**3. Dynamics in the Case of the Positive Coupling**

In this section, we construct a map on *kn*, *xn*, and *in* and make conclusions on dynamics of system (3) in the case of positive coupling (*γ* > 0).

Define *C*(*n*) and *D*(*n*) as

$$\begin{aligned} C(n) &= A(k\_{n\prime}\mathbf{x}\_{n\prime}2T + t\_{n\prime}t\_{n\prime}) + B(k\_{n\prime}\mathbf{x}\_{n\prime}2T + t\_{n\prime}t\_{n\prime}),\\ D(n) &= A(k\_{n\prime}\mathbf{x}\_{n\prime}2T + t\_{n\prime}t\_{n\prime}) - B(k\_{n\prime}\mathbf{x}\_{n\prime}2T + t\_{n\prime}t\_{n\prime}), \end{aligned}$$

where *<sup>n</sup>* <sup>∈</sup> <sup>N</sup>. Suppose that

$$C(n) \neq 0,\tag{26}$$

((26) is condition (17) with *k* = *kn*, *x* = *xn*, *t*<sup>1</sup> = *tn*) and Assumptions 1 and 2 hold for values *kn*, *xn* and *tn* for all *<sup>n</sup>* <sup>∈</sup> <sup>N</sup>. Then, acting like in Section 2, we get that in the case of positive coupling values *ui*(*tn*+1) and *<sup>u</sup>*3−*i*(*tn*+1) have form

$$u\_i(t\_{n+1}) = \frac{\lambda}{4}(\mathcal{C}(n) + o(1))e^{-(t\_{n+1} - t\_n - 2T)}, \\ u\_{3-i}(t\_{n+1}) = \frac{\lambda}{4}(\mathcal{C}(n) + o(1))e^{-(t\_{n+1} - t\_n - 2T)}.$$

Thus, we obtain that, in the case *γ* > 0, values *tn* (*n* = 1, 2, . . .) satisfy

$$t\_{n+1} - t\_n = (1 + o(1)) \ln \lambda \tag{27}$$

at *λ* → +∞.

From (12) and (27), we get that the mapping on *kn*, *xn*, and *in* has form

$$\begin{aligned} k\_{n+1} &= \text{sign}(\mathbb{C}(n)), \\ i\_{n+1} &= \begin{cases} i\_{n\prime} & \text{sign}(\mathbb{C}(n)D(n)) = -1, \\ 3 - i\_{n\prime} & \text{sign}(\mathbb{C}(n)D(n)) = 1, \\ x\_{n+1} &= k\_{n+1} + O(\lambda^{-2\gamma}) \end{cases} \end{aligned} \tag{28}$$

at *λ* → +∞.

It follows from (28) that we have *kn* − *xn* = *o*(1) for all *n* = 2, 3, ... under the condition that Assumptions 1 and 2 and inequality (26) are fulfilled. Thus, starting from the second iteration Assumption 1 should be satisfied for parameters *k* = *kn*, *x* = *kn* + *o*(1), and *t*<sup>1</sup> = *tn*. Let's formulate this assumption for these values of parameters *k*, *x*, and *t*1. Functions *A*(*kn*, *kn* + *o*(1), *t*, *tn*) and *B*(*kn*, *kn* + *o*(1), *t*, *tn*) have form

$$A(k\_n, k\_n + o(1), t, t\_n) = B(k\_n, k\_n + o(1), t, t\_n) + o(1) = 2 \int\_{T + t\_n}^t e^{s - t} F\left(k\_n p e^{t\_n + T - s}\right) ds + o(1).$$

In Assumption <sup>1</sup> value *<sup>t</sup>* ∈ [*tn* + *<sup>T</sup>*, *tn* + <sup>2</sup>*T*], so, for each *<sup>n</sup>* value, ˜*<sup>t</sup>* = *<sup>t</sup>* − *tn* is in the segment [*T*, 2*T*]. Since

$$\int\_{T+t\_n}^t e^{s-t} F\left(k\_n p e^{t\_n+T-s}\right) ds = \int\_T^t e^{s-T} F\left(k\_n p e^{T-s}\right) ds,$$

then Assumption 1 for any *n* = 2, 3, ... is the same (only *kn* may change, but it takes two values only). Thus, if the following assumption holds, then Assumption 1 holds for all *n* = 2, 3, . . .

**Assumption 3.** *Number of points t* <sup>∗</sup> ∈ [*T*, 2*T*] *such that h*(*k*, *t* ∗) = 0 *is finite. If h*(*k*, *t* ∗) = 0*, then there exists j* <sup>∈</sup> <sup>N</sup> *such that <sup>∂</sup><sup>j</sup> h*(*k*, ˜*t*) *∂*˜*tj* ˜*t*=*t*<sup>∗</sup> *is non-zero. Here, k* <sup>=</sup> <sup>1</sup> *or* <sup>−</sup><sup>1</sup> *and*

$$h(k, \tilde{t}) = \int\_{\tilde{T}}^{\tilde{t}} e^{s - \tilde{t}} F\left(kpe^{T-s}\right) ds.$$

Under Assumption 3, the asymptotics of the solution has form

$$\begin{aligned} u\_{i\_n}(t) &= k\_n p e^{-(t - t\_n)} + o(1), \\ u\_{3 - i\_n}(t) &= k\_n p e^{-(t - t\_n)} + o(1) \end{aligned} \tag{29}$$

on the time segments *t* ∈ [*tn*, *tn* + *T*], where *n* = 2, 3, ... ((29) is Formula (9) with *i* = *in*, *k* = *kn*, *x* = *xn* = *kn* + *o*(1), and *t*<sup>1</sup> = *tn*). On the segments *t* ∈ [*tn* + *T*, *tn* + 2*T*], the main terms of asymptotics of solution is given by the formula

$$\begin{aligned} u\_{i\_n}(t) &= \lambda \left( h(k\_n, t - t\_n) + o(1) \right), \\ u\_{3 - i\_n}(t) &= \lambda \left( h(k\_n, t - t\_n) + o(1) \right) \end{aligned} \tag{30}$$

((30) is Formula (10) with *i* = *in*, *k* = *kn*, *x* = *xn* = *kn* + *o*(1), and *t*<sup>1</sup> = *tn*, where functions *A* and *B* are rewritten in terms of function *h*).

