From Phase Space Representation to Amplitude Equations in a Pattern Forming Experiment

TL;DR

This paper presents a method to reconstruct amplitude equations from slow nonlinear relaxation dynamics in a ferrofluid pattern formation experiment, enabling better understanding of the underlying nonlinear behavior.

Contribution

It introduces a technique to derive amplitude equations directly from experimental relaxation data in a ferrofluid pattern formation system.

Findings

Successfully reconstructed the nonlinear amplitude equation from relaxation dynamics.

Identified the appropriate weakly nonlinear expansion for the hysteretic transition.

Demonstrated the method's effectiveness in a slow ferrofluid pattern formation experiment.

Abstract

We describe and demonstrate a method to reconstruct an amplitude equation from the nonlinear relaxation dynamics in the succession of the Rosensweig instability. A flat layer of a ferrofluid is cooled such that the liquid has a relatively high viscosity. Consequently, the dynamics of the formation of the Rosensweig pattern becomes very slow. By sudden switching of the magnetic induction, the system is pushed to an arbitrary point in the phase space spanned by the pattern amplitude and the magnetic induction. Afterwards, it is allowed to relax to its equilibrium point. From the dynamics of this relaxation, we reconstruct the underlying fully nonlinear equation of motion of the pattern amplitude. The measured nonlinear dynamics serves to select the best weakly nonlinear expansion which describes this hysteretic transition.