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.

Á. Corral, Universal finite-time scaling in the transcritical, saddle-node, and pitchfork discrete and continuous bifurcations, Chaos An Interdisciplinary Journal of Nonlinear Science 35(1) (2025) 013106.