Phase Transitions Without Instability: A Universal Mechanism from Non-Normal Dynamics

Abstract: We identify a new universality class of phase transitions that arises in non-normal systems, challenging the classical view that transitions require eigenvalue instabilities. In traditional bifurcation theory, critical phenomena emerge when spectral stability is lost; here, we show that transitions can occur even when all equilibria are spectrally stable. The key mechanism is the transient amplification induced by non-orthogonal eigenvectors: noise-driven dynamics are enhanced not by lowering energy barriers, but by increasing the effective shear of the flow, which renormalizes fluctuations and acts as an emergent temperature. Once the non-normality index $\kappa$ exceeds a critical threshold $\kappa_c$, stable equilibria lose practical relevance, enabling escapes and abrupt transitions despite preserved spectral stability. This pseudo-criticality generalizes Kramers' escape beyond potential barriers, providing a fundamentally new route to critical phenomena. Its implications are broad: in biology, DNA methylation dynamics reconcile long-term epigenetic memory with rapid stochastic switching; in climate, ecology, finance, and engineered networks, abrupt tipping points can arise from the same mechanism. By demonstrating that phase transitions can emerge from non-normal amplification rather than eigenvalue instabilities, we introduce a predictive, compact framework for sudden transitions in complex systems, establishing non-normality as a defining principle of a new universality class of phase transitions.