We assume that the following non-degeneracy condition holds:

$$h(1,2T)h(-1,2T) \neq 0\tag{31}$$

(the fulfillment of this inequality guarantees that the Assumption 2 and (26) are satisfied for all *n* = 2, 3, . . .).

Then, on the segments, a *t* ∈ [*tn* + 2*T*, *tn*+1] solution satisfies equalities

$$\begin{aligned} u\_{i\_n}(t) &= \lambda \left( h(k\_n, 2T) + o(1) \right) e^{-\left(t - t\_n - 2T\right)}, \\ u\_{3-i\_n}(t) &= \lambda \left( h(k\_n, 2T) + o(1) \right) e^{-\left(t - t\_n - 2T\right)}. \end{aligned} \tag{32}$$

at *λ* → +∞ ((32) is Formula (12) with *i* = *in*, *k* = *kn*, *x* = *xn* = *kn* + *o*(1), and *t*<sup>1</sup> = *tn*, where functions *A* and *B* are rewritten in terms of function *h*).

Thus, we have the following theorem:

**Theorem 1.** *Suppose γ* > 0 *and for values of k*<sup>1</sup> *and x*<sup>1</sup> *Assumptions 1, 2, and inequality* (17) *hold. Suppose Assumption 3 and inequality* (31) *hold. Then, for any sufficiently large λ* > 0*, there exists t*<sup>2</sup> = *t*2(*k*1, *x*1) > 0 *such that for all t* > *t*<sup>2</sup> *solution of system* (3) *satisfies Formulas* (29)*,* (30)*, and* (32)*.*

In Figure 1, an example of a solution of system (3) in the case of *γ* > 0 is shown.

Since *<sup>F</sup>* is smooth and *xn*+<sup>1</sup> <sup>−</sup> *kn*+<sup>1</sup> <sup>=</sup> *<sup>O</sup>*(*λ*−2*γ*) at *<sup>λ</sup>* <sup>→</sup> <sup>+</sup>∞, then, in the case *<sup>γ</sup>* <sup>&</sup>gt; <sup>1</sup> <sup>2</sup> , we have the following statement.

**Figure 1.** Example of solution. Values of parameters: *T* = 1, *γ* = 0.1, *p* = 1, *λ* = 10,000. Black line—*u*1(*t*), orange dashed line—*u*2(*t*).

**Corollary 4.** *Suppose γ* > <sup>1</sup> <sup>2</sup> *and for values k*<sup>1</sup> *and x*<sup>1</sup> *Assumptions 1 and 2 hold and inequality* (17) *is true. Suppose Assumption 3 and inequality* (31) *are true. Then, for any sufficiently large λ* > 0*, there exists t*2(*k*1, *x*1) > 0 *such that for all t* > *t*<sup>2</sup> *inequality* |*u*1(*t*) − *u*2(*t*)| = *o*(1) *is true.*

#### **4. Dynamics in the Case of Negative Coupling**

In this section, we assume that <sup>−</sup><sup>1</sup> <sup>2</sup> < *γ* < 0. We construct map on *kn*, *xn*, and *in* for these values of *γ* and make conclusions about dynamics of system (3).

Suppose inequality

$$D(n) \neq 0\tag{33}$$

and Assumptions <sup>1</sup> and <sup>2</sup> for values *kn*, *xn*, and *tn* hold for all *<sup>n</sup>* <sup>∈</sup> <sup>N</sup>. Then, like in Section 2, we obtain that, in the case <sup>−</sup><sup>1</sup> <sup>2</sup> < *<sup>γ</sup>* < 0, values *ui*(*tn*+1) and *<sup>u</sup>*3−*i*(*tn*+1) have the form

$$u\_i(t\_{n+1}) = \frac{\lambda}{4}(D(n) + o(1))e^{-(1+2\gamma)(t\_{n+1} - t\_n - 2T)}, \\ u\_{3-i}(t\_{n+1}) = \frac{\lambda}{4}(-D(n) + o(1))e^{-(1+2\gamma)(t\_{n+1} - t\_n - 2T)}.$$

Thus, we obtain that, in the case of negative coupling,

$$t\_{n+1} - t\_n = \left(\frac{1}{1+2\gamma} + o(1)\right) \ln \lambda \tag{34}$$

at *λ* → +∞. It follows from (12) and (34) that the mapping on *kn*, *xn*, and *in* has form

$$\begin{aligned} k\_{n+1} &= \begin{cases} \text{sign}(D(n)), & \text{sign}(\mathbb{C}(n)D(n)) = -1, \\ -\text{sign}(D(n)), & \text{sign}(\mathbb{C}(n)D(n)) = 1, \end{cases} \\ i\_{n+1} &= \begin{cases} i\_{\mathbb{H}\_{\prime}} & \text{sign}(\mathbb{C}(n)D(n)) = -1, \\ 3 - i\_{\mathbb{H}\_{\prime}} & \text{sign}(\mathbb{C}(n)D(n)) = 1, \end{cases} \\ x\_{\mathbb{H}+1} &= -k\_{\mathbb{H}+1} + O\left(\lambda^{\frac{2\gamma}{1+2\gamma}}\right), \end{aligned} \tag{35}$$

at *λ* → +∞.

