*5.2. Large Coefficient γ-Induced Bifurcations*

Upon condition (83), we study the behavior of solutions of the boundary value problem (79), (80) in a sufficiently small equilibrium state *N*<sup>0</sup> ≡ 1 neighborhood. The characteristic quasipolynomial of a boundary value problem linearized on *N*<sup>0</sup> has the form

$$\varepsilon\lambda = -\varepsilon\tau \exp(-\lambda T) + \exp(ilm - \mu m^2) - 1, \quad 0 < \varepsilon \ll 1, \quad m = 0, \pm 1, \pm 2, \dots \tag{98}$$

As in Section 5.1, we restrict ourselves to the most interesting and important situations depending on the relationship between the parameters *ε*, *h*, and *μ*. In Section 5.2.1, we assume that condition (84) holds: *<sup>h</sup>* <sup>=</sup> <sup>√</sup>*εh*1, *<sup>μ</sup>* <sup>=</sup> *<sup>k</sup>ε*. Below, we demonstrate that Andronov– Hopf bifurcation occurs in (79), (80) even for small values of the delay parameter *<sup>T</sup>* ∼ *<sup>ε</sup>*1/2. The periodic solution bifurcating from the equilibrium state *N*<sup>0</sup> turns out to be rapidly oscillating in time. In Section 5.2.2, the relations *h* = *εh*2, *μ* = *k*1*ε*<sup>2</sup> are assumed to be valid. In this case, the corresponding bifurcation process has an infinite dimension: infinitely many roots of the characteristic Equation (98) tend to the imaginary axis as *ε* → 0. We construct quasinormal forms, i.e., the families of complex parabolic (and degenerate parabolic) boundary value problems whose nonlocal dynamics determine the behavior of the initial boundary value problem (79), (80) solutions in the small neighborhood of *N*<sup>0</sup> for small *ε*. In Section 5.2.3, even more complicated families of quasinormal forms are constructed to determine the dynamic properties of problem (79), (80) under the constraint *μ* = *k*1*ε*2. The conclusions are given in Section 5.2.4.

It is natural to choose the delay time *T* as the main bifurcation parameter. We recall that the bifurcation value for *T* is determined from the equality 2*rT* = *π* in Equation (79). In the cases considered here, we demonstrate that the bifurcation parameter can be asymptotically small.

$$\text{15.2.1. Bifurccation Analysis upon Condition } h = \varepsilon^{1/2} h\_1, \mu = k\varepsilon.$$

In (98), we assume

$$
\lambda = i\omega \quad (\omega > 0), \quad \omega = \varepsilon^{-1/2} h\_1 + \omega\_{2\prime} \quad T = \varepsilon^{1/2} T\_{1\cdots}
$$

Then, we obtain

$$\begin{aligned} r\cos(h\_1 T\_{1m}) &= -\left(k + \frac{1}{2}h\_1^2\right)m^2, \\ r\sin(h\_1 T\_{1m}) &= \omega\_2. \end{aligned} \tag{99}$$

up to *O*(*ε*1/2). If it exists, let *T*1*m*, *m* = 1, 2, ... be the least positive root of Equation (99). Obviously, there is a finite number of such roots. Let *T*<sup>0</sup> <sup>1</sup> be the least of them. Let *<sup>T</sup>*<sup>0</sup> <sup>1</sup> = *T*1*m*<sup>0</sup> . We formulate two statements about the roots of Equation (98) that are simple but cumbersome proofs that are omitted.

**Lemma 6.** *Let T*<sup>1</sup> < *T*<sup>0</sup> <sup>1</sup> *. Then, for all sufficiently small ε, the Equation* (98) *roots separate from zero negative real parts as ε* → 0*.*

**Lemma 7.** *Let T*<sup>1</sup> > *T*<sup>0</sup> <sup>1</sup> *. Then, for all sufficiently small ε, Equation* (98) *has a root separate from the zero positive real part as ε* → 0*.*

Thus, in the context of the above lemmas, the issue of the boundary value problem (79), (80) local dynamics is solved trivially.

