**1. Introduction**

Consider equation

$$
\dot{u}(t) = -\nu u(t) + \lambda F(u(t-T)),
\tag{1}
$$

where *u* is a scalar function, parameters *ν*, *T*, and *λ* are positive, *F*(*u*) is some nonlinear compactly supported function. This equation is a mathematical model in problems of radiophysics and biology. It simulates a generator with nonlinear delayed feedback with a first-order RC low-pass filter (see, for example, [1–3]). Such generators are used in the manufacture of sonars, noise radars, and D-amplifiers [2]. Equation (1) models a biological process where the single state variable *u* decays with a rate *ν* proportional to *u* in the present and is produced with a rate dependent on the value of *u* some time in the past [4]. Such processes arise in a variety of problems in various areas in biology (see Table 1 and references in [4]). In addition, the dynamics of Equation (1) is of general scientific interest [5–13]. The authors find complicated periodic solutions [5–7] and chaos [8] in this model in the case of "step-like" nonlinearity. In Ref. [9], authors study properties of solutions and find a global attractor of model (1) with delayed positive feedback and in the paper [10] existence and stability of relaxation cycle of the multidimensional system (1) in the case of large *λ* is studied. In Refs. [11–13], the authors study properties of solutions of normalized Equation (1) (parameters *ν* = *λ* = 1) in the case of sufficiently large *T* (*T* 1). They deal with equation

$$
\varepsilon \dot{u}(t) = -u(t) + f(u(t-1)),
\tag{2}
$$

where *ε* = 1/*T* and study how the dynamics of this equation when *ε* is small (when *T* is large in (1)) is related with dynamics of this equation in the case *ε* = 0.

In this paper, we deal with a system of two coupled normalized (*ν* = 1) equations of the form (1)

$$\begin{cases} \begin{array}{l} \dot{u}\_1 + \boldsymbol{u}\_1 = \lambda F(\boldsymbol{u}\_1(t - T)) + \gamma (\boldsymbol{u}\_2 - \boldsymbol{u}\_1), \\\ \dot{u}\_2 + \boldsymbol{u}\_2 = \lambda F(\boldsymbol{u}\_2(t - T)) + \gamma (\boldsymbol{u}\_1 - \boldsymbol{u}\_2). \end{array} \tag{3}$$

Here, delay time *T* is a positive constant, a nonlinear sufficiently smooth function *F*(*u*) is compactly supported:

$$F(u) = \begin{cases} f(u), & |u| \le p\_{\prime} \\ 0, & |u| > p\_{\prime} \end{cases}$$

where *p* is some positive constant.

We assume that function *f*(*u*) on the segment *u* ∈ [−*p*, *p*] satisfies the conditions:

$$\begin{array}{l} f(p) = f(-p) = 0; \\ f(u) \neq 0 \text{ except for a finite number of points;} \\ \text{if } f(u^\*) = 0, \text{ then } f'(u^\*) \neq 0 \text{ or } f''(u^\*) \neq 0. \end{array} \tag{4}$$

and that coefficient *λ* is large enough: *λ* 1.

This model simulates two coupled *D*-amplifiers or two noise-radars with a large amount of feedback. If coupling parameter *γ* is asymptotically small at *λ* → +∞, then exponentially orbitally stable relaxation cycles coexist in model (3) (see [14,15]). Now, we are interested in nonlocal dynamics of this model in the case *γ* is some nonzero constant and we study how the dynamic properties of the system differ in the cases of positive and negative coupling.

The paper is organized as follows. In Section 2, we introduce some set of initial conditions and integrating by steps system (3) under some non-degeneracy conditions we construct solutions with initial conditions from the chosen set. By formulas of solution, we obtain the operator of translation along the trajectories Π and map describing dynamics of this operator. Using this map, we clarify asymptotics of solutions of system (3) in the case *γ* > 0 in Section 3 and in the case *γ* < 0 in Section 4. In Section 5, as an example, we consider a narrower class of functions *f* and prove that asymptotic formulas of solution given in Sections 2–4 are valid for a wide set of initial conditions (for all initial conditions from this set, non-degeneracy conditions hold) and prove the existence of relaxation cycles in system (3). We show that the dynamics of system (3) is significantly different in the case of positive and negative coupling in Section 6 and, in Section 7, we draw conclusions.

