**3. Chain Dynamics in Case of Diffusional Couplings**

The boundary value problem (15) is examined. Let *δ* = 0 for definiteness. Then, the linearized on the zero equilibrium state boundary value problem has the form

$$\frac{\partial u}{\partial t} = -r u(t - T, \mathbf{x}) + r \gamma \int\_{-\infty}^{+\infty} F(s, \mathbf{c}) u(t - h, \mathbf{x} + s) ds,\tag{44}$$

$$
\mu(t, x + 2\pi) \equiv \mu(t, x). \tag{45}
$$

Here, we assume that function *F*(*s*,*ε*) is given by

$$F(\mathbf{s}, \varepsilon) = F\_{\varepsilon}(\mathbf{s}) - 2F\_0(\mathbf{s}) + F\_{-\varepsilon}(\mathbf{s}).\tag{46}$$

Depending on the parameter *σ*, three fundamentally different events can be distinguished. The first and the simplest of them assumes the parameter *σ* > 0 to be somehow fixed and, naturally, independent of the small parameter *ε*. This case is studied in Section 3.1. In Section 3.2, we assume that there is a value of *σ*<sup>0</sup> > 0 such that

$$
\sigma = \mathfrak{e} \sigma\_0. \tag{47}
$$

The critical case of infinite dimension mentioned above is realized under this condition. Finally, in Section 3.3, we assume the parameter *σ* to be even smaller: *σ* = *o*(*ε*). More precisely, for some fixed *σ*<sup>0</sup> > 0, we consider the relation

$$
\sigma = \varepsilon^2 \sigma\_0. \tag{48}
$$

This case is the most complicated and intriguing. It naturally generalizes the case of 'purely diffusional' couplings for which *σ*∼0.

*3.1. Chain Dynamics for Fixed σ Value*

In Formula (46), we arbitrarily fix the value *σ*<sup>0</sup> > 0. The inequality

$$0 < r < \frac{\pi}{2} \tag{49}$$

is the necessary and sufficient condition for the real parts of all eigenvalues of the characteristic Equation (20) to be negative.

For *r* = *π*/2, Equation (20) has exactly two pure imaginary roots *λ*<sup>±</sup> = ±*iπ*/2, and the real parts of the remaining roots are negative and separated from zero as *ε* → 0. Thus, the conditions of the well-studied Andronov–Hopf bifurcation are satisfied. Let

$$r = \frac{\pi}{2} + \varepsilon^2 r\_1 \tag{50}$$

for the arbitrarily fixed value *r*1.

Then, for *ε* 1, the roots *λ*±(*ε*) of Equation (20) close to *λ*<sup>±</sup> are as follows:

$$\begin{aligned} \lambda\_+(\varepsilon) &= \overline{\lambda}\_-(\varepsilon), \\ \lambda\_+(\varepsilon) &= i\frac{\pi}{2} + \varepsilon^2 \lambda\_{10} + O(\varepsilon^4), \\ \text{where } \lambda\_{10} &= (1 + \frac{\pi^2}{4})^{-1}(\frac{\pi}{2} + i)r\_1. \end{aligned} \tag{51}$$

Under these conditions and for sufficiently small *ε*, the boundary value problem (15) has a two-dimensional stable local integral invariant manifold M(*ε*) in the zero equilibrium state neighborhood, on which this boundary value problem can be written as a special scalar complex ordinary differential equation

$$\frac{d\tilde{\xi}}{d\tau} = \lambda\_{10}\tilde{\xi} + \mathcal{g}\tilde{\xi}|\tilde{\xi}|^2,\tag{52}$$

where *τ* = *ε*2*t* is a slow time, and *ξ*(*τ*) is a slowly varying amplitude in the asymptotic presentations of solutions on the manifold M(*ε*)

$$u = \varepsilon \left[ \tilde{\xi}(\tau) \exp\left(i\frac{\pi}{2}t\right) + \tilde{\xi}(\tau) \exp\left(-i\frac{\pi}{2}t\right) \right] + \varepsilon^2 u\_2(t, \tau) + \varepsilon^3 u\_3(t, \tau) + \dots \tag{53}$$

Here, the functions *uj*(*t*, *τ*) are 4-periodic with respect to *t*. We insert the formal expression (53) into (15) and collect the coefficients at the same powers of *ε*. First, we equate the coefficients at *ε*<sup>2</sup> to obtain

$$
\mu\_2 = \frac{2-i}{5} \mathfrak{J}^2 \exp(i\pi t) + \frac{2+i}{5} \mathfrak{J}^2 \exp(-i\pi t). \tag{54}
$$

At the next step, from the solvability condition of the resulting equation with respect to *u*3, we obtain the necessity of satisfying the relation (52), where

$$g = -\frac{\pi}{2} \left[ 3\pi - 2 + i(\pi + 6) \right] \cdot \left( 10 \left( 1 + \frac{4}{\pi^2} \right) \right)^{-1}. \tag{55}$$

Let us formulate the resulting statements. Their proofs are well-known (see, for example, [19]).

**Theorem 3.** *Let r*<sup>1</sup> < 0*. Then, for all sufficiently small ε, the solution of the boundary value problem* (15) *from some sufficiently small ε-independent equilibrium state u*<sup>0</sup> = 0 *neighborhood tends to zero as t* → ∞*.*

**Theorem 4.** *Let r*<sup>1</sup> > 0*. Then, all the solutions of Equation* (52) *except the zero solution tend to an orbitally stable cycle*

$$\begin{aligned} \xi\_0(\tau) &= \left[ 10 \frac{\pi}{2} r\_1 (3\pi - 2)^{-1} \right]^{\frac{1}{2}} \tilde{\xi}\_0 \exp(i \phi\_0 \tau), \\ \phi\_0 &= 3 \lambda\_{10} + \tilde{\xi}\_0^2 \Im \xi\_{\tau} \end{aligned} \tag{56}$$

*and the solutions (*≡ 1*) from* M(*ε*) *tend to cycle*

$$u\_0(t, \varepsilon) = \varepsilon \left[ \tilde{\varsigma}\_0(\varepsilon^2 t) \exp\left(i\frac{\mathcal{T}}{2}t\right) + \tilde{\varsigma}\_0(\varepsilon^2 t) \exp\left(-i\frac{\mathcal{T}}{2}t\right) \right] + O(\varepsilon^2) \tag{57}$$

*as t* → ∞*.*

Thus, in the considered case, the boundary value problem (15) can have only a homogeneous cycle 1 + *u*0(*t*,*ε*) in a zero neighborhood, which is a logistic Equation (1) under the condition (50). Apparently, the case considered here is of no interest.
