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The quasi-periodic doubling cascade in the transition to weak turbulence

The quasi-periodic doubling cascade is shown to occur in the transition from regular to weakly turbulent behaviour in simulations of incompressible Navier-Stokes flow on a three-periodic domain. Special symmetries are imposed on the flow field in order to reduce the computational effort. Thus we can apply tools from dynamical systems theory such as continuation of periodic orbits and computation of Lyapunov exponents. We propose a model ODE for the quasi-period doubling cascade which, in a limit of a perturbation parameter to zero, avoids resonance related problems. The cascade we observe in the simulations is then compared to the perturbed case, in which resonances complicate the bifurcation scenario. In particular, we compare the frequency spectrum and the Lyapunov exponents. The perturbed model ODE is shown to be in good agreement with the simulations of weak turbulence. The scaling of the observed cascade is shown to resemble the unperturbed case, which is directly related to the well known doubling cascade of periodic orbits.