Let us study the behavior of the solutions of the problem (79), (80) in the *N*<sup>0</sup> neighborhood close to the critical case condition

$$T\_1 = T\_1^0 + \varepsilon^{1/2} T\_{01}.$$

Then, the characteristic Equation (98) has the coupling of the roots *λ*1,2(*ε*) of the form

$$
\lambda\_{1,2}(\mathfrak{e}) = \pm i (\mathfrak{e}^{-1/2} h\_1 + \omega\_2) + O(\mathfrak{e}^{1/2}).
$$

The real parts of the remaining roots of Equation (98) are negative and zero-separated as *ε* → 0. In this case, for small *ε*, the boundary value problem (79), (80) has a twodimensional stable local invariant integral manifold in the (small) *N*<sup>0</sup> neighborhood, on which this boundary value problem can be represented in the form of the scalar complex equation up to the summands of the *O*(*ε*1/2) order:

$$\frac{\partial \mathfrak{J}}{\partial \mathfrak{T}} = a\mathfrak{J} + d|\mathfrak{J}|^2 \mathfrak{J}\_\prime \tag{100}$$

where *τ* = *ε*1/2*t* is a 'slow' time. The solution *N*(*t*, *x*,*ε*) on this manifold is coupled to solutions of Equation (100) by the relation

$$\begin{split} N(t, \mathbf{x}, \varepsilon) &= \varepsilon^{1/4} \Big[ \tilde{\xi}(\tau) \exp(i m\_0 \mathbf{x} + i(\varepsilon^{-1/2} h\_1 m\_0 + \omega\_2)t) + \tilde{\xi}(\tau) \exp(-i m\_0 \mathbf{x} - \mathbf{z}) \\ &i(\varepsilon^{-1/2} h\_1 m\_0 + \omega\_2)t) \Big] + \varepsilon^{1/2} u\_2(t, \tau, \mathbf{x}) + \varepsilon^{3/4} u\_3(t, \tau, \mathbf{x}) + \dots \,\, . \end{split} \tag{101}$$

Here, the functions *uj*(*t*, *τ*, *x*) are periodic with respect to first and third arguments, with periods 2*π*(*ε*−1/2*h*1*m*<sup>0</sup> + *ω*2)−<sup>1</sup> and 2*π*, respectively.

Regarding the dynamics of Equation (100) (and hence the boundary value problem (79), (80) in the *N*<sup>0</sup> neighborhood), one needs to find the coefficients *α* and *d*. To accomplish this, we insert the formal series (101) into (79) and equate the coefficients at the same powers of *ε*. At the second step, we obtain

$$u\_{2}(t,\tau,\mathbf{x}) = u\_{20}|\xi|^{2} + u\_{21}\xi^{2}\exp\left(2im\_{0}\mathbf{x} + 2i(\varepsilon^{-1/2}h\_{1}m\_{0} + \omega\_{2})t\right) + \dots$$

$$u\_{21}\xi^{2}\exp\left(-2im\_{0}\mathbf{x} - 2i(\varepsilon^{-1/2}h\_{1}m\_{0} + \omega\_{2})t\right)$$

$$\begin{array}{rcl} \text{and} &\\ & \mu\_{20} & = & 2 \cos(T\_1^0 h\_1 m\_0), \\ & \mu\_{21} & = & -r \Big( \exp(-2i T\_1^0 h\_1 m\_0)(2i\omega\_2 + 4m\_0^2 k + r \exp(-2i T\_1^0 h\_1 m\_0)) \Big)^{-1}. \end{array}$$

From the solvability condition obtained at the third-step equation with respect to *u*3(*t*, *τ*), we obtain

$$\begin{array}{rcl}d&=&-\;r\left[1+\exp\left(-2i T\_1^0 h\_1 m\_0\right)-r\exp\left(-i T\_1^0 h\_1 m\_0\right)\left(2i\omega\_2 + 4m\_0^2 k + \frac{1}{2}\right)\right] \\ &\quad \;r\exp\left(-2i T\_1^0 h\_1 m\_0\right))^{-1}\Big|\_{\,\,t} \\ \alpha&=&i r h\_1 T\_1^0 m\_0 \exp\left(-i T\_1^0 h\_1 m\_0\right) - \left(k + \frac{1}{2} h\_1^2\right) m\_0^2.\end{array}$$

We note that each of the quantities *α* and *d* can be either positive or negative, depending on the values of the parameters *T*<sup>0</sup> <sup>1</sup> , *m*0, *r*, *k*, and *h*1. As an example, we make one statement about the dynamics of problem (79), (80).

