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Bifurcations and Chaos in a Six-dimensional Turbulence Model of Gledzer

Makoto Umeki
Journal / Anthology

Journal of the Physical Society of Japan
Year: 2006
Volume: 75
Issue: 7

Gledzer's cascade shell model of turbulence with six real variables is studied numerically using mathematica 5.1. The Poincaré plot of the first mode v1 is used to determine periodic, doubly periodic and chaotic solutions and the routes to chaos via both frequency locking and period doubling. 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 a smaller viscosity parameter in the present model indicates that the turbulent state may be very sensitive to the Reynolds number.

*Applied Mathematics > Complex Systems
*Mathematics > Calculus and Analysis > Dynamical Systems

bifurcation, chaos, turbulence, shell model, Mathematica