Normal forms and chaos in thermosolutal convection

The authors investigate the onset of double-diffusive convection in tall thin rolls. If the height/width ratio is large and parameters are such that the system is close to the double-zero (Bogdanov) bifurcation, corresponding to oscillations with zero frequency, then the dynamics is described by a fourth-order system of nonlinear ordinary differential equations. This can be reduced to a third-order system by assuming that the diffusivity ratio is small. The reduced system is shown to contain parameter ranges where there is a heteroclinic orbit connecting two saddle-foci with eigenvalues satisfying Shil'nikov's criterion for the existence of stable chaos in the neighbourhood of the global bifurcation. This result demonstrates that chaotic oscillations can be found arbitrarily close to the onset of two-dimensional thermosolutal convection. Moreover, the third-order system is the appropriate canonical form for studying the Shil'nikov mechanism with Z2 symmetry.