Pseudo-Coherence and Stochastic Synchronization: A Non-Normal Route to Collective Dynamics without Oscillators

Abstract

Collective temporal organization in complex systems is commonly attributed to synchronization, resonance, or proximity to dynamical instabilities. Here we identify a distinct mechanism by which coherent, synchronization-like behavior can emerge in stochastic systems that are linearly stable and contain no intrinsic oscillators. The mechanism arises from non-normal pseudospectral amplification and leads to what we term pseudo-coherence: an intermittent form of collective organization characterized by transient phase alignment, broken time-reversal symmetry, positive entropy production, and drifting spectral peaks. Using a minimal overdamped stochastic model, we show that increasing non-normality drives a sharp pseudo-critical transition. Beyond a well-defined threshold, fluctuations concentrate along a dominant reaction mode, generating intermittent growth of Kuramoto-like order parameters and irreversible probability currents without eigenvalue crossings or Hopf bifurcations. Analytically, we demonstrate that pseudo-critical non-normal dynamics reshapes the imaginary pseudospectrum, amplifying slow fluctuations and producing coherent frequency bands under finite-time observation. These results identify pseudo-coherence as a new route to collective temporal organization in non-equilibrium systems, suggesting that apparent rhythms and synchronization in natural systems may arise from non-normal stochastic amplification rather than intrinsic oscillators.

Figures (7)

• Figure 1: Emergent coherence and synchronization induced by non-normality in an overdamped stochastic system without oscillators. Synthetic time series \ref{['eq:VAR']} for $N=100$, $\alpha=1$, $\beta=0.1$, and $\epsilon=10^{-3}$ (top row) and corresponding Kuramoto-like order parameters \ref{['eq:order_parameter']} (bottom row) for three values of the non-normality index (\ref{['eq:K']}): normal dynamics ($K=0$, left), pseudo-critical regime ($K=K_c$, center), and strongly non-normal regime ($K=10K_c$, right). The system is purely overdamped, linearly stable, driven by uncorrelated noise, and contains no intrinsic oscillators or periodic forcing. The time series are projected onto three geometric groups defined by the sign structure of the reaction mode \ref{['eq:clusters']}: two non-normal clusters (Cluster 1 and Cluster 2) and a residual subspace orthogonal to the non-normal sub-space. The dashed black line corresponds to the global order parameter. Colored background bands indicate successive time windows used as reference in Figure \ref{['fig:synthetic_spec']}.
• Figure 2: Non-normality-driven transition. Mean (left panel) and standard deviation (right panel) of the Kuramoto-like order parameters \ref{['eq:order_parameter']} of the reaction clusters, the residual subspace, and the full system as a function of the normalized non-normality index $K/K_c$ ($N=100$, $\alpha=1$, $\beta=0.1$, $\epsilon=10^{-3}$). The residual and global measures remain nearly constant, while the reaction clusters exhibit a sharp increase in mean and a pronounced peak in variance near $K/K_c \sim 1$, marking the onset of intermittent collective organization.
• Figure 3: Spectral analysis of the synthetic time series of Fig. \ref{['fig:synthetic_series']} for three values of the non-normality index ($K=0$, $K=K_c$, $K=10K_c$).Left panel: Power spectra computed over three typical time windows of fixed length (100 data points); each curve corresponds to a different window. Center-left: Canonical averaged spectrum $\hat{P}(f/f_{\max})$ obtained by rescaling frequencies by the peak frequency of each window and averaging across windows (Savitzky-Golay smoothing applied for visualization). Center-right: Distribution of the peak frequencies $f_{\max}$ over a large ensemble of time windows. Right panel: Power spectra computed over the full time series for the reaction clusters and the residual subspace, using FFT and averaging over components within each cluster. For $K=0$ and $K=K_c$, the tail of the spectrum is proportional to $1/f^2$, compared to to $1/f^4$ for $K=10 K_c$.
• Figure 4: Maximum (over $\tau\in[0,10]$) of the lead-lag imbalance measure \ref{['eq:main_ll_def']} as a function of $K/K_c$.
• Figure 5: Emergence of an intrinsic arrow of time from non-normal dynamics. Lead--lag imbalance $I(\tau)$\ref{['eq:S3_Imbalance_def']} given by (\ref{['eq:ll_imb']}) as a function of the time lag $\tau$, computed from the same synthetic time series shown in the paper, for three values of the non-normality index: normal dynamics ($K=0$), pseudo-critical regime ($K=K_c$), and strongly non-normal regime ($K=10K_c$).