*3.2. Rapidly Oscillating Solutions of Equation* (24)

We shall consider rapidly oscillating solutions of Equation (24).

These are the set of those solutions *u*(*t*, *ϕ*) that become equal to 1 at *t* = 0, and have one "step" on the segment [−1, 0], as shown in Figure 3.

**Figure 3.** Graph of *ϕ*(*s*) ∈ *S*(*τ*1, *τ*2) for *s* ∈ [−1, 0] and *u*(*t*, *ϕ*) for *t* ≥ 0.

We shall describe the corresponding set of initial functions *ϕ*(*s*). We arbitrarily fix values *τ*<sup>1</sup> and *τ*<sup>2</sup> for which

$$-1 < v\_1 < v\_2 < 0.$$

Below, we shall denote by *δ* > 0 an arbitrary sufficiently small quantity that does not depend on *λ* and whose precise value is unimportant. We introduce into the analysis three functions *κ*1,2,3(*s*, *λ*):

$$\begin{aligned} \kappa\_1(s,\lambda) &= \exp\left(\lambda\delta(s-\tau\_1)\right), \quad s \in [-1,\tau\_1];\\ \kappa\_2(s,\lambda) &= \min\left(A\left[1+(A-1)\exp\left(-\lambda\delta(s-\tau\_1)\right)\right]^{-1}, A\left[1+(A-1)\exp(\lambda\delta(s-\tau\_2))\right]^{-1}\right), \\ &s \in (\tau\_1,\tau\_2);\\ \kappa\_3(s,\lambda) &= \max\left(\exp\left(-\lambda\delta(s-\tau\_2)\right), \exp(\lambda\delta s)\right), \quad s \in [\tau\_2,0]. \end{aligned}$$

Finally, by *<sup>S</sup>*(*τ*1, *<sup>τ</sup>*2) we denote the set of functions *<sup>ϕ</sup>*(*s*) from *<sup>C</sup>*[−1,0] that satisfy the conditions

$$\begin{aligned} \varphi(\tau\_1) &= \varphi(\tau\_2) = \varphi(0) = 1; \\ &0 \le \varphi(s) \le \kappa\_1(s, \lambda), \text{ for } s \in [-1, \tau\_1]; \\ \kappa\_2(s, \lambda) &\le \varphi(s, \lambda) \le A\_\prime \text{ for } s \in (\tau\_1, \tau\_2); \\ &0 \le \varphi(s) \le \kappa\_3(s, \lambda) \text{ for } s \in [\tau\_2, 0]. \end{aligned}$$

The first and the second positive roots of the equation *u*(*t*, *ϕ*) = 1 will be denoted by *<sup>t</sup>*1(*ϕ*) and *<sup>t</sup>*2(*ϕ*). We set *<sup>t</sup>*<sup>0</sup> = *<sup>A</sup>*(*<sup>A</sup>* − <sup>1</sup>)−1(*τ*<sup>1</sup> + <sup>1</sup>). Asymptotic analysis of *<sup>u</sup>*(*t*, *<sup>ϕ</sup>*) for *t* ∈ [0, *τ*<sup>2</sup> + 1] leads to the following statement:

**Lemma 5.** *Suppose that the condition*

$$t\_0 < r\_2 + 1\tag{29}$$

*is fulfilled. Then for t*1(*ϕ*) *the following asymptotic (for λ* → ∞*) equality holds:*

$$t\_1(\varrho) = t\_0 + o(1).$$

**Proof.** As in Theorem 2, when constructing the asymptotics of the solution to Equation (24), we use the formula (25). It follows from (25), that on the segment *t* ∈ [0, *τ*<sup>1</sup> + 1] solution has form (26). It follows from (26), that on the interval *t* ∈ (0, *τ*<sup>1</sup> + 1] equality *u*(*t*, *ϕ*) = *A* + *o*(1) holds.

On the segment *t* ∈ [*τ*<sup>1</sup> + 1, *τ*<sup>2</sup> + 1] the function *u*(*t* − 1, *ϕ*) = *A* + *o*(1). That is why on this time segment the solution to Equation (24) satisfies the formula

$$u(t,\varphi) = A\left[1 + (A - 1)\exp\left(\lambda A\left[-\left(\tau\_1 + 1\right) + (A - 1)\left(t - \left(\tau\_1 + 1\right)\right) + o(1)\right]\right)\right]^{-1}.\tag{30}$$

While *τ*<sup>1</sup> + 1 < *t* < *t*0, the right-hand side in formula (30) is asymptotically close to *A* at *λ* → +∞, and for *t* > *t*<sup>0</sup> this expression is *o*(1) at *λ* → +∞. Therefore, if *t*<sup>0</sup> < *τ*<sup>2</sup> + 1, then the value *t*<sup>0</sup> is asymptotically close to *t*1(*ϕ*) at *λ* → +∞.

