Robust Heteroclinic Cycles in Two-Dimensional Rayleigh–bénard Convection Without Boussinesq Symmetry

Abstract

The onset of convection in systems that are heated via current dissipation in the lower boundary or that lose heat from the top boundary via Newton's law of cooling is formulated as a bifurcation problem. The Rayleigh number as usually defined is shown to be inappropriate as a bifurcation parameter since the temperature difference across the layer depends on the amplitude of convection and hence changes as convection evolves at fixed external parameter values. A modified Rayleigh number is introduced that does remain constant even when the system is evolving, and solutions obtained with the standard formulation are compared with those obtained via the new one. Near the 1 : 2 spatial resonance in low Prandtl number fluids these effects open up intervals of Rayleigh number with no stable solutions in the form of steady convection or steadily traveling waves. Direct numerical simulations in two dimensions show that in such intervals the dynamics typically take the form of a nearly heteroclinic modulated traveling wave. This wave may be quasiperiodic or chaotic.

The 1:2 mode interaction and heteroclinic networks in Boussinesq convection

“…It is known [23] that additive noise in a system with heteroclinic attractors typically produces approximately periodic behaviour and this is what we often observe. Similarly, an orbit of a very long period instead of a theoreticaly predicted heteroclinic cycle related to the 1 : 2 mode interaction was observed in non-Boussinesq convection by Mercader et al [15]. They suggested that higher modes present in the PDE's could possibly prevent formation of the structurally stable heteroclinic cycle (or perhaps destabilize it) with the result that a long periodic orbit was present instead.…”

“…(12) with coefficients Eq. (15,16) and (17) are shown on Figure 6. This illustrates the variation of the bifurcation points in Figure 4 with P .…”

Dynamics in the 1:2 spatial resonance with broken reflection symmetry

“…In these spatially extended systems, the equilibria correspond to spatially periodic steady states, and the 180 • rotation connecting them corresponds to translation by half a wavelength. Since then, dynamics that appear to be associated with the presence of attracting structurally stable heteroclinic cycles have been observed in experiments with premixed flames on a circular porous plug burner [15] and in a number of partial differential equations [16][17][18]. One finds that in general the switching time between successive visits to a steady state saturates at a finite value, resulting either in a long-period periodic orbit or sometimes a chaotic orbit [17].…”

“…Since then, dynamics that appear to be associated with the presence of attracting structurally stable heteroclinic cycles have been observed in experiments with premixed flames on a circular porous plug burner [15] and in a number of partial differential equations [16][17][18]. One finds that in general the switching time between successive visits to a steady state saturates at a finite value, resulting either in a long-period periodic orbit or sometimes a chaotic orbit [17]. The origin of this behavior in the partial differential equations is not entirely clear but is believed to be due to numerical error, since similar behavior is also found in the normal form for the 1:2 spatial resonance [19].…”

Nonlinear Marangoni convection in circular and elliptical cylinders

“…25,26 The most dramatic feature of this interaction is the presence, in certain parameter regimes, of structurally stable heteroclinic cycles connecting the m = 2 state with its rotations by / 4. Such cycles have been observed in A = 2.5 containers by Johnson and Narayanan 5 and reproduced within weakly nonlinear theory by Dauby et al;24 see also Ref.…”