**Theorem 13.** *Let α* > 0 *and d* < 0*. Then, for all sufficiently small ε, the boundary value problem* (79)*,* (80) *has a stable periodic solution N*0(*t*, *x*,*ε*) *in the N*<sup>0</sup> *neighborhood, for which*

$$N\_0(t, \mathbf{x}, \varepsilon) = 1 + 2\varepsilon^{1/4}\rho\_0 \cos(m\_0 \mathbf{x} + \varepsilon^{-1/2}h\_1 m\_0 + \omega\_2 + \varepsilon^{1/2}\phi\_0 + O(\varepsilon^{3/4}))t + O(\varepsilon^{1/2}),$$

*where*

$$
\rho\_0 = \left(\Re a (-\Re d)^{-1}\right)^{1/2}, \quad \wp\_0 = \heartsuit a + \rho\_0^2 \heartsuit d.
$$

**Remark 2.** *The results presented here could be obtained by studying the bifurcations from the equilibrium state ξ*<sup>0</sup> ≡ 1 *in the boundary value problem* (89)*,* (90)*.*

5.2.2. High-Mode Bifurcations upon Condition *μ* = *ε*2*k*1, *h* = *εh*<sup>2</sup>

It is assumed here that the conditions *h* = *εh*2, *μ* = *k*1*ε*<sup>2</sup> are satisfied. Under these conditions, we study the behavior of the solutions of the problem (79), (80) from a small *N*<sup>0</sup> neighborhood. First, we consider the characteristic Equation (98). For *T* = 0, its roots have negative real parts (separated from zero as *ε* → 0). Let us demonstrate that there are roots with a close to zero real part as *<sup>T</sup>* ∼ *<sup>ε</sup>*1/2, and the numbers (modes) *<sup>m</sup>* ∼ *<sup>ε</sup>*−1/2 correspond to them. In (98), we assume

$$T = \varepsilon^{1/2} T\_1, \quad m = c\varepsilon^{-1/2} + \theta\_{\varepsilon}, \quad \lambda = i\varepsilon^{-1/2} h\_2 c + i\omega(c),$$

where *<sup>θ</sup><sup>c</sup>* = *<sup>θ</sup>c*(*ε*) ∈ [0, 1) is the value that complements the previous summand *<sup>c</sup>ε*−1/2 to an integer. Then, we obtain the equality

$$i\omega(c) = -r \exp(-ih\_2 T\_1 c) - \left(k\_1 + \frac{1}{2}h\_2^2\right)c^2 \tag{102}$$

(up to *O*(*ε*1/2)). Let us determine the least value of *T*<sup>1</sup> = *T*<sup>0</sup> <sup>1</sup> for which this equation is solvable with respect to *c*, i.e., for some *c* = *c*0, the relation

$$r\cos(h\_2 T\_1^0 \mathbf{c}\_0) = -\left(k\_1 + \frac{1}{2}h\_2^2\right)\mathbf{c}\_0^2\tag{103}$$

holds. We denote by *x*<sup>0</sup> the least positive root of the equation

$$2\mathfrak{x} = -\mathfrak{x}\_{\ast}$$

Then, the following simple statements hold.

