Phase description of nonlinear dissipative waves under space-time-dependent external forcing (Q1957112)

The authors investigate the dynamics of propagating dissipative waves under external forcing which is space-time periodic based on the one model system undergoing separation and chemical reactions \[ \frac{\partial \psi}{\partial t}=\nabla^2[-\nabla^2\psi-\tau+\psi^3]+a_1\psi+a_2\phi+a_3+\Gamma(x,t)\eqno(1) \] \[ \frac{\partial \phi}{\partial t}=b_1\psi+b_2\phi+b_3+\Gamma(x,t) \eqno(2) \] The function \(\Gamma(x,t)=\varepsilon\cos (q_fx-\Omega t)\) stands for the external forces which is moving steadily to the right with the strength \(\varepsilon\), the wave number \(q_f\) and frequency \(\Omega\). The numerical simulations of eqs. (1) and (2) were carried out in one dimension and as a result a phase diagram for the entrained and non-entrained states under the external forcing was obtained. Analyzing the phase equations it is showed that the transition between entrained and non-entrained states occur as a Hopf bifurcation as well as a saddle-node bifurcation.