## **2. Constructing the Asymptotics of Solutions**

Let's find relaxation solutions of (3) and study the dynamics of this system. For this purpose, we consider initial conditions (*u*1(*s*), *<sup>u</sup>*2(*s*))*<sup>T</sup>* <sup>∈</sup> *<sup>C</sup>*[−*T*,0](R2) outside of the strip |*uj*(*s*)| < *p* (*s* ∈ [−*T*, 0], *j* = 1, 2) and construct asymptotics of all solutions of system (3) for this set of initial conditions.

Due to the choice of initial conditions on the segment *t* ∈ [0, *T*], system (3) has the form

$$\begin{cases} \dot{u}\_1 + \mu\_1 = \gamma (\mu\_2 - \mu\_1), \\\ u\_2 + \mu\_2 = \gamma (\mu\_1 - \mu\_2). \end{cases} \tag{5}$$

Moreover, system (3) has form (5) until at least one of the components of the solution comes into the strip |*uj*| < *p*. Thus, for *t* ≥ 0, until at least one of the components of the solution of system (3) for the first time comes into the strip |*uj*| < *p*, a solution of system (3) has form

$$\begin{array}{l} u\_1(t) = \frac{1}{2}(u\_1(0) + u\_2(0))e^{-t} + \frac{1}{2}(u\_1(0) - u\_2(0))e^{-(1+2\gamma)t}, \\ u\_2(t) = \frac{1}{2}(u\_1(0) + u\_2(0))e^{-t} - \frac{1}{2}(u\_1(0) - u\_2(0))e^{-(1+2\gamma)t}. \end{array} \tag{6}$$

It follows from (6) that, in the case *<sup>γ</sup>* <sup>&</sup>lt; <sup>−</sup><sup>1</sup> <sup>2</sup> , there exist solutions of system (3) tending to infinity, and, in the case *<sup>γ</sup>* <sup>=</sup> <sup>−</sup><sup>1</sup> <sup>2</sup> , there exist solutions of system (3) tending to a constant at *t* → +∞. We are interested in relaxation solutions, which is why we assume further that *<sup>γ</sup>* <sup>&</sup>gt; <sup>−</sup><sup>1</sup> 2 .

If *<sup>γ</sup>* <sup>&</sup>gt; <sup>−</sup><sup>1</sup> <sup>2</sup> , then at least one component of a solution eventually comes into the strip |*uj*| < *p* (*j* = 1 or 2). Let *t*<sup>1</sup> ≥ 0 be the first time moment such that some component of the solution (we denote it as *ui*) gets inside the strip |*ui*(*t*)| ≤ *p*:

$$|u\_1(s+t\_1)| \ge p, \quad |u\_2(s+t\_1)| \ge p \text{ for } s \in [-T,0), \tag{7}$$


$$
u\_i(t\_1) = kp, \quad \upharpoonright\_{3-i}(t\_1) = \ge p,\tag{8}$$

where *k* denotes the sign of *ui*(*t*1) (parameter *k* takes values −1 or 1) and *x* is some value such that <sup>|</sup>*x*| ≥ 1. We denote the set of pairs of initial functions (*u*1(*s*), *<sup>u</sup>*2(*s*))*<sup>T</sup>* <sup>∈</sup> *<sup>C</sup>*[−*T*,0](R2) satisfying conditions (7) and (8) as *IC*(*i*, *k*, *x*).

We will integrate system (3) using a method of steps. It follows from (7) that, on the first step (time segment *t* ∈ [*t*1, *t*<sup>1</sup> + *T*]), system (3) has form (5) and the solution has a form

$$\begin{array}{rcl} u\_i(t) &=& \frac{(k+x)p}{2} e^{-(t-t\_1)} + \frac{(k-x)p}{2} e^{-(1+2\gamma)(t-t\_1)},\\ u\_{3-i}(t) &=& \frac{(k+x)p}{2} e^{-(t-t\_1)} + \frac{(x-k)p}{2} e^{-(1+2\gamma)(t-t\_1)}.\end{array} \tag{9}$$

