Local and Global Bifurcations in Finite-Dimensional Center Manifold Equations of Double-Diffusive Convection

A finite dimensional amplitude equation model of 2-dimensional double-diffusive convection near a quadruple-zero (codimension 4) bifurcation point is derived using center manifold reduction. The derivation employs small perturbation-theory to obtain an asymptomatic solution to the 2-dimensional Navier-Stokes equations. The coefficients of the amplitude equations are derived for two parameter regimes corresponding to high and moderate thermal Rayleigh numbers. By numerically approximating the Poincare map of the amplitude equations, local and global bifurcations are detected that lead to birth of strange attractors. Specifically, strange attractors are generated by homoclinic explosions in the Poincare map. For high thermal Rayleigh numbers, this route to chaos in the Poincare map is analogous to that route present in the continuous Shimizu-Morioka and Rucklidge models, where the bifurcation to periodic convection is supercritical. For low thermal Rayleigh numbers, the route to chaos in the Poincare map is shown to be analogous to the route observed in the Lorenz equations. Additionally, the bifurcations of the strange attractors of the Poincare map are studied, and numerical simulations reveal the presence of period doubling regimes and intermittency, as well as exotic bifurcations which include splitting, and interior crises, of strange attractors.