Chaos for a damped and forced KdV equation

The attractor for a damped and forced Korteweg–de Vries equation is studied. For large damping, it is shown that the attractor is trivial, with all solutions converging to a unique fixed point. For small damping, however, the dynamics can be complicated. It is investigated numerically how the attractor changes when the damping term decreases. In particular, a period-doubling cascade of periodic solutions is found, culminating with a chaotic regime, the ratio between consecutive points in this cascade converging to Feigenbaum’s constant. Beyond this cascade, the frequency spectrum of the solution is continuous and one Lyapunov exponent is positive. This seems to be the first work to obtain the onset of temporal chaos for this equation.