The Route to Chaos for the Kuramoto-Sivashinsky Equation

We present the results of extensive numerical experiments of the spatially periodic initial value problem for the Kuramoto-Sivashinsky equation. Our concern is with the asymptotic non-linear dynamics as the dissipation parameter decreases and spatio-temporal chaos sets in. To this end the initial condition is taken to be the same for all numerical experiments a single sine wave is used and the large time evolution of the system is followed numerically. Numerous computations were performed to establish the existence of windows, in parameter space, in which the solution has the following characteristics as the viscosity is decreased A steady fully modal attractor to a periodic in time attractor, to another steady fully modal attractor, to another time-periodic attractor, to a steady tetramodal attractor, to another time-periodic attractor having a full sequence of period-doublings in the parameter space to chaos. Numerous solutions are presented which provide conclusive evidence of the period-doubling cascades which precede chaos for this infinite-dimensional dynamical system. kr