Making Hidden States Visible
Many physical systems can be in more than one stable state: A laser can be on or off, and a magnetic bit up or down. This multistability can appear in nonlinear resonators—such as ferromagnets and genetic toggle switches in cells—that are driven into different steady states, or “branches,” by ramping up and down the driving parameter [1]. This behavior is often pictured using a familiar hysteresis loop that traces the system’s trajectory between a lower branch and higher branch (Fig. 1). It is easy to imagine that additional steady states might coexist with those sampled, but experiments have largely ignored that possibility, assuming instead that slow, quasistatic parameter sweeps reveal all “physically relevant” behavior.
In a new acoustic experiment, Kun Zhang from the Wuhan University in China and colleagues challenge that assumption [2]. They show that a pair of coupled acoustic cavities can host a fully “folded” steady state that is perfectly stable yet invisible to conventional sweeps. This hidden branch can, however, be reached with carefully designed sound pulses, the team shows. These results—combined with those from another recent study [3]—turn the abstract idea of hidden multistability into a concrete and controllable feature of nonlinear resonator networks, which might one day be used to securely store sensitive information.
In the simplest case, a system will respond linearly to a driving input. For example, the sound intensity in a simple cavity will increase as the power to an input speaker is ramped up. There’s only one steady state in this case, characterized by a straight line. But if nonlinear effects are significant, the system can be driven into multiple steady states, depending on both the starting point and the drive protocols. In the textbook picture, this multistability is depicted by an S-shaped response curve, where the top, middle, and bottom branches of the “S” correspond to three separate steady states. The middle state is typically unstable, so the system toggles between the top and bottom states. Specifically, the system traces a familiar hysteresis loop between the two states as the driving parameter is swept back and forth in the usual protocol.
However, a recent theoretical analysis by Meng-Xi Bi from Xi’an University of Posts and Telecommunications in China and colleagues showed that this simple picture can be incomplete [4]. The researchers found combinations of nonlinearities for which the response curve folds more intricately into three stable steady states, while a standard sweep explores only two branches (Fig. 1, right panel). In their terminology, the extra branch is hidden not because it is fragile or short-lived but because its basin of attraction (the set of initial states that relax into a given steady state) does not intersect the trajectory followed by the usual protocol [5]. Bi and colleagues proposed using short pulses in the driving parameter to kick the system out of the usual trajectory, allowing it to settle into the basin of a hidden steady state.
Zhang and colleagues implement this idea in a tabletop acoustic platform consisting of two air-filled cavities coupled acoustically through an air channel [2]. Speakers in each cavity supply the driving sound waves, while microphones record the sound-pressure response. A long-standing challenge in acoustics is that sizeable intrinsic nonlinearities usually require very large sound pressures, which are hard to generate and control. Zhang and colleagues’ approach sidesteps this difficulty by synthesizing an effective Kerr-like response through electroacoustic feedback. Specifically, one cavity is engineered to shift its resonance frequency upward with sound intensity, and the other downward. This combination of effective stiffening and softening produces a nonlinear response curve with three coexisting steady states.
To explore these steady states, Zhang and colleagues drove one cavity at a fixed frequency and slowly ramped the drive amplitude up and down. Initially, they saw only the expected hysteresis loop associated with two stable branches: one at high sound pressure and the other at low sound pressure. Standard sweeps never landed on the third branch. To reveal the hidden state, the researchers superimposed short pulses on the slow sweep that transiently changed the acoustic field and carried the system out of its current basin and into a different basin. When the pulse ended, the system relaxed into the new basin’s stable state, allowing the team to reproducibly transfer the system from a visible branch to the hidden one and to map out its full extent.
In parallel with this on-site-nonlinearity route, my colleagues and I have explored a complementary mechanism [3]. We engineered a pair of coupled acoustic resonators in which the coupling strength itself depends on the oscillation amplitude and can even change sign. In this nonlinearly coupled resonator pair, or “dimer,” the steady-state equation for the resonance amplitudes also supports three stable solutions, of which only two can be visited by a simple up–down sweep. Instead of using short pulses, we rely on the fact that a nonlinear, sign-changing coupling reshapes the effective normal modes of the dimer. By following a multistep, quasiadiabatic path in parameter space, we gradually deformed the landscape so that the trajectory from rest was diverted into the basin of the intermediate state. This experiment and that of Zhang and colleagues thus embody complementary strategies—pulses and tailored ramps—for steering nonlinear systems through parts of their state space that ordinary protocols overlook.
From the standpoint of nonlinear dynamics, there is nothing mysterious about extra stable solutions to a higher-order steady-state equation. What is new is the decision to treat those solutions as physically relevant states whose accessibility can be quantified and controlled. Hidden multistability shifts attention from asking how many steady states exist to asking which steady states are reachable under which protocols. That shift matters for both understanding and design. Hidden steady states could be exploited for multilevel memories or secure encoding schemes in which certain configurations are reachable only via special sequences of operations. Conversely, identifying hidden basins may help define robust operating regimes that reliably avoid them in safety-critical devices.
Looking ahead, the notion of hidden multistability is likely to become even richer in larger networks. Arrays of coupled resonators can host many coexisting basins of attraction and complex switching pathways shaped by gain and loss. In such systems, protocol-dependent accessibility may intertwine with other current themes in wave physics, such as non-Hermitian gain–loss engineering [6], topological protection [7, 8], and explicit time modulation [9]. The two acoustic experiments show that the “extra” steady states folded inside a hysteresis loop are not just mathematical curiosities. They are real, measurable, and manipulable players in driven–dissipative resonator systems—and they invite us to think of multistability not as a fixed landscape but as one that can be navigated in many more ways than a simple up–down sweep would suggest.
References
- D. Kaplan and L. Glass, Understanding Nonlinear Dynamics, Textbooks in Mathematical Science (Springer Verlag, New York, 1995)[Amazon][WorldCat].
- K. Zhang et al., “Experimental observation of hidden multistability in nonlinear systems,” Phys. Rev. Lett. 136, 037201 (2026).
- Z.-X. Chen et al., “Nonlinear coupling induced anomalous state transfer and complete multistate excitation via adiabatic control,” Phys. Rev. Lett. 136, 037202 (2026).
- M.-X. Bi et al., “Folding state within a hysteresis loop: Hidden multistability in nonlinear physical systems,” Phys. Rev. Lett. 132, 137201 (2024).
- D. Dudkowski et al., “Hidden attractors in dynamical systems,” Phys. Rep. 637, 1 (2016).
- K. Takata and M. Notomi, “Photonic topological insulating phase induced solely by gain and loss,” Phys. Rev. Lett. 121, 213902 (2018).
- R. Flores-Calderón et al., “Topological quantum criticality from multiplicative topological phases,” Phys. Rev. Lett. 134, 116602 (2025).
- M. Jürgensen et al., “Quantized fractional Thouless pumping of solitons,” Nat. Phys. 19, 420 (2023).
- E. Galiffi et al., “Photonics of time-varying media,” Adv. Photon. 4, 014002 (2022).