Thus, under Assumptions 1, 2 and (33) on the *n*-th (where *n* ≥ 2) iteration of mapping, we have *kn* + *xn* = *o*(1) at *λ* → +∞. Thus, starting from the second iteration, Assumption 1 should be satisfied for *k* = *kn*, *x* = −*kn* + *o*(1), and *t*<sup>1</sup> = *tn*. Let's formulate this assumption for these values of parameters. Functions *A*(*kn*, −*kn* + *o*(1), *t*, *tn*) and *B*(*kn*, −*kn* + *o*(1), *t*, *tn*) have the form

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

Value *t* in Assumption 1 on the *n*-th iteration of steps described in Section 2 is in the segment [*tn* + *<sup>T</sup>*, *tn* + <sup>2</sup>*T*]; therefore, for each step value, ˜*<sup>t</sup>* = *<sup>t</sup>* − *tn* is in the segment [*T*, 2*T*]. Note that

$$\begin{split} &\int\_{T+t\_n}^{t} \left(\varepsilon^{s-t} + e^{(1+2\gamma)(s-t)}\right) F\left(k\_{\mathrm{n}} p \varepsilon^{(1+2\gamma)(t\_n+T-s)}\right) ds \\ &\qquad + \int\_{T+t\_n}^{t} \left(\varepsilon^{s-t} - e^{(1+2\gamma)(s-t)}\right) F\left(-k\_{\mathrm{n}} p e^{(1+2\gamma)(t\_n+T-s)}\right) ds \\ &= \int\_{T}^{t} \left(\varepsilon^{s-\tilde{t}} + e^{(1+2\gamma)(s-\tilde{t})}\right) F\left(k\_{\mathrm{n}} p e^{(1+2\gamma)(T-s)}\right) ds + \int\_{T}^{t} \left(\varepsilon^{s-\tilde{t}} - e^{(1+2\gamma)(s-\tilde{t})}\right) F\left(-k\_{\mathrm{n}} p e^{(1+2\gamma)(T-s)}\right) ds \end{split}$$

and

$$\begin{aligned} &\int\_{T+t\_n}^{t} \left(\varepsilon^{s-t} - \varepsilon^{(1+2\gamma)(s-t)}\right) F\left(k\_n p e^{(1+2\gamma)(t\_n+T-s)}\right) ds \\ &\qquad + \int\_{T+t\_n}^{t} \left(\varepsilon^{s-t} + \varepsilon^{(1+2\gamma)(s-t)}\right) F\left(-k\_n p e^{(1+2\gamma)(t\_n+T-s)}\right) ds \\ = &\int\_{T}^{t} \left(\varepsilon^{s-t} - \varepsilon^{(1+2\gamma)(s-t)}\right) F\left(k\_n p e^{(1+2\gamma)(T-s)}\right) ds + \int\_{T}^{t} \left(\varepsilon^{s-t} + \varepsilon^{(1+2\gamma)(s-t)}\right) F\left(-k\_n p e^{(1+2\gamma)(T-s)}\right) ds .\end{aligned}$$

Thus, for each *n* = 2, 3, ..., Assumption 1 is the same (only *kn* may change). Thus, if the following assumption holds, then Assumption 1 holds for all *n* = 2, 3, . . ..

**Assumption 4.** *Number of points t* <sup>∗</sup> ∈ [*T*, 2*T*] *such that g*1(*k*, *t* <sup>∗</sup>) = 0 *(g*2(*k*, *t* <sup>∗</sup>) = 0*) is finite. If g*1(*k*, *t* ∗) = 0 *(g*2(*k*, *t* <sup>∗</sup>) = <sup>0</sup>*), then there exists <sup>j</sup>* <sup>∈</sup> <sup>N</sup> *such that <sup>∂</sup><sup>j</sup> g*1(*k*, ˜*t*) *∂*˜*tj* ˜*t*=*t*<sup>∗</sup> *( ∂j g*2(*k*, ˜*t*) *∂*˜*tj* ˜*t*=*t*<sup>∗</sup> *, respectively) is non-zero. Here, k* = 1 *or k* = −1 *and*

$$\begin{split} \mathcal{g}\_{1}(k,\tilde{t}) &= \int\_{\tilde{T}}^{\tilde{t}} \left( e^{\varepsilon - \tilde{t}} + e^{(1+2\gamma)(s-\tilde{t})} \right) F \left( kpe^{(1+2\gamma)(T-s)} \right) ds \\ &+ \int\_{\tilde{T}}^{\tilde{t}} \left( e^{\varepsilon - \tilde{t}} - e^{(1+2\gamma)(s-\tilde{t})} \right) F \left( -kpe^{(1+2\gamma)(T-s)} \right) ds, \\ \mathcal{g}\_{2}(k,\tilde{t}) &= \int\_{\tilde{T}}^{\tilde{t}} \left( e^{\varepsilon - \tilde{t}} - e^{(1+2\gamma)(s-\tilde{t})} \right) F \left( kpe^{(1+2\gamma)(T-s)} \right) ds \\ &+ \int\_{\tilde{T}}^{\tilde{t}} \left( e^{\varepsilon - \tilde{t}} + e^{(1+2\gamma)(s-\tilde{t})} \right) F \left( -kpe^{(1+2\gamma)(T-s)} \right) ds. \end{split}$$