We note that for every *t* ∈ (0, *t*1(*ϕ*)) the relation

$$u(t, \boldsymbol{\varrho}) = A + o(1)$$

is fulfilled. Next, for *t* ∈ (*t*1(*ϕ*), *τ*<sup>2</sup> + 1), we have *u*(*t*, *ϕ*) = *o*(1), with

$$
\mu(\mathfrak{r}\_2 + 1, \mathfrak{q}) = A(A - 1)^{-1} \exp(-\lambda A(A - 1)(\mathfrak{r}\_2 + 1 - t\_0 + o(1))).
$$

We set *t* <sup>0</sup> = *<sup>A</sup>*(*τ*<sup>2</sup> + <sup>1</sup>)+(<sup>1</sup> − *<sup>A</sup>*)*t*0. If

$$t^0 < 1,\tag{31}$$

then for *t* ∈ (*τ*<sup>2</sup> + 1, *t* <sup>0</sup> + *δ*1) (where *δ*<sup>1</sup> is some sufficiently small quantity that does not depend on *λ*) we find that

$$u(t, \boldsymbol{\varrho}) = A / \left[ (A - 1) \exp \left( -\lambda A \left( t - A(\tau\_2 + 1) - (1 - A)t\_0 + o(1) \right) \right) + 1 \right].$$

From this follow both the existence of *t*2(*ϕ*) and the asymptotic formula *t*2(*ϕ*) = *t* <sup>0</sup> + *o*(1). It is easy to show, that if inequality *t* <sup>0</sup> > 1 holds, then *<sup>t</sup>*2(*ϕ*) exists, but *<sup>t</sup>*2(*ϕ*) − *<sup>t</sup>*1(*ϕ*) > 1. So, in the case *t* <sup>0</sup> > 1 we return to the case of the slowly oscillating solutions.

If inequality (31) holds, then as in the preceding section, we introduce into the analysis the operator Π, using the rule Π(*ϕ*(*s*)) = *u*(*s* + *t*2(*ϕ*), *ϕ*).

Note, that *t*2(*ϕ*) corresponds to 0, *t*1(*ϕ*) corresponds to *τ*2, 0 corresponds to *τ*<sup>1</sup> (see Figure 3). That is why *τ*¯1 = 0 − *t* <sup>0</sup> + *<sup>o</sup>*(1) *<sup>τ</sup>*¯2 = *<sup>t</sup>*<sup>0</sup> − *<sup>t</sup>* <sup>0</sup> + *o*(1).

A consequence of the formulas given here is the following statement:

**Theorem 3.** *Let the conditions* (29) *and* (31) *are fulfilled. Then for all sufficiently large λ for the Equation* (24) *the following inclusion holds:*

$$\Pi(\boldsymbol{\varrho}(s)) \in \mathcal{S}(\mathbb{P}\_1 + o(1), \mathbb{P}\_2 + o(1)), s$$

with 
$$\begin{aligned} \bar{\tau}\_1 &= A\tau\_1 - A\tau\_2, \\ \bar{\tau}\_2 &= A^2 (A - 1)^{-1} \tau\_1 - A\tau\_2 + A (A - 1)^{-1}. \end{aligned} \tag{32}$$

The system of Equation (32) is a linear inhomogeneous difference system of equations. After *τ*¯1 and *τ*¯2 have been determined, the situation is repeated, i.e., while for current values of *τ*¯1 and *τ*¯2 inequalities (29) and (31) are fulfilled, using (32) we calculate *τ*¯¯1 and *τ*¯¯2, and so on. This iteration process has a fixed point. By virtue of the fact that the determinant of the linear part in (32) is greater in modulus than 1 the iteration process is divergent, i.e., the equilibrium state in (32) is unstable.

We also remark that the number of periodic solutions of Equation (24) grows without limit as *λ* → ∞. We shall show this. It was established above that for sufficiently large

*λ* Equation (24) has a slowly oscillating periodic solution *u*0(*t*, *λ*) with period *T*(*λ*). This solution for each integer *n* satisfies the equation

$$
\dot{u} = \lambda \left[ 1 - u(t - 1 - nT(\lambda)) \right] u(A - u). \tag{33}
$$

In (33) we replace the time *t* → (1 + *nT*(*λ*))*t*. We then find that the function *un*(*t*, *λ*) = *u*0((1 + *nT*(*λ*))*t*, *λ*) is a solution of the equation

$$
\dot{u} = \lambda (1 + nT(\lambda)) [1 - u(t - 1)] u(A - u). \tag{34}
$$

The period of the function *un*(*t*, *λ*) is equal to *T*(*λ*)[1 + *nT*(*λ*)]−1. In particular, the unstable periodic solution determinable from the fixed points *τ*10, *τ*<sup>20</sup> of the mapping (32) corresponds to the value *n* = 1 in (34), i.e., to the function *u*1(*t*, *λ*).

We shall consider the more general case when on the segment [−1, 0] the initial functions *ϕ*(*s*) take the value 1 exactly (2*n* + 1) times:

$$1 - 1 < \tau\_1 < \tau\_2 < \dots < \tau\_{2n} < 0, \quad \varphi(\tau\_j) = \varphi(0) = 1 \quad (j = 1, \dots, 2n). \tag{35}$$

Following the method proposed above, we arrive at the 2*n*-dimensional map

$$\begin{aligned} \mathfrak{r}\_1 &= \mathfrak{r}\_3 - t^0, \quad t^0 = A \mathfrak{r}\_2 - A \mathfrak{r}\_1, \\ \mathfrak{r}\_2 &= \mathfrak{r}\_4 - t^0, \quad t\_0 = A(A-1)^{-1}(\mathfrak{r}\_1 + 1), \\ &\dots &\dots \\ \mathfrak{r}\_{2n-1} &= -t^0, \quad \mathfrak{r}\_{2n} = t\_0 - t^0. \end{aligned} \tag{36}$$

While we are in the class of initial conditions (conditions (35) are satisfied), we iterate this map. It can be shown that if the equilibrium state in (36) exists, then it is unstable. Since solution *un*(*t*, *λ*(1 + *nT*(*λ*))) corresponds to equilibrium state in map (36), then this periodic solution of (24) is unstable.
