Will a time-varying complex system be stable?

Abstract

Randomly-assembled dynamical systems are theoretically predicted to be unstable upon crossing a critical threshold of complexity, as first shown by May. Yet, empirical complex systems exhibit remarkable stability, indicating the presence of additional mechanisms playing a stabilizing role. The relation between complexity and stability is typically assessed by assuming fixed interactions, whereas real systems often evolve in intrinsically time-dependent states. To understand how this affects stability, we linearize a general non-autonomous dynamics around a reference operating state and model the resulting parameters as stochastic processes, which represent the minimal extension of static random interactions to time-varying ones. We derive exact stability bounds that generalize complexity-stability theory to dynamically varying systems. Notably, we find that temporal variability allows systems to remain stable even when their instantaneous Jacobian would predict instability. We compare our results against a non-linear neural network model, where our theory applies exactly, and the generalized Lotka-Volterra equations, where we numerically find that time-varying interactions systematically postpone the onset of replica-symmetry breaking. Overall, our results indicate that temporal variability systematically improves stability, demonstrating a general mechanism by which complex systems can violate classical complexity-stability bounds.

Figures (6)

• Figure 1: Phase diagram of the linear model Eq. (\ref{['eq:model']}) with annealed interactions as set by Eq. (\ref{['eq:model-correlations']}). Below the bound given by Eq. (\ref{['eq:stability-bound']}) (solid line), the model converges to a stable stationary state, while above it the system is unstable. The two stable phases differ in that, below the May bound $R=1$ (dashed line) in the "Static stability" region, all eigenvalues of the instantaneous Jacobian have negative real part. Instead, above the May bound, some eigenvalues of the Jacobian have positive real part, yet the dynamics is stabilized by sufficiently rapid temporal variability, in the "Dynamic stability" phase. The critical complexity below which the system remains dynamically stable increases as the scale of temporal variability $\tau$ is reduced, and diverges as $\tau\to0$.
• Figure 2: Validation of the stability bound in a neural network model with annealed interactions. In the upper panel we compare the critical complexity threshold Eq. (\ref{['eq:stability-bound']}) with $R=\sigma g$ (solid line) with numerical integration of the model Eq. (\ref{['eq:NN']}) with interactions given by Eq. (\ref{['eq:NN-interactions']}) (circles and triangles). Each realization is classified as stable if it settles into the fixed point $x_i=0$, and unstable if it reaches a non-zero stationary state. The agreement is excellent, as expected since our stability bound applies exactly to this model. The lower panels qualitatively display the dynamics in the two phases.
• Figure 3: Numerical phase diagram of generalized Lotka-Volterra equations with annealed interactions. For each parameter set, we simulate two replicas of the GLV dynamics Eq. (\ref{['eq:GLV']}) which share the same disorder realization $\alpha_{ij}(t)$ but start from different initial conditions. The system is classified as replica-symmetric (RS) if the distance $d(t)$ [Eq. (\ref{['eq:d(t)']})] vanishes at long times, and replica-symmetry-breaking (RSB) otherwise. The dashed line marks the stability boundary for the quenched case zenari2026generalized, while the dotted line marks the numerical boundary separating the RS and RSB phases. As in Fig. \ref{['fig:phase-diagram']}, the lower green region indicates static stability, while the intermediate blue area indicates dynamic stability (see text). Consistent with the linear model, temporal variability extends the stable RS regime beyond the quenched limit. The response function is $J(x)=2x/(2+x)$ and the correlation time is $\tau=1$. The lower panels qualitatively show single species trajectories for the two replicas in each phase.
• Figure 4: Qualitative explanation for the stabilization mechanism in time-varying systems. Left: The instantaneous spectrum of the Jacobian matrix $M_{ij}(t)$ is constant, and a fraction of this spectrum can have positive real part (red points). Right: Dynamics of perturbations at different times (curved arrow). The stable (green dashed lines) and unstable (red solid lines) directions continuously shift as the Jacobian matrix evolves, which effectively averages out the divergence. The panels are illustrative, as eigenvectors are generally complex.
• Figure 5: Numerical simulations of the linear model Eq. (\ref{['eq:model']}) of the main text with parameters $M_{ij}(t)$ and $h_i(t)$ having different correlation times $\tau_M$ and $\tau_h$, as prescribed by Eq. (\ref{['eq:different-Mh']}). The left panel shows $\tau_M=1$, and the right panel shows $\tau_M=10$. The solid line is the theoretical result given by Eq. (\ref{['eq:stability-bound']}) of the main text with $\tau=\tau_M$. Each simulation is classified as stable if it reaches a finite stationary state or unstable if it diverges. As expected, stability is independent of the parameter $\tau_h$, which results in a constant stability line as a function of this parameter.