*2.1. Case of Small γ Values*

Under the conditions *rT* < *π*/2 and (23), the roots of Equation (22) have separated from zero negative real parts as *μ* → 0. For *rT* > *π*/2, Equation (22) has a root with a positive real part separated from zero as *μ* → 0. The local dynamics of (11), (12) are not considered in these cases.

We assume that the conditions

$$
\mu r\_0 T\_0 = \frac{\pi}{2}, \quad r = r\_0 + \mu r\_1, \quad T = T\_0 + \mu T\_1, \quad \gamma = \mu \gamma\_1 \tag{24}
$$

hold for some positive *r*<sup>0</sup> and *T*0. In this case, *vk* = exp(*iπ*(2*T*0)−1*t* + *ikx*) is the solution of the linear boundary value problem (21) as well as

$$\begin{split} v(t, \mathbf{x}) &= \sum\_{k=-\infty}^{+\infty} \tilde{\varsigma}\_{k} \exp\left(i\pi (2T\_{0})^{-1}t + ik\mathbf{x}\right) = \\ &\exp\left(i\pi (2T\_{0})^{-1}t\right) \sum\_{k=-\infty}^{+\infty} \tilde{\varsigma}\_{k} \exp(ik\mathbf{x}) = \exp\left(i\pi (2T\_{0})^{-1}t\right)\tilde{\varsigma}(\mathbf{x}). \end{split} \tag{25}$$

Then, we seek the solution to the nonlinear boundary value problem (11), (12) in the form

$$\begin{aligned} \mu(t, \mathbf{x}, \boldsymbol{\mu}) &= \mu^{1/2} \Big( \overline{\boldsymbol{\xi}}(\mathbf{r}, \mathbf{x}) \exp \left( i \pi (2T\_0)^{-1} t \right) + \overline{c \overline{c}} \Big) + \\ & \quad \mu \boldsymbol{\mu}\_2(t, \mathbf{r}, \mathbf{x}) + \mu^{3/2} \boldsymbol{\mu}\_3(t, \mathbf{r}, \mathbf{x}) + \dots \end{aligned} \tag{26}$$

where *τ* = *μt* is the 'slow' time, *ξ*(*τ*, *x*) is the unknown complex amplitude, the functions *uj*(*t*, *τ*, *x*) are 4*T*0-periodic with respect to *t* and 2*π*-periodic with respect to *x*. Here, *cc* means the complex conjugate to the previous term expression.

We insert the formal expression (26) into (11) and collect the coefficients at the same powers of *μ*. At the first step, we equate the coefficients of *μ*1/2 and obtain an identity. Then, we collect the coefficients at the first power of *μ* and obtain the equation for *u*<sup>2</sup>

$$\dot{u}\_2 = -r u\_2(t - T, \mathbf{x}) - r \left[ \xi \exp\left( -i \frac{\pi}{2} + i \frac{\pi}{2T\_0} t \right) + \bar{\xi} \exp\left( i \frac{\pi}{2} - i \frac{\pi}{2T\_0} t \right) \right] \cdot \left( \xi \exp\left( i \frac{\pi}{2T\_0} t \right) + \xi \right).$$

Hence,

$$\mu\_2 = A \mathfrak{J}^2 \exp(i\pi (T\_0)^{-1} t) + \overline{c} \overline{c}, \quad A = -\exp(-i r\_0) \left( 2(i + \exp(-2i T\_0)) \right)^{-1}.$$

At the third step, we obtain the equation for determining *u*3, the solvability condition of which in the indicated class of functions is formulated as

$$\begin{split} \frac{\partial \tilde{\xi}}{\partial \tau} &= b\_{\tilde{\xi}}^{\tilde{\pi}} + \gamma\_0 \left( M(\tilde{\xi}) - \tilde{\xi} \right) + \beta \tilde{\xi} |\tilde{\xi}|^2, \\ \tilde{\xi}(\tau, \varkappa + 2\pi) &\equiv \tilde{\xi}(\tau, \varkappa). \end{split} \tag{27}$$

Here, *M*(*ξ*) stands for the mean value with respect to *x* ∈ [0, 2*π*] of the function *ξ*(*τ*, *x*):

$$M(\xi) = \frac{1}{2\pi} \int\_0^{2\pi} \xi(\mathbf{r}, \mathbf{x}) d\mathbf{x}.$$

The following equalities hold for the coefficients in (27):

$$\begin{array}{rcl} b & = & \left(1 + \frac{\pi^2}{4}\right)^{-1} \left[\left(\frac{\pi}{2} + i\right)r\_1 + \lambda\_0^2 T\_1 \left(1 - i\frac{\pi}{2}\right)\right]\_\prime \\ \gamma\_0 & = & \gamma\_1 r\_0 \exp\left(-i\pi h (2T\_0)^{-1}\right) \cdot \left[i\pi (2T\_0)^{-1} - r\_0 \exp\left(-i\pi (2T\_0)^{-1}\right)\right]^{-1}, \\ \beta & = & -\lambda\_0 [3\pi - 2 + i(\pi + 6)] \left(10 \left(1 + \frac{4}{\pi^2}\right)\right)^{-1}, \end{array}$$

In the considered case, the next statement indicates the boundary value problem (27) to be the QNF for the boundary value problem (11), (12).

