Fig. 3

Left panel: Traveling wave solution \(\theta _{s}^{(f)}(t)=f(t+\alpha s)\) for \(\alpha= 0.2\) and \(s =1,\ldots,30\) vs. time t, together with the forcing signal f (which satisfies condition (C) in Theorem 2.1, so as to ensure that the wave is stable—see text). Subsequent panels: Example of a trajectory \(\theta_{s}^{(f)}(t)\) for \(s=1,\ldots,30\) and disjoint consecutive time windows. The initial phases have been chosen randomly and the forcing signal f is the same as in the left panel. Clearly, the convergence \(\theta_{s}^{(f)}(t) \to f(t+\alpha s)\) occurs at the first site \(s=1\) and then propagates down the chain, as described in the text; the higher the value of s is, the longer the transient time \(\theta_{s}^{(f)}(t)\) needs in order to get close to \(f(t+\alpha s)\)