A numerical method based on wavelet approximations is developed to solve the diffusion equations that arise from kinetic theory models of polymer dynamics and for coupling with continuum calculations for simulating complex flows of polymer solutions. The dynamics of a liquid crystalline polymer solution in a simple, homogeneous shear flow is computed by using the Doi model with orientational-dependent rotational diffusivity as a function of Deborah number, De, and different initial conditions. Stochastic methods were previously used for computing this flow, and aperiodic behavior was found at high De. Calculations with the wavelet-Galerkin method demonstrate a stable limit cycle in the same parameter regime. The existence of the time-periodic solution is traced to a supercritical Hopf bifurcation at De = De(Hopf); the value of De(Hopf) scales with (U - U-c)(alpha), where U is the strength of the intermolecular potential. The exponent in this correlation, alpha, depends on the form of the mean-field intermolecular potential and can be used to characterize the anisotropic environment experienced by the polymer molecules. (C) 2003 Elsevier B.V. All rights reserved.