Illusions of Criticality: Crises Without Tipping Points

Abstract

Abrupt shifts in ecosystems, brains, markets, and climate are often diagnosed as signs of approaching a tipping point, i.e. a critical bifurcation where stability is lost. Here we reveal a broader and more deceptive mechanism: pseudo-bifurcations. In stochastic non-normal systems, asymmetric interactions produce transient episodes of apparent instability despite long-term stability. We show analytically, numerically, and with empirical evidence from brain dynamics during epileptic seizures that pseudo-bifurcations reproduce the full set of early-warning signals usually taken as proof of proximity to tipping points, including critical slowing down, increased variance, and dimensional collapse. Crucially, these false alarms can occur well before any true bifurcation, systematically biasing crisis diagnosis. This discovery reframes how abrupt transitions are interpreted across disciplines: what has long been attributed to ``criticality'' may instead reflect the hidden geometry of non-normal dynamics. By uncovering this illusion of criticality, we call for a fundamental reassessment of how crises are identified, predicted, and managed in natural, social, and technological systems.

Figures (12)

• Figure 1: (top) Schematic representation of equation \ref{['eq:basic_dynamics']} in its reduced form \ref{['eq:dim_red']}. (bottom) Associated transient dynamics for $||\mathbf{x}||_2$ both as a function of time (main) and in the reduced $(n,r)$-plane (inset).
• Figure 2: (Top) Behavior of the Ornstein-Uhlenbeck parameter $\theta_t$ (\ref{['eq:theta_root']}) for different levels of non-normality $\kappa$. For $\kappa > \kappa_c$, the typically positive mean-reversion coefficient $\theta_t$ transiently becomes negative, thereby inducing repulsive growth away from the stable fixed point. (Bottom) Phase diagram illustrating the system's regimes as a function of the leading eigenvalue $\lambda_+=-\alpha +1$ from the reduced model \ref{['eq:dim_red']} and the condition number $\kappa$, which quantifies non-normality. As the bifurcation at $\lambda_+ = 0$ is approached at fixed $\kappa$, the system first traverses the pseudo-critical regime as $\kappa_c$ becomes smaller than $\kappa$.
• Figure 3: (a-c) Mean-field variable, effective eigenvalue $\lambda_\text{eff}$ and variance defined via \ref{['eq:MF_AR1']} from a simulated system \ref{['eq:basic_dynamics']} with $N=10$ dimensions: normal case (grey line, $\kappa=1$) and non-normal case (black line, $\kappa=30$) with otherwise identical conditions. (e-g) Same as (a-c) but only for the non-normal system for a specific excerpt for better resolution. The vertical dashed line highlights a pseudo-bifurcation at which the effective eigenvalue crosses zero. (d) Reduced dynamics \ref{['eq:dim_red']} of Prop.\ref{['pr:dim_red']} in the $(n,r)$-plane. While the non-normal system (yellow to red) displays distinct transient repulsion and contraction along the reaction mode $r$, the normal system (green to blue) shows no such trends (as also highlighted in the re-scaled inset figure). (h) Complementary cumulative distribution (CCD) of projection along the reaction mode $r$ from subplot (d). The CCD of the non-normal system (dark line) is a mixture of two distributions: one dominated by noise, and the other dominated by quasi-deterministic loops. The second distribution is absent in the normal case (grey line). The probability distribution function (PDF) of the mean-field (inset) shows no such distinction between the normal and non-normal systems. (i) Autocorrelation of the mean-field dynamics. (j) Lead-lag correlation between the system's alignment along the non-normal mode and the mean-field dynamics, with a clear indication that the former leads the later when the system is non-normal.
• Figure 4: The five top panels show time series driven by time-dependent control parameters. For the non-normal system (Reaction, top), the control parameter is the condition number $\kappa$. For the normal forms (Fold, Transcritical, and Supercritical/Subcritical Pitchfork), the control parameter governs the system's approach to the bifurcation (see Supplementary Material for details). All systems are subjected to additive i.i.d. Gaussian noise with amplitude $\delta = 10^{-2}$. The two bottom panels display, over a rolling window of 30 time units, the evolution of the effective eigenvalue and the variance for each simulation.
• Figure 5: (Top Left) Mean-field state of the 23 EEG channels before, during (red), and after an epileptic seizure. Inset plots show a high-resolution version of a single second before (blue) and during (red) the seizure. (Bottom Left) Condition number $\kappa$ of the EEG connectome estimated over a rolling window of 40 seconds (light grey line). An additional moving average over a time scale of 10 seconds is applied, resulting in the smoother black line. (Top Right) Loops, similar to Figure \ref{['fig:simulations']}(d), along the non-normal and reaction components during an excerpt of non-seizure (left/blue; stochastic loops) and seizure (right/red; quasi-deterministic loops). (Bottom Right) Lead-lag correlation between the system's exposure to the non-normal component and the mean-field state during a non-seizure (blue frame) and a seizure (red frame) period. The purple vertical line indicates when the correlation reaches its maximum. During the seizure, activity along the non-normal component consistently precedes the mean-field excursions by approximately 0.1 seconds.

Theorems & Definitions (16)

• Definition 1
• Definition 2
• Definition 3
• Definition 4
• Proposition 1
• Proposition 2
• Proposition 3
• Definition 5
• Proposition 4
• Proposition 5