*4.1. Construction of Slowly Oscillating Solutions*

We shall consider the construction of slowly oscillating periodic solutions. We introduce into the analysis a set of initial conditions consisting of all those functions *<sup>ϕ</sup>*(*s*) ∈ *<sup>C</sup>*[−1,0] for which

$$\begin{aligned} \varphi(0) &= 1, \quad 0 \le \varphi(s) \le 1; \\\varphi(s) &\le \exp(\lambda \delta s), \end{aligned}$$

where *δ* > 0 is some sufficiently small but fixed value. We investigate the asymptotic form of all solutions *u*(*t*, *ϕ*) with initial conditions from *S*. We denote by *t*1(*ϕ*) and *t*2(*ϕ*) the first and second positive roots of equation *u*(*t*, *ϕ*) = 1. In the case when

$$t\_2(\varrho) - 1 > t\_1(\varrho),\tag{38}$$

we can determine the translation along the trajectories operator Π: Π(*ϕ*(*s*)) = *u*(*s* + *t*2(*ϕ*), *ϕ*). From the inequality (38) we find that Π(*ϕ*(*s*)) ∈ *S*. If for all *ϕ*(*s*) ∈ *S* we have Π(*ϕ*(*s*)) ∈ *S*, i.e., Π*S* ⊂ *S*, we can conclude that there exists an attractor with initial conditions from *S* and that there exists a fixed point *ϕ*0(*s*) ∈ *S*. The solution *u*0(*t*, *λ*) = *u*(*t*, *ϕ*0(*s*)) is thereby periodic, with period *T*(*λ*) = *t*2(*ϕ*0). For the expressions *t*1,2(*ϕ*) it is possible to obtain

asymptotic formulas, and this means that we can formulate conditions on the parameters of the problem that ensure fulfilment of the inequality (38). We confine ourselves here to giving only sufficient conditions for fulfilment of this inequality.

**Lemma 6.** *Let*

$$1 - (1 - \mathfrak{a})A > 0\tag{39}$$

*and*

$$1 - \varkappa A < 0.\tag{40}$$

*Then for all sufficiently large λ for Equation* (37) *we have the following inclusion:*

Π*S* ⊂ *S*.

**Proof.** For each *t* ∈ (0, *h*] the asymptotic equality

$$\mu(t,\boldsymbol{\varrho}) = A \left[ 1 + (A - 1) \exp \left( -\lambda A \left( t + o(1) \right) \right) \right]^{-1}$$

holds. But if *t* ∈ [*h*, 1], then

$$u(t,q) = A\left[1 + (A-1)\exp\left(-\lambda A\left((1-(1-a)A)t + h(1-a)A + o(1)\right)\right)\right]^{-1}.$$

In particular,

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

It follows from inequality (39) that *u*(*t*, *ϕ*) = *A* + *o*(1) on the segment *t* ∈ [*h*, 1 + *δ*1], where *δ*<sup>1</sup> is some positive and small (but independent on *λ*) constant.

Therefore, for *t* > 1 while *u*(*t*, *ϕ*) = *A* + *o*(1) formula

$$u(t,\boldsymbol{\varrho}) = A \left[ 1 + (A - 1) \exp \left( -\lambda A \left( 1 - (1 - a)A(1 - h) + (1 - A)(t - 1) + o(1) \right) \right) \right]^{-1}$$

holds. From this formula we find that *t*1(*ϕ*) exists and that

$$t\_1(\varphi) = 1 + \left(1 - (1 - a)A(1 - h)\right)(A - 1)^{-1} + o(1), \quad (t\_1(\varphi) > 1).$$

Starting from *τ* = *t*1(*ϕ*) we obtain that on the interval *t* ∈ (*t*1(*ϕ*), *t*1(*ϕ*) + *h*] equality *u*(*t*, *ϕ*) = *o*(1) holds and that on the segment *t* ∈ [*t*1(*ϕ*) + *h*, *t*1(*ϕ*) + 1] formula

$$u(t,\boldsymbol{\varrho}) = A \left[ 1 + (A - 1) \exp \left( -\lambda A \left( (1 - A)h + (1 - \alpha A)(t - h - t\_1(\boldsymbol{\varrho})) + o(1) \right) \right) \right]^{-1}$$

is true. It follows from (40) that *u*(*t*, *ϕ*) = *o*(1) on the segment *t* ∈ [*t*1(*ϕ*) + *h*, *t*1(*ϕ*) + 1]. Then for *t* > *t*1(*ϕ*) + 1 while *u*(*t*, *ϕ*) = *o*(1) we have the equality

$$u(t,\boldsymbol{\varrho}) = A \left[ 1 + (A - 1) \exp \left( -\lambda A \left( (1 - A)h + (1 - aA)(1 - h) + (t - 1 - t\_1(\boldsymbol{\varrho})) + o(1) \right) \right) \right]^{-1}.\tag{41}$$

From Formula (41) we obtain both the existence of *t*2(*ϕ*) and inequality *t*2(*ϕ*) − *t*1(*ϕ*) > 1.

It follows from this that Π(*ϕ*(*s*)) ∈ *S* and Π*S* ⊂ *S*. The Lemma is proven.

**Theorem 4.** *Let inequalities* (39) *and* (40) *hold. Then for all sufficiently large λ Equation* (37) *has an asymptotically orbitally stable slowly oscillating 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*

$$\begin{aligned} t\_1(\lambda) &= 1 + \left(1 - (1 - a)A(1 - h)\right)(A - 1)^{-1} + o(1), \\ T(\lambda) &= t\_2(\lambda) = 2 + \left(1 - (1 - a)A(1 - h)\right)(A - 1)^{-1} + (A - 1)h - (1 - aA)(1 - h) + o(1). \end{aligned}$$

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

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

The proof of existence of the periodic solution in Theorem 4 follows from the proof of the Lemma 6, and proof of stability of this solution 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.
