**2. Dynamics of Fully Connected Spatially Distributed Chain**

In this section, the local dynamics of the boundary value problem (11), (12) are studied. The linearized at zero boundary value problem

$$\frac{\partial v}{\partial t} = -r\upsilon(t - T, \mathbf{x}) + r\gamma \left[ \mathcal{M} \left( \upsilon(t - h, \mathbf{s}) - \upsilon(t - h, \mathbf{x}) \right) \right], \tag{21}$$

$$\upsilon(\mathbf{r}, \mathbf{x} + 2\pi) \equiv \upsilon(\mathbf{r}, \mathbf{x})$$

has the characteristic equation

$$\lambda = -r \exp(-\lambda T) + r\gamma \exp(-\lambda h)[\delta\_k - 1], \quad k = 0, \pm 1, \pm 2, \dots, \tag{22}$$

where

$$
\delta\_k = \begin{cases} 1, & k = 0, \\ 0, & k \neq 0. \end{cases}
$$

We obtain this equation from (21) by substituting the elementary Euler solutions *vk* = exp(*ikx* + *λt*) (*k* = 0, ±1, ±2, . . .).

The case of the small parameter *γ* is studied in Section 2.1. We fix some *γ*<sup>1</sup> in such a way that

$$
\gamma = \mu \gamma\_1 \quad \text{and} \quad 0 < \mu \ll 1. \tag{23}
$$

The general case is studied in Section 2.2.