Thus, under Assumption 4, the asymptotics of the solution has form

$$\begin{array}{l} \mu\_{i\_{\rm li}}(t) = k\_{\rm II} p \varepsilon^{-(1+2\gamma)(t-t\_{\rm n})} + o(1), \\ \mu\_{3-i\_{\rm n}}(t) = -k\_{\rm n} p \varepsilon^{-(1+2\gamma)(t-t\_{\rm n})} + o(1) \end{array} \tag{36}$$

on the segments *t* ∈ [*tn*, *tn* + *T*] ((36) is Formula (9) with *i* = *in*, *k* = *kn*, *x* = *xn* = −*kn* + *o*(1), and *t*<sup>1</sup> = *tn*). On the segments, the *t* ∈ [*tn* + *T*, *tn* + 2*T*] solution satisfies equalities

$$\begin{aligned} u\_{i\_n}(t) &= \frac{\lambda}{2} \left( \mathcal{g}\_1(k\_{n\prime}t - t\_n) + o(1) \right), \\ u\_{3-i\_n}(t) &= \frac{\lambda}{2} \left( \mathcal{g}\_2(k\_{n\prime}t - t\_n) + o(1) \right) \end{aligned} \tag{37}$$

((37) is Formula (10) with *i* = *in*, *k* = *kn*, *x* = *xn* = −*kn* + *o*(1), and *t*<sup>1</sup> = *tn*, where functions *A* and *B* are rewritten in terms of functions *g*<sup>1</sup> and *g*2).

Suppose that the following non-degeneracy condition holds:

$$\begin{array}{c} g\_1(1,2T)g\_1(-1,2T)g\_2(1,2T)g\_2(-1,2T) \neq 0, \\ g\_1(1,2T) \neq g\_2(1,2T), \\ g\_1(-1,2T) \neq g\_2(-1,2T), \end{array} \tag{38}$$

(the fulfillment of these inequalities leads to fulfillment of Assumption 2 and inequality (33) for all *n* = 2, 3, ...). Thus, under condition (38) on the segments *t* ∈ [*tn* + 2*T*, *tn*+1], we have the following asymptotics of solution:

$$\begin{split} u\_{i\_{n}}(t) &= \frac{\lambda}{2} \begin{pmatrix} 2T \\ \int\_{T}^{2} \varepsilon^{\varepsilon} \left( F \left( k\_{n} p e^{(1+2\gamma)(T-s)} \right) + F \left( -k\_{n} p e^{(1+2\gamma)(T-s)} \right) \right) ds + o(1) \end{pmatrix} e^{t\_{n} - t} \\ &+ \frac{\lambda}{2} \begin{pmatrix} 2T \int\_{T}^{(1+2\gamma)s} \left( F \left( k\_{n} p e^{(1+2\gamma)(T-s)} \right) - F \left( -k\_{n} p e^{(1+2\gamma)(T-s)} \right) \right) ds + o(1) \end{pmatrix} e^{(1+2\gamma)(t\_{n} - t)}, \\ u\_{3-i\_{n}}(t) &= \frac{\lambda}{2} \begin{pmatrix} 2T \\ \int\_{T}^{2} \varepsilon^{\varepsilon} \left( F \left( k\_{n} p e^{(1+2\gamma)(T-s)} \right) + F \left( -k\_{n} p e^{(1+2\gamma)(T-s)} \right) \right) ds + o(1) \end{pmatrix} e^{t\_{n} - t} \\ &- \frac{\lambda}{2} \begin{pmatrix} 2T \int\_{T}^{(1+2\gamma)s} \left( F \left( k\_{n} p e^{(1+2\gamma)(T-s)} \right) - F \left( -k\_{n} p e^{(1+2\gamma)(T-s)} \right) \right) ds + o(1) \end{pmatrix} e^{(1+2\gamma)(t\_{n} - t)} \end{split} \tag{39}$$

((39) is Formula (12) with *i* = *in*, *k* = *kn*, *x* = *xn* = −*kn* + *o*(1), and *t*<sup>1</sup> = *tn*, where functions *A* and *B* are rewritten in terms of function *F*).

We obtain the following result on dynamics of system (3).

**Theorem 2.** *Suppose* <sup>−</sup><sup>1</sup> <sup>2</sup> < *γ* < 0 *and for values of k*<sup>1</sup> *and x*<sup>1</sup> *Assumptions 1, 2, and inequality* (20) *hold. Suppose Assumption 4 and inequalities* (38) *hold. Then, for any sufficiently large λ* > 0*, there exists t*<sup>2</sup> = *t*2(*k*1, *x*1) > 0 *such that for all t* > *t*<sup>2</sup> *solution of system* (3) *satisfies Formulas* (36)*,* (37)*, and* (39)*.*

In Figure 2, an example of the solution in the case of <sup>−</sup><sup>1</sup> <sup>2</sup> < *γ* < 0 is shown.

**Figure 2.** Example of solution. Values of parameters: *T* = 0.9, *γ* = −0.2, *p* = 1, *λ* = 10,000. Black line—*u*1(*t*), orange dashed line—*u*2(*t*).

#### **5. Example**

In this section, we show how method described in Sections 2–4 works in the case when function *f* satisfies conditions (4) and inequality

$$\inf(u) > 0 \text{ if } 0 < |u| < p \tag{40}$$

and initial conditions satisfy inequalities

$$\begin{aligned} \text{kx} > 0 &\text{ if } \gamma > 0, \\ \text{kx} < 0 &\text{ if } -\frac{1}{2} < \gamma < 0 \end{aligned} \tag{41}$$

(here *k* and *x* are defined as in Section 2).

As in Section 2, we construct asymptotics of all solutions of system (3) with initial conditions outside of the strip |*uj*| < *p* (*j* = 1, 2) and satisfying inequality (41). Let *t*<sup>1</sup> and *i* be defined as in Section 2. Then, the following lemmas hold.

