Universal finite-time scaling in the transcritical, saddle-node, and pitchfork discrete and continuous bifurcations

TL;DR

This paper extends the finite-time scaling laws for bifurcations to include pitchfork bifurcations in both discrete and continuous systems, enabling direct analysis of transient dynamics without prior fixed point knowledge.

Contribution

It reformulates and generalizes previous scaling laws to cover additional bifurcation types, notably pitchfork bifurcations, providing a universal description of transient behavior.

Findings

Scaling laws accurately describe transient dynamics near bifurcations.

Finite-time bifurcation diagrams can be obtained without knowing stable fixed points.

Universal applicability to various bifurcation types.

Abstract

Bifurcations are one of the most remarkable features of dynamical systems. Corral et al. [Sci. Rep. 8(11783), 2018] showed the existence of scaling laws describing the transient (finite-time) dynamics in discrete dynamical systems close to a bifurcation point, following an approach that was valid for the transcritical as well as for the saddle-node bifurcations. We reformulate those previous results and extend them to other discrete and continuous bifurcations, remarkably the pitchfork bifurcation. In contrast to the previous work, we obtain a finite-time bifurcation diagram directly from the scaling law, without a necessary knowledge of the stable fixed point. The derived scaling laws provide a very good and universal description of the transient behavior of the systems for long times and close to the bifurcation points.