*3.1. Asymptotic Behavior of Slowly Oscillating Solutions*

We shall consider the asymptotic behavior of a slowly oscillating relaxation periodic solution of Equation (4).

In (4) it is convenient to replace the time *t* → *tT* and to again denote the product *λT* by *λ*. Then Equation (4) takes the form

$$
\dot{u} = \lambda [1 - u(t - 1)] u(A - u). \tag{24}
$$

Let us introduce some notation.

By *<sup>S</sup>* we denote the set of all functions *<sup>ϕ</sup>*(*s*) ∈ *<sup>C</sup>*[−1,0] that satisfy the conditions (7).

Let *u*(*t*, *ϕ*) be the solution of (24) with an initial condition *ϕ*(*s*) specified at *t* = 0, i.e., *u*(*s*, *ϕ*) = *ϕ*(*s*) for *s* ∈ [−1, 0]. We shall construct the asymptotics for *λ* → ∞ of all solutions *u*(*t*, *ϕ*) with *ϕ*(*s*) ∈ *S*.

**Theorem 2.** *For all sufficiently large λ Equation* (24) *has an asymptotically orbitally stable periodic (with period T*(*λ*)*) solution u*0(*t*, *λ*)*, for which u*0(0, *λ*) = *u*0(*t*1(*λ*), *λ*) = *u*0(*t*2(*λ*), *λ*) = 1*, where tj*(*λ*) *are successive positive roots of the equation u*0(*t*, *λ*) = 1*, and*

$$\begin{aligned} t\_1(\lambda) &= 1 + (A - 1)^{-1} + o(1), \\ T(\lambda) &= t\_2(\lambda) = A + 1 + (A - 1)^{-1} + o(1). \end{aligned}$$

*For every t from the intervals* (0, *t*1(*λ*)) *and* (*t*1(*λ*), *t*2(*λ*))*, respectively, the following equalities are fulfilled:*

$$u\_0(t, \lambda) = \begin{cases} A + o(1), & t \in (0, t\_1(\lambda)), \\ o(1), & t \in (t\_1(\lambda), t\_2(\lambda)). \end{cases}$$

**Proof.** We use the following formula for the solutions *u*(*t*, *ϕ*) of Equation (24):

$$u(t,\boldsymbol{\varrho}) = Au(\boldsymbol{\tau},\boldsymbol{\varrho})\exp\left(\lambda A \int\_{\boldsymbol{\tau}}^{t} \left(1 - u(\boldsymbol{s} - 1, \boldsymbol{\varrho})\right) d\boldsymbol{s}\right).$$

$$\left[A - u(\boldsymbol{\tau}, \boldsymbol{\varrho}) + u(\boldsymbol{\tau}, \boldsymbol{\varrho})\exp\left(\lambda A \int\_{\boldsymbol{\tau}}^{t} \left(1 - u(\boldsymbol{s} - 1, \boldsymbol{\varrho})\right) d\boldsymbol{s}\right)\right]^{-1},\tag{25}$$

where *t* and *τ* are arbitrary values such that 0 < *τ* < *t*.

We shall formulate some simple statements:

**Lemma 2.** *Let t* ∈ (0, 1]*. Then*

$$u(t,\varphi) = A\left[1 + (A - 1)\exp\left(-\lambda A \left(1 + o(1)\right)t\right)\right]^{-1}.\tag{26}$$

The proof of (26) follows from (25) with *τ* = 0.

Setting *τ* = 1 in (25), we immediately arrive at the following statement:

**Lemma 3.** *Let t* ∈ (1, 1 + (*<sup>A</sup>* − <sup>1</sup>)−<sup>1</sup> + *<sup>δ</sup>*]*, where <sup>δ</sup>* > <sup>0</sup> *is some small but fixed value. Then*

$$u(t,\boldsymbol{\varphi}) = A \left[ 1 + (A - 1) \exp\left( \lambda A \left[ (A - 1) \left( 1 + o(1) \right) (t - 1) - 1 \right] \right) \right]^{-1}.\tag{27}$$

Below, we denote by *t*1(*ϕ*), *t*2(*ϕ*), . . . the first, second, . . . positive roots of the equation *u*(*t*, *ϕ*) = 1.

From (27) we arrive at the conclusion that the value *t*1(*ϕ*) exists and

$$\begin{aligned} t\_1(\boldsymbol{\varrho}) &= 1 + (A - 1)^{-1} + o(1), \\ u(t, \boldsymbol{\varrho}) &= A + o(1) \text{ for every } t \in (0, t\_1(\boldsymbol{\varrho})), \\ u(t, \boldsymbol{\varrho}) &= o(1) \text{ for every } t \in (t\_1(\boldsymbol{\varrho}), t\_1(\boldsymbol{\varrho}) + 1). \end{aligned}$$

The last equality is true, because formula (27) holds on the interval (*t*1(*ϕ*), *t*1(*ϕ*) + 1).