**Lemma 3.** *If initial conditions fulfill* (41)*, then functions ui*(*t*) *and <sup>u</sup>*3−*i*(*t*) *do not change their signs on the segment t* ∈ [*t*1, *t*<sup>1</sup> + *T*] *and for all t* ∈ [*t*1, *t*<sup>1</sup> + *T*] *inequalities*

$$\begin{aligned} u\_i(t)u\_{3-i}(t) &> 0 \text{ if } \gamma > 0, \\ u\_i(t)u\_{3-i}(t) &< 0 \text{ if } -\frac{1}{2} < \gamma < 0 \end{aligned} \tag{42}$$

*hold.*

**Proof.** Consider the case *k* = 1. If *γ* > 0, then *x* ≥ 1 > 0. For these values of *k*, *x*, and *γ* system of inequalities,

$$\begin{cases} \left| k + \mathbf{x} \right| > \left| \mathbf{x} - k \right|, \\\ e^{-(t - t\_1)} \ge e^{-(1 + 2\gamma)(t - t\_1)} \end{cases} \tag{43}$$

holds. Since *ui*(*t*) and *<sup>u</sup>*3−*i*(*t*) have form (9), *<sup>k</sup>* + *<sup>x</sup>* > 0 and (43) holds, then we get that

$$
u\_i(t) > 0, \ u\_{3-i}(t) > 0\tag{44}$$

on the interval *<sup>t</sup>* <sup>∈</sup> [*t*1, *<sup>t</sup>*<sup>1</sup> <sup>+</sup> *<sup>T</sup>*]. If <sup>−</sup><sup>1</sup> <sup>2</sup> < *γ* < 0, then *x* ≤ −1 < 0. This is why we obtain that

$$\begin{cases} \left| k + \mathbf{x} \right| < \left| \mathbf{x} - k \right|\_{\prime} \\ e^{-(t - t\_1)} \le e^{-(1 + 2\gamma)(t - t\_1)} \end{cases} \tag{45}$$

It follows from (9), *k* − *x* > 0, and (45) that

$$
u\_i(t) > 0, \ u\_{3-i}(t) < 0\tag{46}$$

on the interval *t* ∈ [*t*1, *t*<sup>1</sup> + *T*].

Consider the case *k* = −1. If *γ* > 0, then *x* ≤ −1 < 0. Then, from (9), *k* + *x* < 0, and (43), we obtain that

$$
\mu\_i(t) < 0, \ u\_{3-i}(t) < 0 \tag{47}
$$

on the interval *<sup>t</sup>* <sup>∈</sup> [*t*1, *<sup>t</sup>*<sup>1</sup> <sup>+</sup> *<sup>T</sup>*]. In addition, in the case <sup>−</sup><sup>1</sup> <sup>2</sup> < *γ* < 0, we get that *x* ≥ 1 > 0 and from (9), *k* − *x* > 0, and (45), we get

$$
u\_i(t) < 0, \ u\_{3-i}(t) > 0 \tag{48}$$

on the interval *t* ∈ [*t*1, *t*<sup>1</sup> + *T*].

It follows from (44), (46)–(48) that inequalities (42) hold.

**Lemma 4.** *If function ui*(*t*) *comes into the strip* |*ui*(*t*)| < *p at the point t* = *t*1*, then (1) x satisfies inequality*

$$|\mathbf{x}| \le |1 + 1/\gamma|;\tag{49}$$

*(2) function ui is in the strip* |*ui*(*t*)| < *p for all t* ∈ (*t*1, *t*<sup>1</sup> + *T*]*.*

**Proof.** It follows from (9) that

$$u\_i'(t) = -\frac{(k+x)p}{2}e^{-(t-t\_1)} - (1+2\gamma)\frac{(k-x)p}{2}e^{-(1+2\gamma)(t-t\_1)}\,\tag{50}$$

therefore

$$
\mu\_i'(t\_1) = -\left(\frac{k+x}{2}p + (1+2\gamma)\frac{k-x}{2}p\right).
$$

Consider the case *k* = 1. For *k* = 1 value, *ui*(*t*1) is equal to *p*. If this function comes into the strip |*ui*(*t*)| < *p* at the point *t* = *t*1, then derivative *u i* (*t*1) is non-positive. For *k* = 1 inequality, *u i* (*t*1) ≤ 0 is equivalent to 1 + *γ* ≥ *γx*. It follows from condition (41) that *γx* > 0 in the case *k* = 1. Thus, in the case *k* = 1, inequality (49) holds.

Consider the case *k* = −1. For *k* = −1 value *ui*(*t*1) = −*p* and if this function comes into the strip |*ui*(*t*)| < *p* at the point *t* = *t*1, then derivative *u i* (*t*1) is non-negative. For *k* = −1 condition, *u i* (*t*1) ≥ 0 is equivalent to inequality −1 − *γ* ≤ *γx*. From (41), we get that *γx* < 0, so inequality (49) is true in this case, too.

It follows from (41) and (49) that in the case *γ* > 0 system of inequalities

$$\begin{cases} |k+x| \ge |(1+2\gamma)(x-k)|, \\ e^{-(t-t\_1)} > e^{-(1+2\gamma)(t-t\_1)} \end{cases} \tag{51}$$

holds and in the case <sup>−</sup><sup>1</sup> <sup>2</sup> < *γ* < 0 system of inequalities

$$\begin{cases} |k+x| \le |(1+2\gamma)(x-k)|, \\ e^{-(t-t\_1)} < e^{-(1+2\gamma)(t-t\_1)} \end{cases} \tag{52}$$

is true on the interval *t* ∈ (*t*1, *t*<sup>1</sup> + *T*]

Using (50)–(52), and (41), we obtain that

$$\begin{cases} u\_i'(t) < 0 \text{ if } k = 1, \\ u\_i'(t) > 0 \text{ if } k = -1 \end{cases} \tag{53}$$

on the interval *t* ∈ (*t*1, *t*<sup>1</sup> + *T*]. Combining (44), (46)–(48) with (53), we get that function *ui*(*t*) is in the strip |*ui*(*t*)| < *p* for all *t* ∈ (*t*1, *t*<sup>1</sup> + *T*].

