**4. Cooperative Dynamics in Chains with Unidirectional Coupling**

We consider the boundary value problem (17) in which the equality

$$F(s, \varepsilon) = F\_{\varepsilon}(s) - F\_0(s)$$

holds for the function *F*(*s*,*ε*). Further, relation (47) holds for the parameter *σ*.

We assume that the equilibrium state *u*<sup>0</sup> = 1 of logistic Equation (1) is asymptotically stable in the absence of couplings. Thus, the values of parameter *r* satisfy the inequality

$$0 < r < \frac{\pi}{2}.\tag{70}$$

Section 4.1 analyses the linearized at zero boundary value problem, and the nonlinear boundary value problem, which is the QNF, is constructed in Section 4.2.

### *4.1. Linear Analysis*

In the considered case, the characteristic equation for the linearized on *u*<sup>0</sup> boundary value problem takes the form

$$(\lambda + r \exp(-\lambda) = d(\exp(iz - \sigma\_0^2 z^2) - 1), \tag{71}$$

where *d* > 0, *z* = *εk*, *k* = 0, ±1, ±2, ... . We study the location of the roots of Equation (71) in order to make a conclusion about the stability of the equilibrium state in the boundary value problem (17).

We present several simple statements about the roots of (71) without proofs.

**Lemma 1.** *For z* ∈ [*π*(2*n* + 1), *π*(2*n* + 2)] *(n* = 0, ±1, ±2, ...*) Equation* (71) *has no roots with a zero real part as d* > 0*.*

**Lemma 2.** *For every z* ∈ (2*πn*, *π*(2*n* + 1))*, there exists d* > 0 *such that Equation* (71) *has a root with a zero real part as d* = *dz.*

We introduce the notation: *<sup>d</sup>*(*r*) = min <sup>−</sup>∞<*z*<<sup>∞</sup> *dz* <sup>=</sup> *dz*(*r*).

Then, the equilibrium state of the boundary value problem (17) is asymptotically stable as *d* ∈ (0, *d*(*r*)). For *d* = *d*(*r*) and *z* = *z*(*r*), Equation (71) has roots *λ*±(*r*) with zero real part: *λ*±(*r*) = ±*iω*(*r*) (*ω*(*r*) > 0).

**Lemma 3.** *The inequalities*

$$0 < z(r) < \pi, \quad \frac{\pi}{2} < \omega(r) < \frac{3\pi}{2}$$

*hold for the values of ω*(*r*) *and z*(*r*)*.*

We consider the questions about the asymptotic behavior of the expressions *d*(*r*), *ω*(*r*) and *z*(*r*) for *r* → 0 and for *r* → *π*/2 separately.

First, let *r* → 0. We denote by *ω*<sup>0</sup> the root from the interval (*π*/2, *π*) of the equation tan *<sup>ω</sup>* = −*ω*−1. Let

$$c\_0 = (1 + \sigma\_0^2)\omega\_0^2(\omega\_0^2 + 4), \quad z\_0 = \omega\_0 c\_0^{-1}.$$

**Lemma 4.** *The asymptotic equalities*

$$d(r) = c\_0 r^{-1} (1 + o(1)), \quad \omega(r) = \omega\_0 + o(1), \quad z(r) = z\_0 r (1 + o(1))$$

*hold as r* → 0*.*

Then, let *r* = *π*/2 − *μ* and 0 < *μ* 1. Now, by *z*<sup>00</sup> ∈ (0, *π*/2), we denote the least root of equation cos *z* = (*π*/4 + 1)−1/2. Then, min *z*<sup>00</sup> = *π*/2(*π*2/4 + 1)−1/2. We assume

$$c\_{00} = \frac{\pi}{2} \left[ \left( \frac{\pi^2}{4} + 1 \right)^{\frac{1}{2}} \exp \left( -\sigma\_0^2 z\_{00}^2 \right) - 1 \right], \quad (c\_0 < 0),$$

$$\omega\_{00} = c\_{00} \frac{\pi}{2} \left( \frac{\pi^2}{4} + 1 \right)^{-\frac{1}{2} \exp \left( -\sigma\_0^2 z\_{00}^2 \right)} - 1.$$

**Lemma 5.** *For all sufficiently small μ, the asymptotic equalities*

$$\begin{aligned} d(r) &= c\_{00}\mu(1+o(1)), \\ \omega(r) &= \frac{\pi}{2} + \omega\_{00}\mu(1+o(1)), \\ z(r) &= z\_{00} + o(1) \end{aligned}$$

*hold.*

The justifications of Lemmas (4) and (5) are quite simple but cumbersome. Therefore, we omit them.

