Finite-time scaling in local bifurcations

TL;DR

This paper introduces finite-time scaling laws for bifurcations in dynamical systems, deriving exact critical exponents and establishing a universal law, linking phase transitions and bifurcations.

Contribution

It analytically derives finite-time scaling laws for transcritical and saddle-node bifurcations, revealing universal behavior and connecting dynamical systems with thermodynamic phase transitions.

Findings

Derived exact finite-time scaling laws for bifurcations

Identified a universal scaling law for the distance to attractor

Established a connection between phase transitions and bifurcations

Abstract

Finite-size scaling is a key tool in statistical physics, used to infer critical behavior in finite systems. Here we use the analogous concept of finite-time scaling to describe the bifurcation diagram at finite times in discrete dynamical systems. We analytically derive finite-time scaling laws for two ubiquitous transitions given by the transcritical and the saddle-node bifurcation, obtaining exact expressions for the critical exponents and scaling functions. One of the scaling laws, corresponding to the distance of the dynamical variable to the attractor, turns out to be universal. Our work establishes a new connection between thermodynamic phase transitions and bifurcations in low-dimensional dynamical systems, and opens new avenues to identify the nature of dynamical shifts in systems for which only short time series are available.