**Lemma 8.** *For T*<sup>0</sup> <sup>1</sup> *and c*0*, the equalities*

$$\begin{array}{rcl}T\_1^0 &=& \frac{1}{2} \mathfrak{x}\_0 \left(k\_1 + \frac{1}{2}h\_2^2\right)^{1/2} (2 - r \cos \mathfrak{x}\_0)^{-1/2},\\\ c\_0 &=& (-r \cos \mathfrak{x}\_0)^{-1/2} \left(k\_1 + \frac{1}{2}h\_2^2\right)^{-1/2}\end{array}$$

*hold.*

**Lemma 9.** *Let T*<sup>1</sup> < *T*<sup>0</sup> <sup>1</sup> *. Then, for all sufficiently small ε, the roots of Equation* (98) *have negative real parts (separated from zero as ε* → 0*).*

**Lemma 10.** *Let T*<sup>1</sup> > *T*<sup>0</sup> <sup>1</sup> *. Then, for all sufficiently small ε, Equation* (98) *has a root with a positive real part (separated from zero as ε* → 0*).*

In the context of Lemmas 9 and 10, the behavior of the solutions of the problem (79), (80) in small (*ε*-independent) neighborhood of *N*<sup>0</sup> is determined in the standard way.

Below, we assume that a case close to critical in the stability problem of *N*<sup>0</sup> is realized. Let, for some arbitrary constant *T*11, the equality

$$T\_1 = T\_1^0 + \varepsilon^{1/2} T\_{11}$$

hold. In this case, infinitely many roots 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. Thus, the critical case of an infinite dimension is realized in the *N*<sup>0</sup> stability problem. Let us apply the technique of [26,28] to study the local dynamics of problem (79), (80) for small *ε*.

First, we note that all modes that correspond to roots of Equation (98) close to the imaginary axis are asymptotically large and have the leading asymptotic term *c*0*ε*−1/2. In this regard, we consider all modes with numbers *m* = *m*(*ε*) for which

$$m(\varepsilon) = c\_0 \varepsilon^{-1/2} + \theta\_0 + b \varepsilon^{-1/4} + \theta\_1. \tag{104}$$

where *θ*<sup>0</sup> = *θc*<sup>0</sup> , *b* is arbitrarily fixed, and *θ*<sup>1</sup> = *θ*(*b*,*ε*) ∈ [0, 1) complements the value of *bε*−1/4 to an integer. The roots *λ* = *λm*(*ε*) of Equation (98) with numbers *m*(*ε*) satisfy the asymptotic equalities

$$
\lambda = i\hbar \varepsilon\_0 \varepsilon^{-1/2} + i\hbar \omega b \varepsilon^{-1/4} + i\omega (c\_0) + i\varepsilon^{1/4} \Delta b + \varepsilon^{1/2} \lambda\_2 + O(\varepsilon^{3/4}).
$$

Here, the following designations are accepted:

$$\Delta = r h\_2 T\_1^0 \exp\left(-i h\_2 T\_1^0 c\_0\right) + i2c\_0 \left(k\_1 + \frac{1}{2} h\_2^2\right), \quad \text{Si}\Delta = 0,$$

$$\Delta\_2 = -\sigma b^2 - 2c\_0 \theta\_0 \left(k\_1 + \frac{1}{2} h\_2^2\right) + r \exp\left(-i h\_2 T\_1^0 c\_0\right) \left(i\omega(c\_0) T\_1^0 + i h\_2 c\_0 T\_{11}\right),$$

where

$$
\sigma = \left(k\_1 + \frac{1}{2}h\_2^2\right)\left(1 + \frac{1}{2}c\_0^2h\_2^2(T\_1^0)^2\right) + i\omega(c\_0)\frac{1}{2}h\_2^2(T\_1^0)^2 \dots
$$

Another notation is used below. Let Ω = Ω(*ε*) be the set of all such values of *b* for which the values of *<sup>b</sup>ε*−1/4 are integers from −<sup>∞</sup> to <sup>∞</sup>.