Since function *ui* is inside the strip |*ui*| < *p* for *t* ∈ [*t*1, *t*<sup>1</sup> + *δ*], then, for *t* ∈ [*t*<sup>1</sup> + *T*, *t*<sup>1</sup> + 2*T*], we have that *<sup>F</sup>*(*ui*(*<sup>t</sup>* − *<sup>T</sup>*)) is not identically equal to 0. In addition, *<sup>F</sup>*(*u*3−*i*(*<sup>t</sup>* − *<sup>T</sup>*)) may be identically equal to 0 or not (it depends on value of *x*). Then, on the second step (*t* ∈ [*t*<sup>1</sup> + *T*, *t*<sup>1</sup> + 2*T*]), we consider system (3) as an inhomogeneous system of ordinary differential equations (here functions *F*(*ui*(*t* − *T*)) and *<sup>F</sup>*(*u*3−*i*(*<sup>t</sup>* − *<sup>T</sup>*)) are known from the previous step and we consider them as inhomogeneity). Thus, the following formula for solution of system (3) holds:

$$\begin{array}{l} u\_i(t) = \frac{(k+x)p}{2}e^{-(t-t\_1)} + \frac{(k-x)p}{2}e^{-(1+2\gamma)(t-t\_1)} + \frac{\lambda}{2}A(k,x,t,t\_1),\\ u\_{3-i}(t) = \frac{(k+x)p}{2}e^{-(t-t\_1)} + \frac{(x-k)p}{2}e^{-(1+2\gamma)(t-t\_1)} + \frac{\lambda}{2}B(k,x,t,t\_1),\end{array} \tag{10}$$

where

$$\begin{split} A(k,x,t,t\_{1}) &= \int\_{T+t\_{1}}^{t} \left( e^{s-t} + e^{(1+2\gamma)(s-t)} \right) F\left( \frac{(k+x)p}{2} e^{t\_{1}+T-s} + \frac{(k-x)p}{2} e^{(1+2\gamma)(t\_{1}+T-s)} \right) ds \\ &+ \int\_{T+t\_{1}}^{t} \left( e^{s-t} - e^{(1+2\gamma)(s-t)} \right) F\left( \frac{(k+x)p}{2} e^{t\_{1}+T-s} + \frac{(x-k)p}{2} e^{(1+2\gamma)(t\_{1}+T-s)} \right) ds, \\ B(k,x,t,t\_{1}) &= \int\_{T+t\_{1}}^{t} \left( e^{s-t} - e^{(1+2\gamma)(s-t)} \right) F\left( \frac{(k+x)p}{2} e^{t\_{1}+T-s} + \frac{(k-x)p}{2} e^{(1+2\gamma)(t\_{1}+T-s)} \right) ds \\ &+ \int\_{T+t\_{1}}^{t} \left( e^{s-t} + e^{(1+2\gamma)(s-t)} \right) F\left( \frac{(k+x)p}{2} e^{t\_{1}+T-s} + \frac{(x-k)p}{2} e^{(1+2\gamma)(t\_{1}+T-s)} \right) ds. \end{split}$$

Let's introduce the following conditions on the functions *A* and *B*:

**Assumption 1.** *Number of points t* <sup>∗</sup> ∈ [*t*<sup>1</sup> + *T*, *t*<sup>1</sup> + 2*T*] *for which A*(*k*, *x*, *t* <sup>∗</sup>, *t*1) = 0 *(B*(*k*, *x*, *t* <sup>∗</sup>, *t*1) = 0*) is finite. If A*(*k*, *x*, *t* <sup>∗</sup>, *t*1) = 0 *(B*(*k*, *x*, *t* <sup>∗</sup>, *<sup>t</sup>*1) = <sup>0</sup>*), then there exists <sup>j</sup>* <sup>∈</sup> <sup>N</sup> *such that <sup>∂</sup><sup>j</sup> A*(*k*, *x*, *t*, *t*1) *∂tj <sup>t</sup>*=*t*<sup>∗</sup> <sup>=</sup> <sup>0</sup> *( ∂j B*(*k*, *x*, *t*, *t*1) *∂tj <sup>t</sup>*=*t*<sup>∗</sup> <sup>=</sup> <sup>0</sup>*, respectively).*

**Assumption 2.** *Inequality A*(*k*, *x*, *t*<sup>1</sup> + 2*T*, *t*1)*B*(*k*, *x*, *t*<sup>1</sup> + 2*T*, *t*1) = 0 *holds.*

Under Assumption 2, we obtain that

$$\begin{aligned} u\_i(t\_1 + 2T) &= \frac{\lambda}{2} \Big( A(k, \mathbf{x}, t\_1 + 2T, t\_1) + o(1) \Big), \\ u\_{3-i}(t\_1 + 2T) &= \frac{\lambda}{2} \Big( B(k, \mathbf{x}, t\_1 + 2T, t\_1) + o(1) \Big) \end{aligned} \tag{11}$$

at *<sup>λ</sup>* → +<sup>∞</sup> and that both functions *ui*(*t*) and *<sup>u</sup>*3−*i*(*t*) at the point *<sup>t</sup>* = *<sup>t</sup>*<sup>1</sup> + 2*<sup>T</sup>* are outside of the strip |*uj*| < *p*.

