Continuation of bifurcations of cycles in dissipative PDEs

Abstract

A new efficient methodology for the continuation of the codimension-one bifurcations of periodic orbits, including pitchfork bifurcations present in reflection--symmetric systems, has been developed. It is based on the combination of Newton-Krylov techniques applied to extended systems, and the integration of systems of variational equations up to second order. The extended systems are adapted from those usually found in the literature for fixed points of maps. A deflation term is needed in some cases to remove the trivial +1 multiplier of the periodic orbits. It will be shown that to evaluate the action of the Jacobian it is only necessary to integrate systems of ODEs of dimension at most four times that of the original system. This minimizes the computational cost.

Two main tools are required to implement the algorithms presented here, a generic continuation code, and a time integrator for the particular problem at hand including the required variational equations.

The thermal convection of a mixture of two fluids in a two-dimensional rectangular box is used as test problem. It is known that the onset of convection is oscillatory below a certain negative value of one of the parameters (the separation ratio), giving rise to a rich dynamics. A non-trivial diagram of periodic orbits is first deployed, by varying the Rayleigh number. Some of the bifurcations found on the main branch of periodic orbits are followed by adding as second parameter the Prandtl number. Several codimension-two points are found, and a double-Hopf bifurcation is studied in more detail. Finally the boundaries of a resonance region, found on a branch of invariant tori, are also continued.