Symmetric periodic orbits and global dynamics of tori in an O(2) equivariant system:: Two-dimensional thermal convection

Numerical simulations of two-dimensional Boussinesq thermal convection in a long cylindrical annulus with radial gravity and heating are used to study the influence of the reflection and rotation symmetries of the system on the sequence of local and global bifurcations leading to complex time dependent behavior. From the results of the linear stability analysis of symmetric periodic orbits, it is shown how, via gluing bifurcations, some quasi-periodic flows recover, as sets, symmetries lost in previous bifurcations. It is also shown how the same mechanism gives rise to a temporal chaotic attractor consisting of random switches between the symmetry-conjugate quasi-periodic orbits. At higher Rayleigh numbers, a chaotic-drifting behavior is found when a circle of invariant tori loses stability. In addition, detailed information about the Floquet multipliers and eigenfunctions of the periodic orbits involved in this dynamics is supplied.