**Lemma 1.** *If Assumptions 1 and 2 hold, then on the segment t* ∈ [*t*<sup>1</sup> + 2*T*, *t*<sup>1</sup> + 3*T*] *functions ui*(*t*) *and <sup>u</sup>*3−*i*(*t*) *have the form*

$$\begin{array}{l} u\_{i}(t) = \frac{\lambda}{4} (A(k, \mathbf{x}, 2T + t\_{1}, t\_{1}) + B(k, \mathbf{x}, 2T + t\_{1}, t\_{1}) + o(1))e^{-(t - t\_{1} - 2T)} \\ + \frac{\lambda}{4} (A(k, \mathbf{x}, 2T + t\_{1}, t\_{1}) - B(k, \mathbf{x}, 2T + t\_{1}, t\_{1}) + o(1))e^{-(1 + 2\gamma)(t - t\_{1} - 2T)} \\ u\_{3-i}(t) = \frac{\lambda}{4} (A(k, \mathbf{x}, 2T + t\_{1}, t\_{1}) + B(k, \mathbf{x}, 2T + t\_{1}, t\_{1}) + o(1))e^{-(t - t\_{1} - 2T)} \\ - \frac{\lambda}{4} (A(k, \mathbf{x}, 2T + t\_{1}, t\_{1}) - B(k, \mathbf{x}, 2T + t\_{1}, t\_{1}) + o(1))e^{-(1 + 2\gamma)(t - t\_{1} - 2T)}. \end{array} \tag{12}$$

**Proof.** Let *t* ∈ [*t*<sup>1</sup> + 2*T*, *t*<sup>1</sup> + 3*T*]. On this segment, we consider system (3) as a system of inhomogeneous linear ordinary differential equations (on this time segment we consider known functions *<sup>λ</sup>F*(*ui*(*<sup>t</sup>* − *<sup>T</sup>*)) and *<sup>λ</sup>F*(*u*3−*i*(*<sup>t</sup>* − *<sup>T</sup>*)) as inhomogeneity). Therefore, a solution of this system on the time segment *t* ∈ [*t*<sup>1</sup> + 2*T*, *t*<sup>1</sup> + 3*T*] has the form of a sum of particular integral (PI) and complementary function (CF, solution of linear part of system (3)–system (5)) with constants determined from the initial conditions (11):

$$\begin{aligned} \boldsymbol{\mu}\_{i}(t) &= \boldsymbol{\mu}\_{i\_{\mathcal{CF}}}(t) + \boldsymbol{\mu}\_{i\_{\mathcal{PI}}}(t), \\ \boldsymbol{\mu}\_{3-i}(t) &= \boldsymbol{\mu}\_{(3-i)\_{\mathcal{CF}}}(t) + \boldsymbol{\mu}\_{(3-i)\_{\mathcal{PI}}}(t). \end{aligned}$$

Let's find asymptotics of particular integral of this system at *λ* → +∞. A particular integral of the system (3) on the time segment *t* ∈ [*t*<sup>1</sup> + 2*T*, *t*<sup>1</sup> + 3*T*] has the form

$$\begin{split} u\_{i p\_l}(t) &= \frac{\lambda}{2} \int\_{t\_1 + 2T}^t \left( e^{s - t} + e^{(1 + 2\gamma)(s - t)} \right) F(u\_i(s - T)) + (e^{s - t} - e^{(1 + 2\gamma)(s - t)}) F(u\_{3 - i}(s - T)) ds, \\ u\_{(3 - i)\_{p\_l}}(t) &= \frac{\lambda}{2} \int\_{t\_1 + 2T}^t \left( e^{s - t} - e^{(1 + 2\gamma)(s - t)} \right) F(u\_i(s - T)) + (e^{s - t} + e^{(1 + 2\gamma)(s - t)}) F(u\_{3 - i}(s - T)) ds. \end{split} \tag{13}$$