**Lemma 5.** *If function f satisfies* (4) *and* (40)*, initial conditions satisfy* (41) *and*

$$|\mathbf{x}| < |1 + 1/\gamma|,\tag{54}$$

*then Assumptions 1–4 hold.*

**Proof.** Consider some function *f*(*u*), satisfying conditions (4) and (40).

Let us prove that for this function Assumption 1 holds. From Lemmas 3 and 4, we obtain that *ui*(*t*) is in the strip |*ui*(*t*)| < *p* and it does not change sign on the interval *t* ∈ (*t*1, *t*<sup>1</sup> + *T*]. This is why from condition (40) we get that the first summands in *A*(*k*, *x*, *t*, *t*1) and *B*(*k*, *x*, *t*, *t*1) are non-zero. Thus, from formulas (44), (46)–(48), and assumption (40), we obtain that the following inequalities hold

$$\begin{array}{llll} A(k, \mathbf{x}, t, t\_1) > 0, & B(k, \mathbf{x}, t, t\_1) > 0 & \text{if} & k = 1, \ \gamma > 0\\ A(k, \mathbf{x}, t, t\_1) > 0, & B(k, \mathbf{x}, t, t\_1) < 0 & \text{if} & k = 1, \ -\frac{1}{2} < \gamma < 0\\ A(k, \mathbf{x}, t, t\_1) < 0, & B(k, \mathbf{x}, t, t\_1) < 0 & \text{if} & k = -1, \ \gamma > 0\\ A(k, \mathbf{x}, t, t\_1) < 0, & B(k, \mathbf{x}, t, t\_1) > 0 & \text{if} & k = -1, -\frac{1}{2} < \gamma < 0 \end{array} \tag{55}$$

on the interval *t* ∈ (*t*<sup>1</sup> + *T*, *t*<sup>1</sup> + 2*T*]. Thus, we have proved that under condition (40) functions *A*(*k*, *x*, *t*, *t*1) and *B*(*k*, *x*, *t*, *t*1) are non-zero on the interval *t* ∈ (*t*<sup>1</sup> + *T*, *t*<sup>1</sup> + 2*T*]. If *t* <sup>∗</sup> = *t*<sup>1</sup> + *T*, then *A*(*k*, *x*, *t* <sup>∗</sup>, *t*1) = *B*(*k*, *x*, *t* <sup>∗</sup>, *<sup>t</sup>*1) = 0. Derivatives *<sup>∂</sup><sup>j</sup> A*(*k*, *x*, *t*, *t*1) *∂tj <sup>t</sup>*=*t*1+*<sup>T</sup>* <sup>=</sup> 0 for *<sup>j</sup>* <sup>=</sup> 1, 2 and derivatives *<sup>∂</sup><sup>j</sup> B*(*k*, *x*, *t*, *t*1) *∂tj <sup>t</sup>*=*t*1+*<sup>T</sup>* <sup>=</sup> 0 for *<sup>j</sup>* <sup>=</sup> 1, 2, 3. Expressions

$$\frac{\partial^3 A(k, \mathbf{x}, t, t\_1)}{\partial t^3} \Big|\_{t = t\_1 + T} = 2f'''(kp) \left( \frac{k + \mathbf{x}}{2} p + (1 + 2\gamma) \frac{k - \mathbf{x}}{2} p \right)^2$$

and

$$\frac{\partial^4 B(k, \mathbf{x}, t, t\_1)}{\partial t^4} \Big|\_{t = t\_1 + T} = 2\gamma f''(kp) \left( \frac{k + \mathbf{x}}{2} p + (1 + 2\gamma) \frac{k - \mathbf{x}}{2} p \right)^2,$$

are non-zero: under condition (54) last factor in these derivatives is non-zero and *f* (*kp*) = 0 because of (4) (if *<sup>x</sup>* <sup>=</sup> <sup>±</sup>(<sup>1</sup> <sup>+</sup> 1/*γ*), then for all *<sup>j</sup>* <sup>∈</sup> <sup>N</sup> expressions *<sup>∂</sup><sup>j</sup> A*(*k*, *x*, *t*, *t*1) *∂tj <sup>t</sup>*=*t*1+*<sup>T</sup>* and *<sup>∂</sup><sup>j</sup> B*(*k*, *x*, *t*, *t*1) *∂tj t*=*t*1+*T* equal zero). Consequently, Assumption 1 holds under condition (54). This assumption holds for *x* = ±*k* + *o*(1) at *λ* → +∞, so Assumptions 3 and 4 hold.

Since the system of inequalities (55) is true for *t* = *t*<sup>1</sup> + 2*T*, then Assumption 2 holds.

Note that if function *ui*(*t*) comes to the strip |*ui*(*t*)| < *p*, then *x* satisfies inequality (49), and for all *x* such that (54) hold, Assumption 1 is true. Thus, only for two values of parameter *x* : *x*1,2 = ±(1 + 1/*γ*) is Assumption 1 false.

