Bifurcations of tori and phase locking in a dissipative system of differential equations

Abstract In a system of ordinary differential equations, obtained through a seven-mode truncation of the plane incompressible Navier-Stokes equations, a two-dimensional torus undergoes first two period-doubling bifurcations and then a transition to a strange attractor. This strange attractor, of Liapunov dimension larger than three in a wide parameter interval, is characterized by a power spectrum which retains the two fundamental frequencies of the original torus superimposed on a broad, jagged background. As the Liapunov dimension goes down towards two, an interesting phenomenon of phase locking occurs, which gives rise to an alternation of chaotic and periodic behavior.