Suppose a particular integral (13) is non-zero. This integral on some segment is non-zero only if functions *<sup>F</sup>*(*ui*(*<sup>s</sup>* − *<sup>T</sup>*)) or *<sup>F</sup>*(*u*3−*i*(*<sup>s</sup>* − *<sup>T</sup>*)) are non-zero on this segment. Function *<sup>F</sup>*(*ui*(*<sup>t</sup>* − *<sup>T</sup>*)) (*F*(*u*3−*i*(*<sup>t</sup>* − *<sup>T</sup>*))) is non-zero only if |*ui*(*<sup>t</sup>* − *<sup>T</sup>*)| < *<sup>p</sup>* (|*u*3−*i*(*<sup>t</sup>* − *<sup>T</sup>*)| < *<sup>p</sup>*). For sufficiently large values of *λ* this condition holds only if *A*(*k*, *x*, *t* − *T*, *t*1) (*B*(*k*, *x*, *t* − *T*, *t*1) respectively) is in the neighborhood of zero. Function *A*(*k*, *x*, ·, *t*1) (*B*(*k*, *x*, ·, *t*1)) is continuous; consequently, there exists point *t* <sup>∗</sup> ∈ [*t*<sup>1</sup> + *T*, *t*<sup>1</sup> + 2*T*] such that *A*(*k*, *x*, *t* <sup>∗</sup>, *t*1) = 0 (*B*(*k*, *x*, *t* <sup>∗</sup>, *t*1) = 0, respectively).

Consider the point *t* <sup>∗</sup> ∈ [*t*<sup>1</sup> + *T*, *t*<sup>1</sup> + 2*T*] such that *A*(*k*, *x*, *t* <sup>∗</sup>, *t*1) = 0. It follows from Assumption 1 that there exist *<sup>j</sup>* <sup>∈</sup> <sup>N</sup> such that *<sup>∂</sup><sup>j</sup> A*(*k*, *x*, *t*, *t*1) *∂tj <sup>t</sup>*=*t*<sup>∗</sup> <sup>=</sup> 0. Let *<sup>q</sup>* be the minimum from these numbers *<sup>j</sup>*. Consequently, it follows from (10) that, in the neighborhood of *t* ∗, we have

$$u\_i(t - T) = \frac{(k + x)p}{2} \varepsilon^{-(t - T - t\_1)} + \frac{(k - x)p}{2} \varepsilon^{-(1 + 2\gamma)(t - T - t\_1)}$$

$$+ \frac{\lambda}{2} \left(\frac{\partial^q A(k, x, t^\*, t\_1)}{\partial t^q} + o(1)\right) \frac{(t - T - t^\*)^q}{q!}.\tag{14}$$

Let's estimate "time of living" Δ*t* <sup>∗</sup> of function *ui*(*t* − *T*) in the strip |*ui*| < *p* in the neighborhood of the point *t* − *T* = *t* ∗ ("time of living" means here length of the maximal interval of values *t* such that *t* ∗ belongs to this segment and inequality |*ui*(*t*)| < *p* is true for all points *t* from this segment). From (14), under the condition that *λ* is sufficiently large, we get that Δ*t* <sup>∗</sup> <sup>≤</sup> *<sup>M</sup>*1*λ*<sup>−</sup> <sup>1</sup> *<sup>q</sup>* , where *M*<sup>1</sup> = *M*1(*k*, *x*, *γ*) is some positive value. From Assumption 1, we know that number of points *t* ∗ such that *A*(*k*, *x*, *t* <sup>∗</sup>, *t*1) = 0 is finite, which is why there exists *Q* = *qmax*—maximum from values *q* for all points *t* ∗. Then, on the whole segment *t* − *T* ∈ [*t*<sup>1</sup> + *T*, *t*<sup>1</sup> + 2*T*] "time of living" Δ*ttotal* of function *ui*(*t* − *T*) in the strip <sup>|</sup>*ui*<sup>|</sup> <sup>&</sup>lt; *<sup>p</sup>* has estimate <sup>Δ</sup>*ttotal* <sup>≤</sup> *<sup>M</sup>*2*λ*<sup>−</sup> <sup>1</sup> *<sup>Q</sup>* , where *M*<sup>2</sup> = *M*2(*k*, *x*, *γ*) is some positive value. Similarly, for function *<sup>u</sup>*3−*i*(*<sup>t</sup>* <sup>−</sup> *<sup>T</sup>*), we have estimate <sup>Δ</sup>*ttotal* <sup>≤</sup> *<sup>M</sup>*3*λ*<sup>−</sup> <sup>1</sup> *<sup>P</sup>* , where *M*<sup>3</sup> and *P* are some positive values. Function *F* is bounded, which is why, for a particular integral (13), we have the following estimate:

$$|u\_{i\_{P\bar{I}}}(t)| \leq M\lambda^{\frac{\max\{P,Q\}-1}{\max\{P,Q\}}}, \quad |u\_{(3-i)\_{P\bar{I}}}(t)| \leq M\lambda^{\frac{\max\{P,Q\}-1}{\max\{P,Q\}}},$$

where *M* is some positive value, *t* ∈ [*t*<sup>1</sup> + 2*T*, *t*<sup>1</sup> + 3*T*].

