Browse Theses

Low dimensional invariant manifolds of dynamical systems include fixed points, periodic solutions, connecting orbits and invariant tori. Considerable work has been done on the computation and bifurcation analysis of all these manifolds with the exception of the invariant tori. The importance of computing an invariant tori can be observed in the numerous dynamical systems including dissipative partial differential equations in which they occur. Tori appear mostly in the bifurcation sequence from a steady state to a chaotic solution. One is interested in following the torus to a breakdown and see what it bifurcates to. The results of this study can give us some insight into Ruelle Takens scenario of transition to turbulence in fluid dynamics. In this thesis we investigate the partial differential equation approach to compute invariant tori using orthogonal collocation discretization. We introduce an adaptive grid refinement scheme for several problems in order to study the breakdown of the torus. We also implement the Hadamard graph transform approach which is used in computing attracting invariant manifolds. We find the results of this method to be similar to that of the collocation method. Finally, we did some visualization work using computer graphics to enable us observe the geometry and the flow on the torus, as well as help us in the study of the breakdown.