  The cascade-shell model of turbulence with six real variables originated by
Gledzer is studied numerically using Mathematica 5.1. Periodic, doubly-periodic
and chaotic solutions and the routes to chaos via both frequency-locking and
period-doubling are found by the Poincar\'e plot of the first mode $v_1$. The
circle map on the torus is well approximated by the summation of several
sinusoidal functions. The dependence of the rotation number on the viscosity
parameter is in accordance with that of the sine-circle map. The complicated
bifurcation structure and the revival of a stable periodic solution at the
smaller viscosity parameter in the present model indicates that the turbulent
state may be very sensitive to the Reynolds number.
