An infinite period bifurcation arising in roll waves down an open inclined channel.

The discussion in a previous paper on roll waves is completed by showing how the limit cycles created at small amplitude by a Hopf bifurcation are destroyed. It is shown that there is an infinite period bifurcation creating stable limit cycles at finite amplitude. The conditions under which such a bifurcation coming out of a separatrix loop from a saddle point in the plane can occur are first derived (under the assumption that the Reynolds number is small). The complete evolution of the limit cycles is then deduced. In the subcritical case it is found that there is just one stable limit cycle, created at small amplitude by a Hopf bifurcation and destroyed at finite amplitude by an infinite period bifurcation. In the supercritical case it is shown that there are two limit cycles, one unstable (created by a Hopf bifurcation) which finally merge and are then both destroyed.