Folding State within a Hysteresis Loop: Hidden Multistability in Nonlinear Physical Systems

Identifying hidden states in nonlinear physical systems that evade direct experimental detection is important as disturbances and noises can place the system in a hidden state with detrimental consequences. We study a cavity magnonic system whose main physics is photon and magnon Kerr effects. Sweeping a bifurcation parameter in numerical experiments (as would be done in actual experiments) leads to a hysteresis loop with two distinct stable steady states, but analytic calculation gives a third folded steady state “hidden” in the loop, which gives rise to the phenomenon of hidden multistability. We propose an experimentally feasible control method to drive the system into the folded hidden state. We demonstrate, through a ternary cavity magnonic system and a gene regulatory network, that such hidden multistability is in fact quite common. Our findings shed light on hidden dynamical states in nonlinear physical systems which are not directly observable but can present challenges and opportunities in applications.

Figure 1

Schematic illustration of the cavity magnonic system and nonlinear steady-state dynamic behaviors of the system. (a) A YIG sphere and a Kerr medium are embedded in a microwave cavity. The microwave transmission is measured by a VNA via ports 1 and 2, and the cavity photons are strongly driven by an MV via port 1. The YIG sphere is placed in a uniformly biased magnetic field $\mathbf{H}$ that aligns the magnetization and tunes the magnon frequency. (b) Theoretically predicted and numerically calculated magnon frequency shift ${\mathrm{\Delta }}_m/2\pi$ versus the driving power $P_d$ for ${\omega }_d/2\pi =10.038\mathrm{GHz}$. The black and red triangle curves, respectively, correspond to the forward and backward sweeps of the driving power $P_d$. The blue solid and dashed curves, respectively, denote the analytically obtained stable and unstable steady states.

Figure 2

Distinct dynamical regimes and hysteresis loops. (a) Driving powers $P_d^{A-D}$ at the switching points versus the driving frequency ${\omega }_d/2\pi$. Regions I–V correspond to bistability, hidden multistability, multistability, two bistability, and again bistability, respectively. The four vertical downward black arrows from left to right denote the frequency values ${\omega }_d/2\pi =10.039$, 10.040, 10.041, and 10.042 GHz, respectively. (b)–(e) Hysteresis loops with or without an FHS for these four frequency values.

Figure 3

Detecting FHS numerically (or experimentally) through application of a suitable control driving pulse. (a) Forward control pulse with parameters $P_0=175\mathrm{mW}$, $P_l=125\mathrm{mW}$, $t_0=1\mathrm{\mu }\mathrm{s}$, and $\mathrm{\Delta }t=1\mathrm{\mu }\mathrm{s}$. (b) Time evolution of the system into the FHS. (c),(d) Similar to (a),(b) but for reversed-pulse control with parameters $P_0=175\mathrm{mW}$, $P_l=75\mathrm{mW}$, $t_0=1\mathrm{\mu }\mathrm{s}$, and $\mathrm{\Delta }t=1\mathrm{\mu }\mathrm{s}$. (e) Sweep of the FHS for system initiated at point $F$ in Fig. 1.