A solution of linear part of system (3) satisfying initial conditions (11) on this segment has form

$$\begin{array}{l} u\_{i\_{\Gamma}}(t) = \frac{\lambda}{4} (A(k, \mathbf{x}, 2T + t\_{1}, t\_{1}) + B(k, \mathbf{x}, 2T + t\_{1}, t\_{1}) + o(1))e^{-(t - t\_{1} - 2T)} \\ + \frac{\lambda}{4} (A(k, \mathbf{x}, 2T + t\_{1}, t\_{1}) - B(k, \mathbf{x}, 2T + t\_{1}, t\_{1}) + o(1))e^{-(1 + 2\gamma)(t - t\_{1} - 2T)} \\ u\_{(3-i)\_{\Gamma}}(t) = \frac{\lambda}{4} (A(k, \mathbf{x}, 2T + t\_{1}, t\_{1}) + B(k, \mathbf{x}, 2T + t\_{1}, t\_{1}) + o(1))e^{-(t - t\_{1} - 2T)} \\ - \frac{\lambda}{4} (A(k, \mathbf{x}, 2T + t\_{1}, t\_{1}) - B(k, \mathbf{x}, 2T + t\_{1}, t\_{1}) + o(1))e^{-(1 + 2\gamma)(t - t\_{1} - 2T)} \end{array}$$

Thus, a complementary function gives us the leading term of asymptotics of solution of system (3) on the segment *t* ∈ [*t*<sup>1</sup> + 2*T*, *t*<sup>1</sup> + 3*T*] and thus a solution on this segment has form (12).

**Corollary 1.** *The leading term of asymptotics of solution of system* (3) *coincides with solution of system* (5) *with initial conditions* (11) *on the segment t* ∈ [*t*<sup>1</sup> + 2*T*, *t*<sup>1</sup> + 3*T*]*.*

Let's study asymptotics of solutions of system (3) for values *t* > *t*<sup>1</sup> + 3*T*. While both components of solution are outside of the strip |*uj*| < *p* (*j* = 1, 2), system (3) has form (5) and solution has form (12). If some component of solution comes to the strip |*uj*| < *p* at the point *t* = *t*<sup>0</sup> > *t*<sup>1</sup> + 2*T*, then on the next step *t* ∈ [*t*<sup>0</sup> + *T*, *t*<sup>0</sup> + 2*T*] nonlinearity *F* is non-zero and the leading term of asymptotics of solution may change. Whether it changes or not is determined by the values of the functions

$$\begin{aligned} \mathcal{G}\_{\pm}(t) &= \left( A(k, \mathbf{x}, 2T + t\_1, t\_1) + B(k, \mathbf{x}, 2T + t\_1, t\_1) \right) \\ &\pm \left( A(k, \mathbf{x}, 2T + t\_1, t\_1) - B(k, \mathbf{x}, 2T + t\_1, t\_1) \right) e^{-2\gamma(t - t\_1 - 2T)} \end{aligned}$$

in the neighborhood of the point *t*0.

Note that, in terms of functions *G*<sup>+</sup> and *G*<sup>−</sup> on the segment *t* ∈ [*t*<sup>1</sup> + 2*T*, *t*0], we have the following representation of functions *ui* and *u*3−*i*:

$$u\_i(t) = \frac{\lambda}{4} \left( G\_+(t) + o(1) \right) e^{-(t - t\_1 - 2T)},\tag{15}$$

$$u\_{3-i}(t) = \frac{\lambda}{4} \left( G\_{-}(t) + o(1) \right) e^{-\left(t - t\_1 - 2T\right)}.\tag{16}$$

There exists two principally different cases when function *ui*(*t*) (or *<sup>u</sup>*3−*i*(*t*)) comes into the strip |*uj*| < *p* at the point *t* = *t*<sup>0</sup> > *t*<sup>1</sup> + 2*T*:


Note that, for some functions *F* and values of parameters *k*, *x*, and *γ*, Case 1 does not take place. Suppose we have function *F* and values of parameters *k*, *x*, and *γ* such that this Case occurs. Then, we have the following Lemma.

