Effects of quenched disorder on critical transitions in pattern-forming systems

TL;DR

This study investigates how quenched disorder influences critical transitions in pattern-forming systems, revealing increased stability, gradual transitions, hysteresis, and early warning indicators through models of pattern formation and vegetation dynamics.

Contribution

It provides the first detailed analysis of quenched disorder effects on pattern-forming systems using the Swift-Hohenberg and vegetation models.

Findings

Disorder increases the durability of patterned states.

Transitions become more gradual with disorder.

Cross-correlation can serve as an early warning indicator.

Abstract

Critical transitions are of great interest to scientists in many fields. Most knowledge about these transitions comes from systems exhibiting the multistability of spatially uniform states. In spatially extended and, particularly, in pattern-forming systems, there are many possible scenarios for transitions between alternative states. Quenched disorder may affect the dynamics, bifurcation diagrams and critical transitions in nonlinear systems. However, only a few studies have explored the effects of quenched disorder on pattern-forming systems, either experimentally or by using theoretical models. Here, we use a fundamental model describing pattern formation, the Swift-Hohenberg model and a well-explored mathematical model describing the dynamics of vegetation in drylands to study the effects of quenched disorder on critical transitions in pattern-forming systems. We find that the disorder affects the patterns formed by introducing an interplay between the imposed pattern and the self-organized one. We show that, in both systems considered here, the disorder significantly increases the durability of the patterned state and makes the transition between the patterned state and the uniform state more gradual. In addition, the disorder induces hysteresis in the response of the system to changes in the bifurcation parameter well before the critical transition occurs. We also show that the cross-correlation between the disordered parameter and the dynamical variable can serve as an early indicator for an imminent critical transition.