Bifurcation of a periodic solution of the NavierStokes equations into an invariant torus
Abstract
This paper extends the study of the bifurcation of timeperiodic solutions of the NavierStokes equations. Timeperiodic motions that are driven by timeperiodic boundary data or by prescribed timeperiodic 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 NavierStokes equations fit into this framework. The solutions which bifurcate from the given timeperiodic 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.
 Publication:

Archive for Rational Mechanics and Analysis
 Pub Date:
 March 1975
 DOI:
 10.1007/BF00280153
 Bibcode:
 1975ArRMA..58...35I
 Keywords:

 Flow Stability;
 NavierStokes Equation;
 Numerical Stability;
 Periodic Functions;
 Toruses;
 Eigenvalues;
 Linear Transformations;
 Operators (Mathematics);
 Oscillating Flow;
 Turbulent Flow;
 Fluid Mechanics and Heat Transfer