A DIRECT METHOD FOR COMPUTATION OF SIMPLE BIFURCATIONS

In the present study, we focus on the computational analysis of partia l differential equations with emphasis on the stability of the equilib rium states and on their bifurcations. In practical applications, it i s not sufficient to obtain an equilibrium solution at a point in the p arameter space. The equilibrium solution branches, their stability cha racteristics, and particularly the critical points of transition from one slate to another (e.g., bifurcation points), are required for unde rstanding the physics of the problem. In principle, the linear stabili ty of an equilibrium state can be investigated by solving an eigenvalu e problem, and consequently, the points of bifurcations can be detecte d. We review alternative techniques for detecting bifurcation points w hich are direct and numerically efficient and, therefore, more practic al. 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 ev olution approach, the test function approach, and the direct method. W e will then extend the direct method for a more practical and efficien t implementation. With this technique, we compute the sequence of tran sitions from steady state to chaotic flow in a two-dimensional lid-dri ven cavity of aspect ratio 0.8, 1.0, and 1.5. We demonstrate the effec tiveness of this technique by computing interesting new dynamics in th is 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 lea ding the system to a time periodic state. We construct the destabilizi ng disturbance structure and conclude that the first bifurcation of th e primary steady state is due to the centrifugal instability of the pr imary eddy. The mechanism of transition to chaos is low-dimensional. T he transition to chaos occurs after a secondary Hopf bifurcation. (C) 1995 Academic Press, Inc.