Universality and Hysteresis in Slow Sweeping of Bifurcations

TL;DR

This paper investigates how slow parameter changes near bifurcations in dynamical systems lead to universal hysteresis behaviors, highlighting the breakdown of adiabatic approximation and deriving explicit asymptotic expressions.

Contribution

It reveals universal hysteresis trajectories near bifurcations and provides explicit formulas for these trajectories and the hysteresis loop area, considering structural asymmetry.

Findings

Universal upsweep and downsweep trajectories depend on a structural asymmetry parameter.

Explicit asymptotic expressions for the trajectories are derived.

The hysteresis loop area is calculated as a function of sweep rate and asymmetry.

Abstract

Bifurcations in dynamical systems are often studied experimentally and numerically using a slow parameter sweep. Focusing on the cases of period-doubling and pitchfork bifurcations in maps, we show that the adiabatic approximation always breaks down sufficiently close to the bifurcation, so that the upsweep and downsweep dynamics diverge from one another, disobeying standard bifurcation theory. Nevertheless, we demonstrate universal upsweep and downsweep trajectories for sufficiently slow sweep rates, revealing that the slow trajectories depend essentially on a structural asymmetry parameter, whose effect is negligible for the stationary dynamics. We obtain explicit asymptotic expressions for the universal trajectories, and use them to calculate the area of the hysteresis loop enclosed between the upsweep and downsweep trajectories as a function of the asymmetry parameter and the sweep rate.