Spatially quasiperiodic convection and temporal chaos in two-layer thermocapillary instabilities.

Abstract

This paper describes an amplitude equation analysis of the interactions between waves with wave number ${\mathit{k}}_{1}$ (and phase speed ${\mathrm{\ensuremath{\omega}}}_{\mathit{c}}$/${\mathit{k}}_{1}$) and stationary convection with wave number ${\mathit{k}}_{2}$. These two modes may bifurcate almost simultaneously from the conductive state of a two-layer B\'enard system, when the ratio of layer thicknesses is near a particular value (codimension-2 singularity). When ${\mathit{k}}_{2}$\ensuremath{\ne}2${\mathit{k}}_{1}$ (nonresonant case) and the first bifurcation occurs for steady convection, a secondary bifurcation to a spatially quasiperiodic and time-periodic mixed mode is obtained when increasing the driving gradient. No stable small-amplitude solution exists when the Hopf bifurcation is the first one. The occurrence of either of these two possibilities depends on the thickness ratio. When ${\mathit{k}}_{2}$=2${\mathit{k}}_{1}$ (resonant case), the system presents a much wider variety of dynamical behaviors, including quasiperiodic relaxation oscillations and temporal chaos. The discussion of the resonant system concentrates on a scenario of transition to chaos consisting of an infinite sequence of ``period-doubling'' homoclinic bifurcations of stable periodic orbits, for which the left-right symmetry of the convective system plays an essential role. For increasing constraint, a reverse cascade is observed, for which quadratic nonlinearities in the Ginzburg-Landau equations are shown to entirely determine the dynamics (cubic and higher-order terms may be neglected near the codimension-2 point). \textcopyright{} 1996 The American Physical Society.