Bifurcation of a periodic solution of the Navier-Stokes equations into an invariant torus

This paper extends the study of the bifurcation of time-periodic solutions of the Navier-Stokes equations. Time-periodic motions that are driven by time-periodic boundary data or by prescribed time-periodic body forces are considered. The analysis is based on the framework used by Ruelle and Takens (1971) to investigate dynamic systems. It is found that the periodic solutions of the Navier-Stokes equations fit into this framework. The solutions which bifurcate from the given time-periodic solutions lie on a torus; if the periodic solutions is visualized as a closed orbit in a function space, then the bifurcating solutions can be seen as lying on some torus surrounding the closed orbit. The torus is invariant for solutions whose trajectories never leave the torus. A stable invariant torus means that if a solution is started sufficiently close to the torus, the distance between the torus and the solution will eventually go to zero.