Hidden Multistability in Dynamical Systems

Updated 9 November 2025

Hidden multistability is the coexistence of multiple stable attractors in a nonlinear dynamical system whose basins are isolated from equilibrium neighborhoods.
It emerges when standard parameter sweeps and equilibrium-based methods fail to reach attractors, necessitating targeted perturbations or specific initial conditions.
Experimental and theoretical studies in circuits, quantum systems, and biochemical networks validate its significance in understanding complex dynamical behaviors.

Hidden multistability is the coexistence of multiple stable invariant sets ("attractors") of a nonlinear dynamical system, which, unlike conventional ("visible") multistability, are not accessible by standard parameter continuation or by initializing near fixed points. These hidden states manifest when their basins of attraction do not intersect neighborhoods of any equilibrium or are not encountered under adiabatic parameter sweeps, requiring either intricate initial condition preparation or deliberate non-adiabatic perturbations to be reached. Hidden multistability has been rigorously analyzed in continuous and discrete dynamical systems, stochastic reaction networks, quantum master equations, and experimentally in nonlinear coupled-cavity platforms, illustrating its fundamental significance for both mathematical theory and practical applications in physics, engineering, and biology.

1. Precise Definition and Classification

The formal notion of a hidden attractor is grounded in basin topology. For an $n$ -dimensional autonomous system

$\dot x = f(x), \qquad x \in \mathbb{R}^n,\ f:\mathbb{R}^n \to \mathbb{R}^n,$

let $\mathcal{E} = \{x^* : f(x^*) = 0\}$ be the set of equilibria and $A \subseteq \mathbb{R}^n$ a compact invariant set attracting an open subset of initial conditions $B(A) = \{ x_0 : \lim_{t \to \infty} \phi(t, x_0) \in A \}$ . The attractor $A$ is called

self-excited if $B(A) \cap U(x^*) \neq \emptyset$ for some $x^* \in \mathcal{E}$ and every sufficiently small neighborhood $U(x^*)$ ;
hidden if $B(A) \cap U(x^*) = \emptyset$ for every $x^* \in \mathcal{E}$ and sufficiently small $U(x^*)$ .

A system is multistable if it admits two or more mutually disjoint basins of distinct attractors. Hidden multistability arises when two or more hidden attractors coexist, with basins isolated from all equilibrium neighborhoods (Kuznetsov, 2015, Escalante-González et al., 2019). This structural classification extends to stochastic and quantum systems, as well as parameter-dependent systems where certain stable branches ("hidden branches") cannot be traversed via continuous variation of control parameters (Zhang et al., 6 Nov 2025, Ma et al., 20 Aug 2024).

2. Mechanisms and Theoretical Criteria

The emergence of hidden multistability fundamentally relies on the system's inability to reach all attractors via traditional (local) bifurcation theory:

Non-accessibility via equilibrium neighborhoods: If all trajectories launched near equilibria tend to other attractors or escape, any attractor whose domain cannot be accessed this way is hidden.
Adiabatic inaccessibility in parameter space: In nonlinear driven–dissipative systems with polynomial nonlinearities of order $d>3$ , folded S-shaped and butterfly-shaped bifurcation manifolds yield stable branches invisible to parameter sweeps. Mathematically, root-counting discriminants and fold (saddle-node) bifurcation conditions,

$P(x, \lambda) = 0, \quad \frac{\partial P}{\partial x}(x, \lambda) = 0,$

delineate multi-stable regions, while discriminants determine intervals where extra real roots (hidden states) appear (Ma et al., 20 Aug 2024).

In chemical reaction networks, algebraic geometry provides necessary and sufficient conditions for multistationarity. For instance, in small zero-one (Boolean) networks, the minimal occurrence of (nondegenerate) hidden multistability corresponds to dimension-$3$ systems whose steady-state polynomials admit multiple real positive roots not accounted for by the sign of the Jacobian determinant (Jiao et al., 17 Jun 2024).

In discrete stochastic gene toggles, hidden multistability arises from noise-induced splitting of the stationary probability distribution into multiple isolated peaks, corresponding to attractor basins that have no deterministic analog (Strasser et al., 2011).

Structural symmetry and geometry play a crucial role: e.g., in topological Kuramoto models, hidden multistability emerges when cycle boundary sizes exceed four, as determined by winding-number constraints (Bačić et al., 7 Oct 2025).

3. Canonical Examples and Experimental Realization

A broad range of systems realizing hidden multistability include:

Chua circuits: Three-dimensional Chua circuits with saturated nonlinearity can display hidden chaotic attractors coexisting with a stable equilibrium, with no initial condition in any equilibrium neighborhood leading to the hidden state (Kuznetsov, 2015).
Piecewise-linear switched systems: By designing switching surfaces to rupture heteroclinic connections, one can isolate the basin of a hidden attractor while self-excited attractors persist elsewhere (Escalante-González et al., 2019). Physically, these structures can be engineered for systems with arbitrary numbers of coexisting hidden/self-excited attractors.
Rydberg atom lattices: Driven–dissipative arrays of Rydberg atoms, described by high-degree mean-field polynomials, show hidden multistability through the presence of tristable regions not explored in standard spectroscopic sweeps, but revealed by preparing the system in appropriate initial conditions (Ma et al., 20 Aug 2024).
Acoustic Kerr-coupled cavities: Experimental demonstration of hidden multistability has been achieved in double oscillator platforms with competing self-focusing and self-defocusing Kerr nonlinearities, where conventional slow parameter sweeps produce only observable bistability, but pulsed excitations can transfer the system among fully hidden stable branches (Zhang et al., 6 Nov 2025).
Stochastic bistable gene networks: In two-stage gene toggle switches, hidden multistability explains the existence of "primed" cell states not captured in deterministic models, with the attractors revealed only in the full master equation (Strasser et al., 2011).
Disentanglement-induced multistability in quantum open systems: Nonlinear master equations with a dissipator promoting disentanglement lead to degenerate self-consistency equations and multiple diagonal (in energy) steady states inaccessible in the standard linear theory, thus enabling hidden multistability among quantum states (Buks, 27 May 2024).

4. Analytical and Computational Detection Methods

The localization and verification of hidden multistability necessitates specialized global and computational techniques:

Homotopy (continuation) methods: Track an attractor under smooth parameter deformation from a known self-excited regime into parameter sets where the attractor becomes hidden, confirming its non-accessibility from equilibria (Kuznetsov, 2015, Kuznetsov et al., 2015).
Perpetual-point approach: In continuous systems, find points where the acceleration vanishes but velocity does not, which can lie in the basins of hidden attractors, facilitating their identification.
Semi-algebraic and real-root classification: For polynomial dynamical systems, enumerate networks or motifs and analyze the discriminant variety to classify parameter regions that support multiple nondegenerate steady states (Jiao et al., 17 Jun 2024).
Data-driven methods: Model-free techniques, notably parameter-aware Echo State Networks (ESNs), can infer hidden attractors and their basins from time series of an accessible attractor and extrapolate to parameter values and initial conditions revealing coexisting, otherwise hidden, dynamics (Roy et al., 2022).
Direct basin sampling and pulse protocols: In experiments and numerical explorations, rapid perturbations (e.g., pulses) can initiate transitions across basin boundaries, mapping the domain of hidden branches not accessed in adiabatic sweeps (Zhang et al., 6 Nov 2025).

5. Dynamics, Bifurcations, and Phase Diagrams

Hidden multistability is fundamentally tied to global bifurcations and the topology of the phase space:

Fold and cusp catastrophes: In mean-field models with high-order feedback, multistability regions are organized by fold (saddle-node) bifurcations, where hidden branches appear as folded-in wells of an effective potential. The boundaries among visible, semi-hidden, and fully hidden multistable regimes are precisely defined by polynomial discriminants and their associated bifurcation lines (Ma et al., 20 Aug 2024).
Irreversibility and non-reciprocal transitions: In systems such as Doubochinski's pendulum, transitions between multistable modes through saddle-node bifurcations can be fundamentally irreversible, with routes between certain attractors forbidden by smooth parameter sweeps unless external intervention is applied (Luo et al., 2019).
Structural cascades: In oscillator networks with higher-order topology, the number and structure of multistable states are determined by the product of allowed winding numbers for cycles associated with cell boundaries, leading to cascades of hidden states as the cycle size increases (Bačić et al., 7 Oct 2025).

6. Applications and Broader Implications

Hidden multistability is of principal relevance across multiple domains:

Cellular decision-making: Minimal motifs supporting hidden multistability can be embedded as "core multistability atoms" in larger biochemical networks (e.g., phosphorylation cascades), underlying robust decision-making and irreversible switching in cell fate (Jiao et al., 17 Jun 2024).
Quantum information: Disentanglement-induced hidden multistability offers a mechanism for emergent stable quantum states not in the span of conventional master equation solutions, with implications for quantum memory and phase transitions (Buks, 27 May 2024).
Information storage and cryptography: Hidden states can encode information securely, as their retrieval requires nontrivial sequences of perturbations rather than simple parameter adjustments, a feature physically realized in acoustic Kerr platforms (Zhang et al., 6 Nov 2025).
Engineering and safety: In electronics, power grids, and control systems (PLLs, rotors), hidden attractors can mark critical zones of instability, loss of synchrony, or undesired transitions, demanding refined analysis and control protocols beyond conventional equilibrium-based design (Kuznetsov, 2015).

Table: Types of Hidden Multistability Across Systems

Physical Context	Origin of Hiddenness	Typical Detection Method
Nonlinear circuits	Isolated basins via nonlinearity	Continuation, basin sampling
Stochastic biochemistry	Noise-induced multiple minima	Quasi-potential, full CME solution
Driven-dissipative atoms	Folded branches in polynomial	Parameter sweep + deliberate pulsing
Coupled oscillators	Higher-order topological cycles	Winding-number analysis, loop constraints
Quantum open systems	Nonlinear master equation	Self-consistency solution, bifurcation

Hidden multistability thus constitutes a universal organizing principle for complex dynamical systems, encapsulating states that are structurally or stochastically shielded from typical local analysis, parameter tracking, or small-noise perturbations, and is now an indispensable concept at the intersection of mathematical theory, experimental physics, and technological design.