Global bifurcations to strange attractors in a model for skew varicose instability in thermal convection

Abstract

We present a global bifurcation study of a four-dimensional system of differential equations, proposed by F.H. Busse and coworkers, modeling instabilities of convection rolls in the Rayleigh-Bénard experiment. The Rayleigh and Prandtl numbers are two natural parameters on which the system depends. We focus on a detailed mathematical study, combining numerical pathfollowing and bifurcation analysis, of chaotic dynamics and transitions to chaotic dynamics. Numerical continuation makes clear how homoclinic and heteroclinic bifurcations organize the bifurcation diagram in the parameter plane. Combined with a theoretical bifurcation analysis this explains the development of patterns and the creation of chaotic spatio-temporal dynamics in the model. The organizing centers, such as heteroclinic cycles with resonance conditions among eigenvalues and homoclinic loops with geometric degeneracies (inclination flips), are identified and their unfoldings are analyzed. This ties the creation of strange attractors of various geometric structures to codimension-two global bifurcations.