TRANSITION TO CHAOS IN A SHELL MODEL OF TURBULENCE WITH THE CHANGE OF THE VISCOSITY

Abstract

In this paper, a shell model for the energy cascade in three-dimensional turbulence at varying the viscosity is studied. When the viscosity parameter ν is large enough, the phase orbit of each shell in the system is a stable fixed point. As ν diminishes down to a critical value, the point becomes unstable and a stable periodic orbit appears via a supercritical Hopf bifurcation. For the quasiperiodic solution of the system, each shell has a own dynamic behavior. At ν = 6.88×10-5, the system initiates a complex scenario of bifurcations, where the phase orbits of un and un+3 begin to form a intersected section. Farther decreasing ν, we observe a transition from quasiperiodic solution to chaos through a sequence of Neimark–Sacker bifurcations. The behaviors can be vividly shown in the change of the average energy function E with ν. At last, the transition to chaos in the ν–δ parameter space is also obtained.