Then we construct asymptotics of solution on the interval *t* > *t*1(*ϕ*) + 1. We set *τ* = *t*1(*ϕ*) and take into account that *u*(*s* − 1) = *A* + *o*(1) on the interval *s* ∈ (*t*1(*ϕ*), *t*1(*ϕ*) + 1). We use equality *u*(*s* − 1) = *o*(1) for *t*1(*ϕ*) + 1 < *s* < *t*. It follows from formula (25) that this equality holds for all *t* > *t*1(*ϕ*) + 1 while

$$\int\_{t\_1(\varphi)}^t \left(1 - \mu(s - 1, \varphi)\right) ds < 0.$$

We obtain the following result:

**Lemma 4.** *For <sup>t</sup>* ∈ [<sup>2</sup> + (*<sup>A</sup>* − <sup>1</sup>)−1, *<sup>A</sup>* + <sup>1</sup> + (*<sup>A</sup>* − <sup>1</sup>)−<sup>1</sup> + *<sup>δ</sup>*]*, where <sup>δ</sup>* > <sup>0</sup> *is some small but fixed value*

$$u(t,\varphi) = A \exp\left(\lambda A \left(1 - A + t - \left(2 + (A - 1)^{-1}\right) + o(1)\right)\right).$$

$$\left[A - 1 + \exp\left(\lambda A \left(1 - A + t - \left(2 + (A - 1)^{-1} + o(1)\right)\right)\right)\right]^{-1}.\tag{28}$$

From (28) the existence of *t*2(*ϕ*) and the asymptotic equalities

$$t\_2(\varphi) = A + 1 + (A - 1)^{-1} + o(1),$$

and

$$u(t, \boldsymbol{\varrho}) = o(1) \text{ for } t \in (t\_1(\boldsymbol{\varrho}), t\_2(\boldsymbol{\varrho})) $$

follow.

We introduce the translation along the trajectories operator Π, which links the function *u*(*s* + *t*2(*ϕ*), *ϕ*), (*s* ∈ [−1, 0]) to a function *ϕ*(*s*). From the formulas given above we arrive at the conclusion that

$$\Pi(\boldsymbol{\varrho}(\boldsymbol{s})) = \boldsymbol{\mathfrak{u}}\left(\boldsymbol{s} + t\_{\mathcal{D}}(\boldsymbol{\varrho}), \boldsymbol{\varrho}\right) \in \mathcal{S}\_{\prime}$$

meaning that Π*S* ⊂ *S*.

From this and from Ref. [23] it follows that this operator has in *S* a fixed point *ϕ*0(*s*), i.e.,

$$
\Pi(\varrho\_0(s)) = \varrho\_0(s).
$$

This means that the solution *u*0(*t*, *λ*) = *u*(*t*, *ϕ*0(*s*)) is periodic, with period *T*(*λ*) = *t*2(*ϕ*0(*s*)). The proof of the stability of *u*0(*t*, *λ*) is rather cumbersome. In the more complicated situation discussed in Refs. [7,29] a detailed proof was given of the stability of the solution constructed there, and so we shall not give the proof here.

It is possible to obtain the asymptotic expansion of *u*0(*t*, *λ*) to any degree of precision. Here we note only that

$$\begin{aligned} \max\_{t} \boldsymbol{u}\_{0}(t, \lambda) &= \boldsymbol{u}\_{0}(1, \lambda) = A[1 + \left(A - 1 + o(1)\right) \exp(-\lambda A))]^{-1}, \\ \min\_{t} \boldsymbol{u}\_{0}(t, \lambda) &= \boldsymbol{u}\_{0}(t\_{1}(\varphi\_{0}) + 1, \lambda) = A(A - 1)^{-1} \exp\left(\lambda A(1 - A)\right) \left(1 + o(1)\right). \end{aligned}$$

It is interesting to compare the principal characteristics of *u*0(*t*, *λ*) with the corresponding characteristics of the stable periodic solution *U*0(*t*, *λ*) of Equation (1) for *λ* → ∞. To obtain formulas for max*<sup>t</sup> U*0(*t*, *λ*) and min*<sup>t</sup> U*0(*t*, *λ*) one should replace *T* by 1 in formulas (2) and (3) (see introduction of this paper and Ref. [7]). In Figure 2 we give graphs of periodic functions *u*0(*t*, *λ*) and *U*0(*t*, *λ*).

**Figure 2.** Form of the functions *u*0(*t*, *λ*), *U*0(*t*, *λ*).
