Phase description of nonlinear dissipative waves under space-time-dependent external forcing

Based on the model system undergoing phase separation and chemical reactions, we investigate the dynamics of propagating dissipative waves under external forcing which is periodic both in space and time. A phase diagram for the entrained and non-entrained states under the external forcing is obtained numerically. Theoretical analysis in terms of phase description of the traveling waves is carried out to show that the transition between the entrained and the non-entrained states by changing the external frequency occurs either through a saddle-node bifurcation or through a Hopf bifurcation and that these two bifurcation lines are connected at a Bogdanov-Takens bifurcation point.