5.2.4. Case of 'Intelligently' Small Parameters *h* and *μ*

To complete the picture, we briefly consider the simplest situation when both parameters *h* and *μ* are 'intelligently' small:

$$
\hbar \mathbf{h} = \varepsilon^2 h\_0, \quad \mu = \varepsilon^2 k\_0. \tag{113}
$$

The bifurcation value of the delay coefficient *T* = *T*<sup>0</sup> satisfies the equality *T* = *π*(2*r*)−1. Let *T* = *T*<sup>0</sup> + *εT*<sup>01</sup> in (79), (80). Then, infinitely many roots *λ<sup>m</sup>* = *λm*(*ε*) of Equation (98) tend to the imaginary axis as *ε* → 0, and there are no roots with a positive zero-separated as *ε* → 0 real part. Therefore, the critical case of an infinite dimension is realized here, too. Let us introduce the formal series

$$\begin{split} N &= 1 + \varepsilon^{1/2} \Big[ \exp \left( i \frac{\pi}{2T\_0} t \right) \tilde{\xi}(\mathbf{r}, \mathbf{x}) + \exp \left( -i \frac{\pi}{2T\_0} t \right) \bar{\xi}(\mathbf{r}, \mathbf{x}) \Big] + \\ &+ \varepsilon \boldsymbol{\mu}\_2(t, \mathbf{r}, \mathbf{x}) + \varepsilon^{3/3} \boldsymbol{\mu}\_2(t, \mathbf{r}, \mathbf{x}) + \dots , \end{split} \tag{114}$$

where *τ* = *εt*, and *uj*(*t*, *τ*, *x*) are periodic with respect to first and third arguments with 4*T*<sup>0</sup> and 2*π* periods, respectively. We insert (114) into (79) and perform standard operations. At the third step, we arrive at the boundary value problem for determining the unknown, slowly varying amplitude *ξ*(*τ*, *x*):

$$\frac{\partial \tilde{\xi}}{\partial \tau} = \left(1 + i\frac{\pi}{2}\right)^{-1} \left[k\_0 \frac{\partial^2 \tilde{\xi}}{\partial \mathbf{x}^2} + h\_0 \frac{\partial \tilde{\xi}}{\partial \mathbf{x}} + r\_0^2 T\_{11}\right] + \mathcal{g} |\tilde{\xi}|^2 \tilde{\xi} \tag{115}$$

$$
\mathfrak{F}(\mathfrak{r}, \mathfrak{x} + 2\pi) \equiv \mathfrak{F}(\mathfrak{r}, \mathfrak{x}),
\tag{116}
$$

where

$$\mathbf{g} = -r[3\pi - 2 + i(\pi + 6)]\left(10\left(1 + \frac{\pi^2}{4}\right)\right)^{-1}\mathbf{J}$$

The coupling between solutions of the problem (115), (116) and asymptotic with respect to residual solutions of the problem (79), (80) is determined by Formula (114). We note that for a periodic solution of (115), (116) of the form *ξ*0(*τ*, *x*) = *Const* · exp(*iωτ* + *ikx*), one can formulate a stronger result about the existence (and inheritance of the stability properties) of a periodic solution of the problem (79), (80), which is close to

$$\left[\varepsilon^{1/2}\left[\tilde{\xi}\_0(\mathbf{r},\mathbf{x})\exp\left(i\left(\frac{\pi}{2T\_0} + O(\varepsilon)\right)t\right) + \tilde{\xi}\_0(\mathbf{r},\mathbf{x})\exp\left(-i\left(\frac{\pi}{2T\_0} + O(\varepsilon)\right)t\right)\right]\right]$$

as *ε* → 0.
