Discrete traveling waves in a relay system of Mackey-Glass equations with two delays (Q2047676)

Consider a ring circuit of \(m\) identical Mackey-Glass generators of the form \[\dot{u}_{j}=-\beta u_{j}+\frac{\alpha\left(u_{j}(t-1)+u_{j-1}(t-\tau)\right)}{1+\left(u_{j}(t-1)+u_{j-1}(t-\tau)\right)^{\gamma}}, \quad u_{0} \equiv u_{m}, \quad j=1, \ldots, m\tag{1} \] with positive parameters \(\alpha,\beta,\gamma,\tau\). Letting \(\gamma \to \infty\) one obtains the limiting equation \[\dot{u}_{j}=-\beta u_{j}+\alpha\left(u_{j}(t-1)+u_{j-1}(t-\tau)\right) F\left(u_{j}(t-1)+u_{j-1}(t-\tau)\right), \quad u_{0} \equiv u_{m}, \quad j=1, \ldots, m \tag{2}\] with \[ F(u) := \lim _{\gamma \rightarrow+\infty} \frac{1}{1+u^{\gamma}}= \begin{cases}1, & 0<u<1, \\ 1 / 2, & u=1, \\ 0, & u>1.\end{cases} \] The motivation to study system (2) is twofold: on the one hand, \(\gamma\gg 1\) is a realistic assumption in applications, and on the other hand, equation (2) can be regarded as a relay circuit analogue of system (1). The main result of the paper shows that for each positive integer \(m\geq 2\), there exists a range of values of the parameters such that for fixed \(\alpha, \beta,\tau\) and \(m\), there exists a discrete traveling wave solution of (2), that is, there is a \(\Delta>0\) for which system (2) has a periodic solution of the form \(u_j(t) = u_\ast (t + j\Delta)\). The key observation is that discrete traveling wave solutions correspond to periodic solutions of the scalar version (i.e.\ \(m=1\)) of equation (2). The proof is constructive: the exact formula for \(u_\ast\) is obtained.