**Lemma 2.** *Suppose some component of solution comes into the strip* |*uj*| < *p at the point t* = *t*<sup>0</sup> > *t*<sup>1</sup> + 2*T and Formula* (12) *is valid for the leading term of asymptotics of solution on the segment t* ∈ [*t*<sup>1</sup> + 2*T*, *t*0]*. If there exists a point from an asymptotically small at λ* → +∞ *neighborhood of the point t* = *t*<sup>0</sup> *such that the second multiplier in* (15) *or* (16) *is equal to zero, then asymptotics of solution on the segment t* ∈ [*t*<sup>0</sup> + *T*, *t*<sup>0</sup> + 2*T*] *has form* (12)*.*

**Proof.** First, note that, if the second multiplier in (15) or (16) is equal to zero at some point from the small neighborhood of the point *t* = *t*0, then there exists value *t*<sup>∗</sup> such that |*t*<sup>∗</sup> − *t*0| = *o*(1) at *λ* → +∞ and *G*+(*t*∗)*G*−(*t*∗) = 0.

Each equation *G*+(*t*) = 0 and *G*−(*t*) = 0 has at most one root and, if one equation has a root, then another equation has no roots. This root does not depend on *λ*, and it follows from Assumption 2 that if *G*+(*t*∗) = 0 (*G*−(*t*∗) = 0), then *G* +(*t*∗) = 0 (*G* <sup>−</sup>(*t*∗) <sup>=</sup> 0, respectively).

Assume without loss of generality that function *ui* comes into the strip |*ui*| < *p* at the point *t* = *t*<sup>0</sup> and *G*+(*t*∗) = 0. Acting like in the proof of Lemma 1, we obtain that "time of living" Δ*t*<sup>∗</sup> of function *ui*(*t*) in the strip <sup>|</sup>*ui*<sup>|</sup> <sup>&</sup>lt; *<sup>p</sup>* in the neighborhood of the point *<sup>t</sup>* <sup>=</sup> *<sup>t</sup>*<sup>∗</sup> has estimate <sup>Δ</sup>*t*<sup>∗</sup> <sup>≤</sup> *constλ*−1. This is why a particular integral of the system (3) on the segment *t* ∈ [*t*<sup>0</sup> + *T*, *t*<sup>0</sup> + 2*T*] has estimate

$$|u\_{i\_{Pl}}(t)| \le const\_{1\prime} \quad |u\_{(3-i)\_{Pl}}(t)| \le const\_{2\prime}$$

and a complementary function has estimate

$$|u\_{i\mathbb{C}F}(t)| \ge \text{const}\_3 \lambda\_\prime \quad |u\_{(3-i)\_{\mathbb{C}F}}(t)| \ge \text{const}\_4 \lambda\_\prime$$

where *const*<sup>3</sup> > 0 and *const*<sup>4</sup> > 0.

Thus, on the segment *t* ∈ [*t*<sup>0</sup> + *T*, *t*<sup>0</sup> + 2*T*], Formula (12) is valid.

For the further reasoning, we need a notation of the time moment of leaving the strip |*uj*| < *p* in Case 1 (if this Case occurs). We denote it as *tleave*. It follows from Lemma 2 that *tleave* < *t*<sup>∗</sup> + *T*. If Case 1 does not take place, then we define *tleave* = *t*<sup>1</sup> + 2*T*. Thus, there exists a constant *Mt*.*l*. > 0 independent on *λ* such that *tleave* < *Mt*.*l*.

Lemma 2 implies the following statement.

**Corollary 2.** *For all <sup>t</sup>* > *tleave, both functions ui*(*t*) *and <sup>u</sup>*3−*i*(*t*) *are outside of the strip* |*uj*| < *<sup>p</sup> until Case 2 occurs.*

Let's study Case 2 in more detail.

