Dynamics of nearly inviscid Faraday waves in almost circular containers

Abstract

Parametrically driven surface gravity-capillary waves in an elliptically distorted circular cylinder are studied. In the nearly inviscid regime, the waves couple to a streaming flow driven in oscillatory viscous boundary layers. In a cylindrical container, the streaming flow couples to the spatial phase of the waves, but in a distorted cylinder, it couples to their amplitudes as well. This coupling may destabilize pure standing oscillations, and lead to complex time-dependent dynamics at onset. Among the new dynamical behavior that results are relaxation oscillations involving abrupt transitions between standing and quasiperiodic oscillations, and exhibiting 'canards'.

Geometric singular perturbation theory in biological practice

“…Best et al (2005) for an application to a pacemaker network, or Higuera et al (2005) for an application to Faraday waves.…”

Relaxation oscillations in a nearly inviscid Faraday system

“…The presence of relaxation oscillations, let alone of canards, is unusual in fluid dynamics, and we believe that our study represents the frrst time that ducks have been found in water. A fortheoming paper describes our results in greater detail [6].…”

Time-Periodic Forcing of Spatially Localized Structures

“…Two types of canards have been identified: phase canards, in which the spatial extent of the localized pattern changes abruptly as an additional wavelength is nucleated or annihilated on either side, and amplitude canards, in which the amplitude temporarily drops to the amplitude of the unstable lower branch of spatially periodic states before abruptly increasing to the amplitude of the stable upper states. Canards are normally considered to be a property of finite-dimensional systems, although there are indications that they should be observable in pattern-forming (i.e., spatially extended) systems and in particular in the Faraday system [16]. It is therefore of particular interest to present clear evidence for such orbits in a partial differential equation.…”