**Lemma 6.** *If function f satisfies* (4) *and* (40)*, then inequalities* (26) *and* (31) *are true in the case γ* > 0 *and inequalities* (33) *and* (38) *hold in the case* <sup>−</sup><sup>1</sup> <sup>2</sup> < *γ* < 0*.*

**Proof.** It follows from Lemma 5 that *A*(*k*, *x*, *t*<sup>1</sup> + 2*T*, *t*1) and *B*(*k*, *x*, *t*<sup>1</sup> + 2*T*, *t*1) have the same sign in the case *<sup>γ</sup>* <sup>&</sup>gt; 0 and the opposite signs in the case <sup>−</sup><sup>1</sup> <sup>2</sup> < *γ* < 0. Therefore, in the case *γ* > 0 (−<sup>1</sup> <sup>2</sup> < *γ* < 0) inequality (26) (inequality (33)respectively) holds for all *n* = 1, 2, 3, .... Thus, inequalities (31) and (38) are fulfilled because they are equivalent to Assumption 2 and conditions (26) and (33) for *n* = 2, 3, . . ..

Thus, we have proved that all assumptions in Theorems 1 and 2 are true if function *f* satisfies (4) and (40) and for *x*<sup>1</sup> conditions (41) and (54) hold. Therefore, for class of functions *f* considered in this section, the following theorems are true.

**Theorem 3.** *Suppose γ* > 0 *and inequalities* (41) *and x*<sup>1</sup> = ±(1 + 1/*γ*) *hold. Then, for any sufficiently large λ* > 0 *there exists t*<sup>2</sup> = *t*2(*k*1, *x*1) > 0 *such that for all t* > *t*<sup>2</sup> *solution of system* (3) *satisfies Formulas* (29)*,* (30) *and* (32)*.*

**Theorem 4.** *Suppose* <sup>−</sup><sup>1</sup> <sup>2</sup> < *γ* < 0 *and inequalities* (41) *and x*<sup>1</sup> = ±(1 + 1/*γ*) *hold. Then, for any sufficiently large λ* > 0 *there exists t*<sup>2</sup> = *t*2(*k*1, *x*1) > 0 *such that for all t* > *t*<sup>2</sup> *solution of system* (3) *satisfies Formulas* (36)*,* (37) *and* (39)*.*

**Remark 1.** *If x*<sup>1</sup> = ±(1 + 1/*γ*)*, then Assumption 1 is not true, so Theorems 3 and 4 are not proven. However, probably, they are true because for all initial conditions in the neighborhood of these values they are true.*

Consider the map (28). If we take set {1} × [1, 1 + 1/*γ* − *δ*] (where *δ* is a small positive constant (0 < *δ* < 1/*γ*)) of pairs (*k*, *x*), then it follows from Lemmas 3–6 that the image of this set under the map (28) is set {1} × [1, 1 + *a*], where *a* = *o*(1) at *λ* → +∞. Therefore, there exists at least one fixed point of the operator of translation along the trajectories and positive relaxation cycle of system (3) corresponds to this fixed point (if *k*<sup>1</sup> and *x*<sup>1</sup> fulfill (41) and function *f* satisfies (40), then in the case of positive coupling solution of system (3) does not change its sign). Similarly, there exists at least one negative relaxation cycle of system (3) in the case of positive coupling.

In Figure 3, there are examples of two coexisting relaxation cycles of system (3).

**Figure 3.** Two coexisting relaxation cycles of the system (3). Values of parameters: *T* = 1, *γ* = 0.4, *p* = 1, *λ* = 10,000. Black line—*u*1(*t*), orange dashed line—*u*2(*t*).

If <sup>−</sup><sup>1</sup> <sup>2</sup> < *γ* < 0, then it follows from (35) that *xn*+<sup>1</sup> = −*kn*+<sup>1</sup> + *o*(1) at *λ* → +∞. It follows from Lemmas 3–6 that for all (*kn*, *xn*) ∈ {−1} × [1, 1 + 1/*γ* − *δ*] and (*kn*, *xn*) ∈ {1} × [−1 − 1/*γ* + *δ*, −1] Theorem <sup>4</sup> is true. Therefore, there exists at least one *<sup>q</sup>* <sup>∈</sup> <sup>N</sup>, such that image of the set {−1} × [1, 1 + 1/*γ* − *δ*] (or {1} × [−1 − 1/*γ* + *δ*, −1]) under the *q*-th iteration of map (35) belongs to the set {−1} × [1, 1 <sup>+</sup> 1/*<sup>γ</sup>* <sup>−</sup> *<sup>δ</sup>*] (or {1} × [−<sup>1</sup> <sup>−</sup> 1/*<sup>γ</sup>* <sup>+</sup> *<sup>δ</sup>*, <sup>−</sup>1] respectively). Thus, in the case of <sup>−</sup><sup>1</sup> <sup>2</sup> < *γ* < 0, there exists at least one relaxation cycle.

Thus, the following statement holds.

**Corollary 5.** *Suppose conditions* (4) *and* (40) *are true. Then, in the case γ* > 0*, there exists at least two relaxation cycles of system* (3) *and in the case of* <sup>−</sup><sup>1</sup> <sup>2</sup> < *γ* < 0 *there exists at least one relaxation cycle of system* (3)*.*

#### **6. Dependence of Dynamics of System** (3) **on the Sign of Coupling**

In this section, we show how asymptotics and difference *tn*+<sup>1</sup> − *tn* (analog of period) of solutions of system (3) depends on the value *<sup>γ</sup>* in the case *<sup>γ</sup>* <sup>&</sup>gt; 0 and in the case <sup>−</sup><sup>1</sup> <sup>2</sup> < *γ* < 0 (in this section below, we discuss only such solutions of system (3) for those assumptions of Theorem 1 or 2 fulfill).

