| I. The complex Ginzburg-Landau (CGL) equation is studied via numerical simulations. The method used is constructed by splitting the CGL equation into two exactly integrable equations and alternating the exact solutions using fractional time steps. In both one and two dimensions, the behavior of chaotic solutions is examined near the nonlinear Schroedinger (NLS) limit. In 1D, no large deviations occur, probability density functions remain purely Gaussian, and no inertial-type range appears in the wavenumber spectrum. Thus, only "soft turbulence" occurs. Estimates are obtained for the dimension of the inertial manifold and the attractor dimension. Scaling with the domain length shows universality in both the wavenumber and Lyapunov exponent spectra. In 2D, large localized spikes, highly non-Gaussian p.d.f.'s, and an inertial-type range are found near the NLS limit, implying the presence of "hard turbulence." The transition from soft to hard turbulence is found to be gradual. The role of topological defects is examined and Lyapunov exponents calculated for one dissipation level show the expected scaling with the domain area.;IIa. The partial differential equation describing a forced nonlinear beam is studied via numerical simulations using a Galerkin projection. The dynamics are found to be very low-dimensional and this is shown to be the result of the form of the nonlinearity in the p.d.e.;IIb. Extensions of the Lorenz equations are studied numerically and analytically. A translationally invariant model captures the dependence on the initial condition of the horizontal position of the convective rolls. An ad hoc extension leads to horizontal oscillation of the rolls, both periodically and chaotically. |