Further, we fix the value *r*<sup>0</sup> ∈ (0, *π*/2) and arbitrary values *r*<sup>1</sup> and *d*1. Let

$$r = r\_0 + \varepsilon^2 r\_1, \quad d = d(r\_0) + \varepsilon^2 d\_1 \tag{72}$$

in (17).

Below, let *<sup>θ</sup>* = *<sup>θ</sup>*(*ε*) ∈ [0, 1) be the expression that complements the quantity *<sup>z</sup>*(*r*0)*ε*−<sup>1</sup> to an integer. We study the asymptotic behavior of the close to imaginary axis roots of Equation (71). We denote them by *λm*(*ε*) and *λm*(*ε*) (*m* = 0, ±1, ±2, . . .). The equalities

$$\lambda\_m(\varepsilon) = i\omega(r) + \varepsilon i R\_1(\theta + m) + \varepsilon^2 (R\_{20} + (\theta + m)^2 R\_2) + \dots \tag{73}$$

hold, where

$$\begin{array}{rcl} R\_1 &=& (1 - r\_0 \exp(-i\omega(r\_0)))^{-1} d(r\_0) z(r\_0) \times \\ & & (1 + 2i\sigma\_0^2) \exp(-\sigma\_0^2 z^2(r\_0) + i z\_0(r\_0)), \\ R\_{20} &=& (1 - r\_0 \exp(-i\omega(r\_0)))^{-1} \times \\ & & [d\_1 (1 - \exp(-\sigma\_0^2 z^2(r\_0) + i z\_0(r\_0)) - 1) - r\_1 \exp(-i\omega(r\_0))], \\ R\_{21} &=& (1 - r\_0 \exp(-i\omega(r\_0)))^{-1} \left[\frac{1}{2} r\_0 \exp(-i\omega(r\_0)) R\_1^2 + \\ & & d(r\_0) \left(2\sigma\_0^2 z^2(r\_0) - \left(\sigma\_0^2 + \frac{1}{2}\right)\right) \exp(-\sigma\_0^2 z^2(r\_0) + i z(r\_0))\right]. \end{array}$$

It is significant that

$$
\Re R\_1 = 0 \text{ and } \Re R\_2 < 0. \tag{74}
$$

*4.2. Construction of Quasinormal Form*

We introduce the formal series

$$\begin{split} u &= \varepsilon \big( \exp(i\omega(r\_0)t) \big) \sum\_{m = -\infty}^{\infty} \xi\_m(\tau) \exp(i(z(r\_0)\varepsilon^{-1} + \theta + m)\mathbf{x} + \varepsilon i\mathbf{R}\_1(\theta + m)t) + \\ &\exp(-i\omega(r\_0)t) \sum\_{m = -\infty}^{\infty} \xi\_m(\tau) \exp(-i(z(r\_0)\varepsilon^{-1} + \theta + m)\mathbf{x} - \\ &\varepsilon i\mathbf{R}\_1(\theta + m)t)) + \varepsilon^2 u\_2(t, \tau, \mathbf{x}, \varepsilon) + \varepsilon^3 u\_3(t, \tau, \mathbf{x}, \varepsilon) + \dots, \quad \tau = \varepsilon^2 t. \end{split} \tag{75}$$

The above expression can be simplified significantly. Let

$$\xi(\tau, y) = \sum\_{m = -\infty}^{\infty} \xi\_m(\tau) \exp(imy), \quad y = \infty + \varepsilon R\_1 t.$$

Then, it is possible to proceed from (75) to the presentation

$$\begin{aligned} u &= \varepsilon (\exp(i(\omega(r\_0) + \varepsilon R\_1 \theta)t + i(z(r\_0)\varepsilon^{-1} + \theta)\mathbf{x})\xi(\tau, y) + \\ &\exp(-i(\omega(r\_0) + \varepsilon R\_1 \theta)t - i(z(r\_0)\varepsilon^{-1} + \theta)\mathbf{x})\xi(\tau, y)) + \\ &\varepsilon^2 u\_2(t, \tau, x, y) + \varepsilon^3 u\_3(t, \tau, x, y) + \dots \ . \end{aligned} \tag{76}$$

