Nonlinear Sciences – Chaotic Dynamics
Scientific paper
2006-05-19
JPSJ Vol. 75 No. 7 (2006) p. 073401
Nonlinear Sciences
Chaotic Dynamics
19 pages, 12 figures submitted to JPSJ
Scientific paper
10.1143/JPSJ.75.073401
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.
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