ON THE BIFURCATION PHENOMENA OF THE KURAMOTO–SIVASHINSKY EQUATION

Bifurcation phenomena of the Kuramoto–Sivashinsky equation have been studied numerically. The solutions considered are restricted to the invariant subspace of odd functions. One possible route to chaos via a period-doubling cascade is investigated in detail: The four-modal steady-state loses its stability through a Hopf bifurcation and a branch of periodic motions is created. After a symmetry breaking the periodic solution undergoes a period-doubling cascade which ends up in two antisymmetric chaotic attractors. A merging of these antisymmetric attractors to a symmetric one is observed. The chaotic branch depending on the bifurcation parameter is characterized by the values of the Lyapunov exponents. Periodic windows within the chaotic region are also detected. Finally, a further increase of the bifurcation parameter leads to a transition from the attractor into transient chaos.