Onset of spontaneous beating and whirling in the follower force model of an active filament

TL;DR

This paper investigates the onset of spontaneous oscillations in an active filament model, revealing how nonlinear dynamics lead to whirling states through a weakly nonlinear analysis near a supercritical double-Hopf bifurcation.

Contribution

It provides a weakly nonlinear analysis of the bifurcation in the follower force model, complementing previous numerical studies with an amplitude equation approach.

Findings

Identification of the supercritical double-Hopf bifurcation as the onset of spontaneous dynamics.

Derivation of an amplitude equation governing oscillation modulation.

Insights into the nonlinear creation and selection of whirling states.

Abstract

We study the onset of spontaneous dynamics in the follower force model of an active filament, wherein a slender elastic filament in a viscous liquid is clamped normal to a wall at one end and subjected to a tangential compressive force at the other. Clarke, Hwang and Keaveny (Phys. Rev. Fluids, to appear) have recently conducted a thorough investigation of this model using methods of computational dynamical systems; inter alia, they show that the filament first loses stability via a supercritical double-Hopf bifurcation, with periodic 'planar-beating' states (unstable) and 'whirling' states (stable) simultaneously emerging at the critical follower-force value. We complement their numerical study by carrying out a weakly nonlinear analysis close to this unconventional bifurcation, using the method of multiple scales. The main outcome is an 'amplitude equation' governing the slow modulation of small-magnitude oscillations of the filament in that regime. Analysis of this reduced-order model provides insights into the onset of spontaneous dynamics, including the creation of the nonlinear whirling states from particular superpositions of linear planar-beating modes as well as the selection of whirling over planar beating in three-dimensional scenarios.