First, consider the case *γ* > 0. If non-degeneracy condition

$$A(k, \mathbf{x}, 2T + t\_1, t\_1) + B(k, \mathbf{x}, 2T + t\_1, t\_1) \neq 0 \tag{17}$$

holds, then there exist positive constants *cmin*, *cMax*, such that

0 < *cmin* < |*G*±(*t*) + *o*(1)| < *cMax*

in some independent on *<sup>λ</sup>* neighborhood of the point *<sup>t</sup>* <sup>=</sup> *<sup>t</sup>*0. Therefore, <sup>|</sup>*λe*−(*t*0−*t*1−2*T*)<sup>|</sup> <sup>&</sup>lt; *<sup>M</sup>*<sup>4</sup> at *λ* → +∞, where *M*<sup>4</sup> is some positive constant. This is why

$$t\_0 - t\_1 = (1 + o(1)) \ln \lambda \tag{18}$$

at *λ* → +∞. In addition, in the neighborhood of the point *t* = *t*0, solution of system (3) has form

$$\begin{array}{l}\mu\_{i}(t) = \frac{\lambda}{4}(A(k, \mathbf{x}, 2T + t\_{1}, t\_{1}) + B(k, \mathbf{x}, 2T + t\_{1}, t\_{1}) + o(1))e^{-(t - t\_{1} - 2T)}\\\mu\_{3-i}(t) = \frac{\lambda}{4}(A(k, \mathbf{x}, 2T + t\_{1}, t\_{1}) + B(k, \mathbf{x}, 2T + t\_{1}, t\_{1}) + o(1))e^{-(t - t\_{1} - 2T)}.\end{array} \tag{19}$$

Consider the case <sup>−</sup><sup>1</sup> <sup>2</sup> < *γ* < 0. If non-degeneracy condition

$$A(k, \mathbf{x}, 2T + t\_1, t\_1) - B(k, \mathbf{x}, 2T + t\_1, t\_1) \neq 0 \tag{20}$$

holds, then, for some positive constants *dmin* and *dMax* in some independent on *λ* neighborhood of the point *t* = *t*0, we have

$$0 < d\_{\min} < |(G\_{\pm}(t) + o(1))e^{2\gamma(t - t\_1 - 2T)}| < d\_{\max}.$$

Therefore, we obtain that <sup>|</sup>*λe*−(1+2*γ*)(*t*0−*t*1−2*T*)<sup>|</sup> <sup>&</sup>lt; *<sup>M</sup>*<sup>5</sup> at *<sup>λ</sup>* <sup>→</sup> <sup>+</sup>∞, where *<sup>M</sup>*<sup>5</sup> is some positive constant. Consequently,

$$t\_0 - t\_1 = \left( (1 + 2\gamma)^{-1} + o(1) \right) \ln \lambda \tag{21}$$

at *λ* → +∞ and in the neighborhood of the point *t* = *t*<sup>0</sup> solution of system (3) has form

$$\begin{array}{l} u\_{i}(t) = \frac{\lambda}{4} (A(k, \mathbf{x}, 2T + t\_{1}, t\_{1}) - B(k, \mathbf{x}, 2T + t\_{1}, t\_{1}) + o(1))e^{-(1+2\gamma)(t-t\_{1}-2T)} \\ u\_{3-i}(t) = -\frac{\lambda}{4} (A(k, \mathbf{x}, 2T + t\_{1}, t\_{1}) - B(k, \mathbf{x}, 2T + t\_{1}, t\_{1}) + o(1))e^{-(1+2\gamma)(t-t\_{1}-2T)} .\end{array} \tag{22}$$

From Formulas (18) and (21), we get that *t*<sup>0</sup> − *tleave* > *T*. In addition, it follows from Formulas (19) and (22) that if |*uj*(*t*0)| = *p*, then there exists *δ* > 0 such that |*uj*(*t*)| < *p* for all *t* ∈ (*t*0, *t*<sup>0</sup> + *δ*). Thus, there exists *t*<sup>2</sup> (it is equal to *t*<sup>0</sup> from the Case 2), such that

$$t\_2 - t\_1 = \begin{cases} (1 + o(1)) \ln \lambda\_\prime & \gamma > 0, \\ ((1 + 2\gamma)^{-1} + o(1)) \ln \lambda\_\prime & -\frac{1}{2} < \gamma < 0, \end{cases} \tag{23}$$

$$|u\_1(s+t\_2)| > p, \quad |u\_2(s+t\_2)| > p \text{ for all } s \in [-T, 0), \tag{24}$$

and

$$u\_{\overline{i}}(t\_2) = \overline{k}p, \quad u\_{\Im - \overline{i}}(t\_2) = \overline{x}p \tag{25}$$

at *λ* → +∞.

It follows from Lemmas 1 and 2, Corollaries 1 and 2 and from the reasoning given above that the next statement is true.
