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

Abstract

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) and the other stable (created by the infinite period bifurcation) which finally merge and are then both destroyed. The discontinuous roll wave solutions derived by R. F. Dressler ( Communs pure appl. Math. 2, 49-194 (1949)) are compared with the continuous solutions for large values of the Reynolds number. It is shown that there is a difference in the jump condition between Dressler’s solutions and the present ones. It is then shown that this difference could be resolved by a slight modification to the dissipation term, which leaves the basic form of the continuous solutions unaltered. Finally it is then shown that both sets of waves are similar in that they both terminate with a solitary wave.