**Theorem 1.** *Let the conditions* (23) *and* (24) *be satisfied, and let the boundary value problem* (27) *have the solution ξ*(*τ*, *x*) *for τ* ≥ *τ*0*. Then, the function*

$$\mu(t, \mathbf{x}, \mu) = \mu^{1/2} \left( \tilde{\xi}(\tau, \mathbf{x}) \exp \left( i\pi (2T\_0)^{-1} t \right) + \overline{c\overline{c}} \right) + \mu \left( A \tilde{\xi}^2(\tau, \mathbf{x}) \exp(i\pi (T\_0)^{-1} t) + \overline{c\overline{c}} \right)$$

*satisfies the boundary value problem* (11)*,* (12) *up to O*(*μ*3/2)*.*

We note that the conditions of zero solution in (27) asymptotic stability consist in the fulfillment of the inequalities

$$
\Re b < 0, \quad \Re(b - \gamma\_0) < 0. \tag{28}
$$

For *b* > 0, the QNF (27) has the homogeneous cycle *ρ*<sup>0</sup> exp(*iω*0*t*), and

$$\rho\_0 = \left(-\Re b \cdot (\Re \beta)^{-1}\right)^{1/2}, \quad \omega\_0 = \he z \partial + \rho\_0^2 \Im \beta.$$

The same cycle under the same condition exists in the logistic Equation (1). The cycle in Equation (1) is orbitally stable, while the orbital stability in (27) requires the following inequalities to be valid:

$$
\rho\_0(1) \qquad \rho\_0 \Re \beta - \Re \gamma\_0 < 0,\tag{29}
$$

$$
\langle \text{2} \rangle \qquad |\gamma\_0|^2 - 2\rho\_0^2 \Big( \Re \mathcal{B} \cdot \Re \gamma\_0 + \Im \mathcal{B} \cdot \Im \gamma\_0 \Big) > 0. \tag{30}
$$

A value of *γ*<sup>0</sup> can be selected such that either (29) or (30) does not hold. We note that stability (instability) condition fulfillment can be achieved by delay coefficient *h* variation.

In addition to one cycle, the QNF (27) can have the cycles

$$
\rho\_k \exp\left(i\omega\_k t + ik\alpha\right) \quad (k = 0, \pm 1, \pm 2, \dots) \dots
$$

These cycles exist under the condition (*b* − *γ*0) > 0 and

$$\rho\_k = \rho^0 = \left(-\Re(b - \gamma\_0) \cdot (\Re \beta)^{-1}\right)^{1/2}, \quad \omega\_k = \omega^0 = \Im(b - \gamma\_0) + \rho\_k^2 \Im \beta. \tag{31}$$

The above cycles are orbitally stable if the inequalities

(1) (*ρ*0)2*<sup>β</sup>* + *γ*<sup>0</sup> < 0,

(2) |*γ*0| <sup>2</sup> + 2(*ρ*0)<sup>2</sup> *β* · *γ*<sup>0</sup> + *β* · *γ*<sup>0</sup> > 0

hold. Strict unfulfillment of at least one of these inequalities implies instability of all cycles.

The problem of the spatially inhomogeneous step-like cycle's existence is more intriguing. At the first instance, we note that Equation (27) is periodic with respect to *t* and 2*π*-periodic piecewise continuous with respect to *x* solution

$$\rho(\mathfrak{x}) \exp(i\omega^0 t), \quad \rho(\mathfrak{x}) = \begin{cases} \rho^0 \, , & \mathfrak{x} \in (0, \pi), \\ -\rho^0 \, , & \mathfrak{x} \in (\pi, 2\pi). \end{cases}$$

One can construct families of 2*πω*−<sup>1</sup> <sup>0</sup> -periodic with respect to *t* and 2*π*-periodic piecewise continuous with respect to *x* solutions *ρ*(*x*, *k*1, *k*2, *α*) exp(*iω*0*t*), where

$$\rho(\mathbf{x},k\_1,k\_2,\boldsymbol{\alpha}) = \begin{cases} \rho^0 \exp(i2\pi \boldsymbol{n}^{-1}k\_1 \mathbf{x}), & \mathbf{x} \in (0,\boldsymbol{a}), \quad k\_1 = \pm 1, \pm 2, \dots, \\\rho^0 \exp(i2\pi (2\pi - \boldsymbol{a})^{-1}k\_2 \mathbf{x}), & \mathbf{x} \in (\boldsymbol{a}, 2\pi - \boldsymbol{a}), \quad k\_2 = \pm 1, \pm 2, \dots, \end{cases}$$

Obviously, constructions of this kind extend to solutions with an arbitrary number of 'steps'.

