Invariant tori in turbulence and chaos

One approach to understand the chaotic dynamics of turbulence is the study of non-chaotic yet dynamically unstable invariant solutions embedded in the system's chaotic attractor. The significance of zero-dimensional unstable fixed points and one-dimensional unstable periodic orbits capturing time-periodic dynamics is widely accepted for high-dimensional chaotic systems including turbulence, while higher-dimensional invariant tori representing quasi-periodic dynamics have rarely been considered. Fully developed turbulence is spatiotempoarally chaotic and has a large number of positive Lyapunov exponents, so-called hyperchaos. We demonstrate that unstable 2-tori are generically embedded in the hyperchaotic attractor of a dissipative system of ordinary differential equations; that tori can be numerically identified via bifurcations of unstable periodic orbits and that their parametric continuation and characterization of stability properties is feasible. As higher-dimensional tori are expected to be structurally unstable, 2-tori together with periodic orbits and equilibria form a complete set of relevant invariant solutions on which to base a dynamical description of chaos. Our results specifically open avenues toward including tori in a generalized periodic orbit theory aimed at most accurately expressing statistical properties of chaos in terms of expansions over the non-chaotic invariant solutions of the system.