TRAVELING-WAVES OF THE KURAMOTO-SIVASHINSKY EQUATION - PERIOD-MULTIPLYING BIFURCATIONS

We present a detailed bifurcation analysis for the travelling-wave solutions of the Kuramoto-Sivashinsky equation, with an emphasis on periodic solutions. The solutions are described by a 1-parameter, reversible third-order ODE. In two previous papers we described new aspects in the observed bifurcations: the 'noose' bifurcation, and a novel kind of 'Shil'nikov' behaviour. This paper brings everything together, and considers the one remaining new aspect, the connected set of period-multiplying k-bifurcations. We offer a possible explanation for this set by considering a 2-parameter, reversible fourth-order ODE that contains the travelling-wave ODE in a particular limit. It is conjectured that the connected set arises from 1 : n resonances of the eigenvalues of a fixed point.