The functions appearing here, *uj*(*t*, *τ*, *x*, *y*), are periodic with respect to *t*, *x*, and *y*. We insert (76) into (17). Then, performing standard techniques, we determine *u*2(*t*, *τ*, *x*, *y*):

$$\begin{split} u\_{2}(t,\tau,x,y) &= u\_{20}|\tilde{\xi}(\tau,y)|^{2} + u\_{21}\tilde{\xi}^{2}(\tau,y)\exp\left( (2i(\omega(r\_{0}) + \varepsilon R\_{1}\theta)t + 2i(z(r\_{0})\varepsilon^{-1} + \theta)x) + \varepsilon \right) \\ &\quad \bar{u}\_{21}\tilde{\xi}^{2}(\tau,y)\exp\left( (-2i\omega(r\_{0}) + \varepsilon R\_{1}\theta)t - 2i(z(r\_{0})\varepsilon^{-1} + \theta)x) \right) \end{split}$$

where

$$\begin{aligned} \mu\_{20} &= -2\cos\omega(r\_0), \\ \mu\_{21} &= -2r\cos(2\omega(r\_0))[2i\omega(r\_0) + r\_0\exp(-2i\omega(r\_0)) - 1], \\ d(r\_0)(\exp(-2i\omega(r\_0) - 4r\_0^2 z^2(r\_0)) - 1)]^{-1}. \end{aligned}$$

At the next step, we obtain the equation for *u*3(*t*, *τ*, *x*, *y*). From its solvability condition in the indicated class of functions, we arrive at the boundary value problem for *ξ*(*τ*, *y*), determining:

$$\frac{\partial \tilde{\xi}}{\partial \tau} = R\_2 \frac{\partial^2 \tilde{\xi}}{\partial y^2} - i\theta R\_2 \frac{\partial \tilde{\xi}}{\partial y} + (R\_{20} + \theta^2 R^2) \tilde{\xi} + q\tilde{\xi} |\tilde{\xi}|^2,\tag{77}$$

$$
\xi(\pi, y + 2\pi) \equiv \xi(\pi, y). \tag{78}
$$

The equality

$$\begin{aligned} q &= r\_0 \left( 1 - r\_0 \exp(-i\omega(r\_0)) \right)^{-1} \left[ 2\cos(\omega(r\_0)) \left( 1 + \exp(-i\omega(r\_0)) - 1 \right) \right] \\ &\quad \mu\_{21} \left( \exp(i\omega(r\_0)) + \exp(-2i\omega(r\_0)) \right) ]. \end{aligned}$$

holds for the coefficient *q*.

We introduce the notation to formulate the basic result. We arbitrarily fix the value *θ*<sup>0</sup> ∈ [0, 1), and let *εn*(*θ*0) be a sequence such that *εn*(*θ*0) → 0 as *n* → ∞, and the equality *θ*(*εn*(*θ*0)) = *θ*<sup>0</sup> holds for each *n*.

From above, we obtain the following statement.

**Theorem 8.** *Let, for some θ* = *θ*0*, the boundary value problem* (77)*,* (78) *have the bounded solution ξ*0(*τ*, *y*) *as τ* → ∞*, y* ∈ [0, 2*π*]*. Then, under condition* (47) *and for ε* = *εn*(*θ*0)*, the function*

$$\begin{split} u\_0(t, \boldsymbol{x}, \boldsymbol{\varepsilon}) &= \boldsymbol{\varepsilon} (\exp(i(\omega(r\_0) + \boldsymbol{\varepsilon} \boldsymbol{R}\_1 \boldsymbol{\theta}\_0) \boldsymbol{t} + i(\boldsymbol{z}(r\_0) \boldsymbol{\varepsilon}^{-1} + \boldsymbol{\theta}\_0) \boldsymbol{x}) \xi\_0(\boldsymbol{r}, \boldsymbol{y}) + \\ &\quad \exp(-i(\omega(r\_0) + \boldsymbol{\varepsilon} \boldsymbol{R}\_1 \boldsymbol{\theta}\_0) \boldsymbol{t} - i(\boldsymbol{z}(r\_0) \boldsymbol{\varepsilon}^{-1} + \boldsymbol{\theta}\_0) \boldsymbol{x}) \bar{\xi}\_0(\boldsymbol{r}, \boldsymbol{y})), \end{split}$$

$$\tau = \varepsilon^2 t, \quad y = x + \varepsilon \mathcal{R}\_1 t$$

*satisfies the boundary value problem* (17) *up to O*(*ε*2)*.*

Therefore, the boundary value problem (77), (78) is the QNF for (17).
