Figure 2

Reduced dynamics of parametric nonlinear (trigonometric) oscillator: (a, b) Energy landscape and trajectory of uncoupled Ising spins when increasing the feedback gain from below the threshold (\(\alpha =0.8\)) to above the threshold (\(\alpha =1.3\)). (c) Time evolution of the real part of the reduced dynamics with feedback gain \(\alpha =0.8\) (cyan) and \(\alpha =1.3\) (blue), where the dash line indicates flipping the sign of the spins will not affect the bistability. (d) Stability analysis of the uncoupled spins. When the feedback gain is below the threshold, the real part of the reduced dynamics will only have one stable fixed point at \(x_1 = 0\) (gray dot and arrow). When the feedback gain is above the threshold, the stable fixed points at \(x_1 = 0\) become unstable (red ring and arrow), and there exist two symmetric stable fixed points at \(x_2 = -x_3\) (red dot and arrow). (e, f) Projected energy landscape on real axis and trajectory of two uncoupled Ising spins when increasing the feedback gain from below the threshold (\(\alpha =0.8\)) to above the threshold (\(\alpha =1.3\)). (g, h) Projected energy landscape on real axis and trajectory of two coupled Ising spins when increasing the feedback gain from below the threshold (\(\alpha =\beta =0.5\)) to above the threshold (\(\alpha =\beta =0.6\)).