2) í µí°´̃ 1 + í µí°´̃ 1 2 = í µí±“ 1 (í µí°´̃ 1) + í µí±“ í µí± (í µí°´̃ 1 , í µí°´̃ 2), (S.I.1) and í µí±‘í µí°´̃ 2 í µí±‘í µí±‡ − í µí±– (í µí»¿ 2 2 + í µí¼† 22 |í µí°´̃ 2 | 2) í µí°´̃ 2 + í µí°´̃ 2 2 = í µí±“ 2 (í µí°´̃ 2) + í µí±“ í µí± (í µí°´̃ 2 , í µí°´̃ 1) (S.I.2) where í µí»¿ 1,2 = í µí±„(í µí¼” 1,2 2 í µí¼” 0 2 − 1) defines the offset of the natural resonance frequency of each NEMS device í µí¼” 1,2 to a nearby frequency, í µí¼” 0 , í µí°´̃ = í µí±Ž ̃í µí±’ í µí±–í µí¼‘ is the complex slow time oscillator displacement, and the nonlinear coefficient í µí¼† is the relationship of the device displacement to the relative change in the NEMS resonance frequency (there is a factor of 3/8 that has been folded into this constant with respect to references 1 and 2). The terms í µí±“ 1,2 and í µí±“ í µí± are the " oscillator feedback " and " coupling signal " in Figure 1, respectively. For a saturated oscillator feedback with linear, reactive, diffusive coupling these equations become