Local nearby bifurcations lead to synergies in critical slowing down: The case of mushroom bifurcations

The behavior of nonlinear systems near critical transitions has significant implications for stability, transients, and resilience in complex systems. Transient times, τ, become extremely long near phase transitions (or bifurcations) in a phenomenon known as critical slowing down, and are observed in electronic circuits, circuit quantum electrodynamics, ecosystems, and gene regulatory networks. Critical slowing down typically follows universal laws of the form τ∼|μ-μ_{c}|^{β}, with μ being the control parameter and μ_{c} its critical value. For instance, β=-1/2 close to saddle-node bifurcations. Despite intensive research on slowing down phenomena for single bifurcations, both local and global, the behavior of transients when several bifurcations are close to each other remains unknown. Here, we investigate transients near two saddle-node bifurcations merging into a transcritical one. Using a nonlinear gene-regulatory model and a normal form exhibiting a mushroom bifurcation diagram we show, both analytically and numerically, a synergistic, i.e., nonadditive, lengthening of transients due to coupled ghost effects and transcritical slowing down. We also 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 timing. This establishes molecular strategies to generate genetic timers with transients much larger than the typical timescales of the reactions involved.