What is more intriguing, these are the cycles consisting of two different steps with respect to 'amplitude' on the interval [0, 2*π*]. To construct them, we arbitrarily fix the parameters *α* ∈ (0, 2*π*) and *ϕ*1,2 ∈ (0, 2*π*). We assume

$$u\_0(t, \mathbf{x}) = \rho(\mathbf{x}) \exp(i\omega t), \quad \rho(\mathbf{x}) = \begin{cases} \rho\_1 \exp(i\rho\_1), & \mathbf{x} \in (0, \mathfrak{a}), \\ \rho\_2 \exp(i\rho\_2), & \mathbf{x} \in (\mathfrak{a}, 2\pi). \end{cases} \tag{32}$$

Let *ϕ* = *ϕ*<sup>2</sup> − *ϕ*1. We insert (32) into (26). Then, we obtain the system of polynomial equations

$$i\omega \rho\_{\dot{j}} = (b - \gamma \omicron \rho)\rho\_{\dot{j}} + \gamma \omicron P + \beta \rho\_{\dot{j}}^3, \quad (\dot{j} = 1, 2). \tag{33}$$

Here, *P* = (2*π*)−1*γ*<sup>0</sup> *αρ*<sup>1</sup> + (1 − *α*)*ρ*<sup>2</sup> exp(*iϕ*) .

The system (33) represents two complex or four real equations of five real variables *ρ*1, *ρ*2, *α*, *ϕ* and *ω*. We proceed with the real-valued notation of this system:

$$
\rho\_1 \quad = \; \Re(b - \gamma\_0)\rho\_1 + \rho\_1^3 \Re \beta + \mathcal{R}\gamma\_0 P\_\prime \tag{34}
$$

$$
\omega \rho\_1 \quad = \quad \odot (b - \gamma \omicron) \rho\_1 + \rho\_1^3 \heartsuit \beta + \heartsuit \gamma \rho\_0 P,\tag{35}
$$

$$\rho\_2 = \, ^3\mathfrak{R}(b - \gamma\_0)\rho\_2 + \rho\_2^3 \mathfrak{R}\mathfrak{F} + \mathfrak{R}\left(\gamma\_0 \exp(-iq\,)P\right), \tag{36}$$

$$
\omega \rho\_2 \quad = \, \, \S (b - \gamma\_0) \rho\_2 + \rho\_2^3 \S \beta + \S \left( \gamma\_0 \exp(-i\rho)P \right), \tag{37}
$$

First, we provide the expressions for *ρ*<sup>3</sup> 1,2 from (34) and (36):

$$\begin{array}{rcl}\rho\_1^3 &=& \left[\left(1 - \mathfrak{R}(b - \gamma\_0)\right)\rho\_1 - \mathfrak{R}(\gamma\_0 P)\right](\mathfrak{R}\beta)^{-1},\\\rho\_2^3 &=& \left[\left(1 - \mathfrak{R}(b - \gamma\_0)\right)\rho\_2 - \mathfrak{R}(\gamma\_0 \exp\left(-i\varrho\right)P)\right](\mathfrak{R}\beta)^{-1}.\end{array}$$

Then, we substitute them into (35) and (37) instead of *ρ*<sup>3</sup> 1,2, respectively. As a result, we obtain the linear system with the 2 × 2 matrix *B* of the form

$$B\left(\begin{array}{c}\rho\_1\\\rho\_2\end{array}\right) = \omega\left(\begin{array}{c}\rho\_1\\\rho\_2\end{array}\right).$$

If the real positive eigenvalue *ω* = *ω*(*α*, *ϕ*) of this matrix can be determined, we obtain the eigenvector *ρ*<sup>2</sup> = *c*(*α*, *ϕ*)*ρ*1. Taking this equality into consideration in (34) and (36), we obtain the expressions for *ρ<sup>j</sup>* = *ρj*(*α*, *ϕ*) (*j* = 1, 2). Finally, another variable is eliminated via the equality

$$c(\mathfrak{a}, \mathfrak{q}) = \rho\_2(\mathfrak{a}, \mathfrak{q}) \left(\rho\_1(\mathfrak{a}, \mathfrak{q})\right)^{-1}.$$

Numerical investigations are carried out in this way.
