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

This paper describes an amplitude equation analysis of the interactions between waves with wave number k(1) (and phase speed w(c)/k(1)) and stationary convection with wave number k(2). These two modes may bifurcate almost simultaneously from the conductive state of a two-layer Benard system, when the ratio of layer thicknesses is near a particular value (codimension-2 singularity). When k(2) not equal 2k(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 k(2)=2k(1) (resonant case), the system presents a much wider 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 are shown to entirely determine the dynamics (cubic and higher-order terms may be neglected near the codimension-2 point).