Local Nearby Bifurcations Lead to Synergies in Critical Slowing Down: the Case of Mushroom Bifurcations

Abstract

The behavior of nonlinear systems close to critical transitions has relevant implications in assessing complex systems’ stability, transient properties, and resilience. Transient times become extremely long near phase transitions (or bifurcations) in a phenomenon generically known as critical slowing down, observed in electronic circuits, quantum electrodynamics, ferromagnetic materials, ecosystems, and gene regulatory networks. Typically, these transients follow well-defined universal laws of the form τ ∼ |µ − µc| β, describing how their duration, τ, varies as the control parameter, µ, approaches its critical value, µc. For instance, transients’ delays right after a saddle-node (SN) bifurcation, influenced by so-called ghosts, follow β = −1/2. Despite intensive research on slowing down phenomena over the past decades for single bifurcations, both local and global, the behavior of transients when several bifurcations are close to each other remains unknown. Here, we study transients close to two SN bifurcations collapsing into a transcritical one. To do so, we analyze a simple nonlinear model of a self-activating gene regulated by an external signal that exhibits a mushroom bifurcation. We also propose and study a normal form for a system with two SN bifurcations merging into a transcritical one. For both systems, we show analytical and numerical evidence of a synergistic increase in transients due to the coupling of the two ghosts and the transcritical slowing down. We also explore the influence of noise on the transients in the gene-regulatory model. We show that intrinsic and extrinsic noise play opposite roles in the slowing down of the transition allowing us to control the timing of the transition without compromising the precision of the timing. This establishes novel molecular strategies to generate genetic timers with transients much larger than the typical timescales of the reactions involved.