The question of the realizability of the Landau scenario for the development of turbulence

We suggest a phenomenological model for the development of turbulence in the form of the nonlinear Klein–Gordon equation perturbed by nonconservative disturbances. Combining analytic and numerical methods, we establish that the transition to turbulence in this equation can follow both the Landau and the Landau–Sell scenarios. As is known, the first scenario is related to a cascade of bifurcations of stable invariant tori of increasing dimensions. The other scenario is related to a chaotic attractor whose Lyapunov dimension increases indefinitely under variation of a certain control parameter.