Asymptotically exact codimension-four dynamics and bifurcations in two-dimensional thermosolutal convection at high thermal Rayleigh number: Chaos from a quasi-periodic homoclinic explosion and quasi-periodic intermittency

Using a perturbation method, we solve asymptotically the nonlinear partial differential equations that govern double-diffusive convection (with heat and solute diffusing) in a two-dimensional rectangular domain near a critical point in parameter space where the linearized operator has a quadruple-zero eigenvalue. The asymptotic solution near this codimension-four point is found to depend on two slow-time-dependent amplitudes governed by two nonlinearly-coupled Van der Pol-Duffing equations. Through numerical approximation of the 3-dimensional Poincaré map in the four-dimensional state space of the amplitude equations, we detect and analyze the bifurcations of the amplitude equations as the thermal Rayleigh number, RT, is increased (for RS≪RT, the solute Rayleigh number) with all other parameters fixed. The bifurcations observed include: Hopf, pitchfork and Neimark-Sacker bifurcations of limit cycles, symmetric and asymmetric saddle-node bifurcations of 2-tori, and reverse torus-doubling cascades. In addition, chaotic solutions are found numerically to emerge via two different types of routes: (1) a route involving a homoclinic explosion in the Poincaré map and; (2) type-I intermittency routes near saddle-node bifurcations of 2-tori. The homoclinic explosion occurs when two unstable 2-tori form homoclinic connections with a saddle limit cycle, thereby creating a homoclinic butterfly in the Poincaré map that leads to a discrete Lorenz-like attractor.