First, consider the case *γ* > 0. From Formulas (29), (30), and (32), we obtain that components *u*1(*t*) and *u*2(*t*) have the same leading terms of asymptotics on the interval *t* ∈ [*t*2, +∞) and that these leading terms of asymptotics do not depend on *γ*. Thus, from Formulas (9), (10), (12), (29), (30) and (32), we obtain that the leading term of asymptotics of solution of system (3) depends on *γ* only for *<sup>t</sup>* <sup>∈</sup> [0, *<sup>t</sup>*2] (see Figure 4). From Corollary 4, we get that in the case *<sup>γ</sup>* <sup>&</sup>gt; <sup>1</sup> <sup>2</sup> difference *u*1(*t*) − *u*2(*t*) has order *<sup>o</sup>*(1) at *<sup>λ</sup>* <sup>→</sup> <sup>+</sup><sup>∞</sup> for all *<sup>t</sup>* <sup>≥</sup> *<sup>t</sup>*2, so we may say that in the case *<sup>γ</sup>* <sup>&</sup>gt; <sup>1</sup> <sup>2</sup> oscillators *u*1(*t*) and *u*2(*t*) "synchronize" (for smaller values of *γ* oscillators *u*1(*t*) and *u*2(*t*) may "synchronize", too, but in the case *γ* > <sup>1</sup> <sup>2</sup> they must "synchronize").

The leading term of asymptotics of the difference *tn*+<sup>1</sup> − *tn* does not depend on *γ*, too.

Figure 4 illustrates dependence of solutions of system (3) on *γ* in the case *γ* > 0. There are solutions of system (3) with identical function *F*, parameters *λ* and *T*, and initial conditions for different parameters *γ* in Figure 4.

**Figure 4.** Solutions of system (3) for different values of parameter *γ*. Values of parameters: *T* = 2, *p* = 1.5, *λ* = 1000, *k* = 1, *x* = 3, (**a**) *γ* = 0.2; (**b**) *γ* = 0.6; (**c**) *γ* = 1; (**d**) *γ* = 1.5. Black line—*u*1(*t*), orange dashed line—*u*2(*t*).

Now, consider the case <sup>−</sup><sup>1</sup> <sup>2</sup> < *γ* < 0.

From (9), (10), (12), (36), (37), and (39), we get that asymptotics of solutions of system (3) depends crucially on the value of parameter *<sup>γ</sup>* for all *<sup>t</sup>* <sup>≥</sup> 0 in the case <sup>−</sup><sup>1</sup> <sup>2</sup> < *γ* < 0 and that oscillators *u*1(*t*) and *u*2(*t*) are not close to each other (the leading terms of their asymptotics are different for all *t* ≥ *t*2).

It follows from (34) that difference *tn*+<sup>1</sup> − *tn* increases with the decreasing of parameter *γ* (see Figure 5).

Thus, asymptotics and shape of solution and difference *tn*+<sup>1</sup> − *tn* depend crucially on the value of *<sup>γ</sup>* in the case <sup>−</sup><sup>1</sup> <sup>2</sup> < *γ* < 0 (see Figure 5).

Figure <sup>5</sup> illustrates the dependence of solutions of system (3) on *<sup>γ</sup>* in the case <sup>−</sup><sup>1</sup> <sup>2</sup> < *γ* < 0. Solutions of system (3) with identical function *F*, parameters *λ* and *T*, and initial conditions for different parameters *γ* are presented in Figure 5.

**Figure 5.** Solutions of system (3) for different values of parameter *γ*. Values of parameters: *T* = 2, *p* = 1.5, *λ* = 1000, *k* = 1, *x* = −4, (**a**) *γ* = −0.1; (**b**) *γ* = −0.25; (**c**) *γ* = −0.4; (**d**) *γ* = −0.45. Black line—*u*1(*t*), orange dashed line—*u*2(*t*).

#### **7. Conclusions**

In this paper, we have studied the nonlocal dynamics of a system of two coupled generators with delayed feedback and dependence of solutions on the value of coupling.

For a wide set of initial conditions from the phase space of system (3) using method of steps and special constructed finite dimensional map, we get asymptotics of relaxation solutions. We obtain relaxation cycles of system (3).

We prove that the dynamics of system (3) are qualitatively different in case *γ* > 0 and case −1 <sup>2</sup> < *γ* < 0: in the case *γ* > 0, there exists a moment of time *t*<sup>2</sup> after that both components of solution have the same leading term of asymptotics and this leading term does not depend on *γ* if *t* > *t*2, generators *u*1(*t*) and *u*2(*t*) "synchronize" if *γ* > <sup>1</sup> <sup>2</sup> ; in the case of <sup>−</sup><sup>1</sup> <sup>2</sup> < *γ* < 0, the leading term of asymptotics and shape of solution depend on *γ*, oscillators *u*1(*t*) and *u*2(*t*) are not close to each other; the leading term of asymptotics of the value *tn*+<sup>1</sup> − *tn* (this value serves us an analog of period) increase with decreasing of the value *<sup>γ</sup>* in the case <sup>−</sup><sup>1</sup> <sup>2</sup> < *γ* < 0 and remains unchanged with changing *γ* in the case *γ* > 0.

The method of research used in this paper is applicable for systems of higher dimensions (case of *n* identically diffusion coupled oscillators, where *n* > 2) and for systems of *n* (*n* ≥ 2) coupled oscillators with other types of coupling.

**Funding:** Research funded by the Council on grants of the President of the Russian Federation (MK-1028.2020.1).

**Conflicts of Interest:** The author declares no conflict of interest. The funders had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript, or in the decision to publish the results.
