Synchronization — the spontaneous alignment of oscillators — is one of the most studied phenomena in complex systems. Fireflies flash together. Pendulum clocks on the same wall align. Heart pacemaker cells beat in unison. The standard explanation requires oscillators: individual units with an intrinsic rhythm that can be entrained by coupling. No oscillators, no synchronization.
Troude and Sornette (arXiv:2603.07206, March 2026) show that synchronized behavior can emerge in systems with no oscillators, no instability, and no resonance. The system is linearly stable — all eigenvalues have negative real parts, meaning every perturbation decays. There are no Hopf bifurcations, no limit cycles, no inherent oscillatory dynamics. The system, left to its own devices, relaxes to a fixed point. It does not oscillate.
But it synchronizes.
The mechanism is non-normal amplification. When the coupling matrix between system components is non-normal — meaning it doesn't commute with its own transpose — perturbations can grow transiently even though they eventually decay. The growth isn't exponential (that would require an unstable eigenvalue). It's algebraic, driven by the non-orthogonality of the eigenvectors. Noise continuously kicks the system into these transiently growing modes, and the growth concentrates the fluctuations along specific directions in state space. The concentrated fluctuations produce Kuramoto-like order parameters — the standard measure of phase synchronization — that intermittently rise to coherent values.
The authors call this pseudo-coherence. It has the statistical signatures of synchronization: transient phase alignment, broken time-reversal symmetry, positive entropy production, drifting spectral peaks. But there is no oscillator generating the rhythm. The rhythm is a statistical artifact of noise being amplified anisotropically by non-normal dynamics. The system doesn't synchronize because its components are oscillating in phase. It synchronizes because its noise is being channeled into modes that mimic oscillation.
Increasing non-normality produces a sharp transition — a pseudo-critical point — where the coherent episodes become dominant. The transition looks like a phase transition in the order parameter but corresponds to no eigenvalue crossing. The underlying dynamics are unchanged. Only the amplification of noise has passed a threshold.
Troude and Sornette, "Pseudo-Coherence and Stochastic Synchronization: A Non-Normal Route to Collective Dynamics without Oscillators," arXiv:2603.07206 (March 2026).