Flow patterns of a viscous incompressible fluid contained between two coaxial cylinders with the inner cylinder rotating are investigated. A turncated set of nonlinear coupled-mode equations governing temporal development of the velocity amplitude of the rotating fluid is presented with the aid of the Galerkin method, and is integrated numerically for several assigned values of the angular speed of the inner cylinder in order to determine whether its solutions are periodic or chaotic. The calculated results of the power spectral density and the separation distance between two orbits with their initial conditions only slightly different from each other show the existence of highly chaotic numerical solutions above a certain value of the angular speed. The results of these numerical calculations are also compared with those of experimental observations.