A direct method for computation of simple bifurcations

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In the present study, we focus on the computational analysis of partial differential equations with emphasis on the stability of the equilibrium states and on their bifurcations. In practical applications, it is not sufficient to obtain an equilibrium solution at a point in the parameter space. The equilibrium solution branches, their stability characteristics, and particularly the critical points of transition from one state to another (e.g., bifurcation points), are required for understanding the physics of the problem. In principle, the linear stability of an equilibrium state can be investigated by solving an eigenvalue problem, and consequently, the points of bifurcations can be detected. We review alternative techniques for detecting bifurcation points which are direct and numerically efficient and, therefore, more practical. Starting with a large dimension dynamical system, which represents a projection of a set of coupled partial differential equations onto a basis function, we discuss the relative effectiveness of the time evolution approach, the test function approach, and the direct method. We will then extend the direct method for a more practical and efficient implementation. With this technique, we compute the sequence of transitions from steady state to chaotic flow in a two-dimensional lid-driven cavity of aspect ratio 0.8, 1.0, and 1.5. We demonstrate the effectiveness of this technique by computing interesting new dynamics in this relatively simple hydrodynamic system. In particular, we show that depending on the aspect ratio, the first transition from steady state could be through a supercritical or a subcritical Hopf bifurcation leading the system to a time periodic state. We construct the destabilizing disturbance structure and conclude that the first bifurcation of the primary steady state is due to the centrifugal instability of the primary eddy. The mechanism of transition to chaos is low-dimensional. The transition to chaos occurs after a secondary Hopf bifurcation.