Time-stepper based numerical bifurcation analysis: an application to the Taylor-Couette problem

Abstract

We present a numerical bifurcation analysis of the Taylor-Couette problem which utilizes a pre-existing time evolution code as the core computational engine. Such time evolution codes, or time-steppers, suffer from the draw back that bifurcation-theoretic results cannot easily be obtained from them, if at all. By incorporating relatively minor changes to the underlying time-stepper code, we have developed a computational structure around the existing time-stepper that enables us to perform these bifurcation-theoretic tasks. The cylinder geometry studied has radius ratio 0.615 and aspect ratio 2.4. For a fixed inner Reynolds number Ri = 300, three distinct solution branches are analyzed for the range of outer Reynolds number –320 ≤ Ro ≤ 0. These branches exhibit a wide variety of interesting points, including Hopf bifurcation points, symmetry-breaking pitchfork bifurcation points, turning points, and a torus bifurcation point. Unstable steady and time-periodic solutions are also computed.