Let us introduce the formal series

$$\begin{split} N &= 1 + \varepsilon^{1/4} \Big[ \exp((il\eta\_2 \varepsilon^{-1/2} c\_0 + i\omega(c\_0))t + \\ & (i\varepsilon^{-1/2} c\_0 + \theta\_0) \mathbf{x}) \sum\_{b \in \Omega} \tilde{\varsigma}\_b(\tau) \exp ily + \exp(-(il\eta\_2 \varepsilon^{-1/2} c\_0 + i\omega(c\_0))t - \\ & i(\varepsilon^{-1/2} c\_0 + \theta \mathbf{x}) \mathbf{x} by) \sum\_{b \in \Omega} \tilde{\varsigma}\_b(\tau) \exp(-ily) + \theta \mathbf{x} \Big] + \varepsilon^{1/2} u\_2(t, \tau, \mathbf{x}, y) + \\ & \varepsilon^{3/4} u\_3(t, \tau, \mathbf{x}, y) + \dots, \end{split} \tag{105}$$

where

$$\pi = \varepsilon^{1/2} t, \quad y = \varepsilon^{-1/4} \mathfrak{x} + (\varepsilon^{-1/4} h\_2 + \varepsilon^{1/4} \Delta) t.$$

The dependence on the first, third, and fourth arguments of function *uj*(*t*, *τ*, *x*, *y*) is periodic, with periods 2*π*(*h*2*ε*−1/2*c*<sup>0</sup> + *ω*(*c*0))−1, 2*π*(*ε*−1/2*c*<sup>0</sup> + *θ*0, and 2*π*, respectively. We insert (105) into (79) and perform standard operations. At the second step, we obtain

$$
\mu\_2(t, \tau, x, y) = \mu\_{20}(\tau, y) \left| \tilde{\boldsymbol{\xi}} \right|^2 + \mu\_{21}(t, \tau, x, y) \tilde{\boldsymbol{\xi}}^2 + \vec{\boldsymbol{\nu}}\_{21}(t, \tau, x, y) \tilde{\boldsymbol{\xi}}^2
$$

and

$$\begin{array}{rcl} \mu\_{20}(\mathbf{r}, \mathbf{x}) &=& 2(\cos \mathbf{x}\_0) |\xi|^2, \\ \mu\_{21} &=& -r \exp(-2i\mathbf{x}\_0) \left[ 2i\omega (\mathbf{c}\_0) + r \exp(2i\mathbf{x}\_0) + \left(k\_1 + \frac{1}{2}h\_2^2\right) 4c\_0^2 \right]^{-1} \times \\ & & \quad \exp\left(2i\left[ (h\_2 \varepsilon^{-1/2} c\_0 + \omega(\mathbf{c}\_0)) t + (\varepsilon^{-1/2} c\_0 + \theta\_0) \mathbf{x} \right] \right). \end{array}$$

At the third step, from the solvability condition of the resulting equation with respect to *u*3, we obtain the equation for determining the unknown amplitude

$$\begin{split} \xi(\tau, y) &= \left( \sum\_{b \in \Omega} \vec{\xi}\_b \exp \, iby \right) \exp \left( i \left[ r\omega(c\_0) T\_1^0 + h\_2 c\_0 T\_{11} \cos \chi\_0 - \right. \\ & \begin{split} 2c\_0 \theta\_0 \left( k\_1 + \frac{1}{2} h\_2^2 \right) \Big| \right), \\ \frac{\partial \xi}{\partial \tau} &= \sigma \frac{\partial^2 \xi}{\partial y^2} + \beta \xi + d |\xi|^2 \xi, \end{split} \tag{106}$$

where

$$\begin{array}{rcl} \beta &=& r h\_2 c\_0 T\_{11} \sin \chi\_0\\ d &=& -r \left\{ 2(\cos \chi\_0)(1 + \exp(-i \chi\_0)) - \\ &\quad \left( \exp(i \chi\_0) + \exp(-2i \chi\_0) \right) r \exp(-2i \chi\_0) \left[ 2i \omega (c\_0) + 1 \right] \right. \\ &\left. r \exp(-2i \chi\_0) + \left( k\_1 + \frac{1}{2} h\_2^2 \right) 4c\_0^2 \right]^{-1} \end{array}$$

We note that there are no boundary conditions for Equation (106). The point is that the function with respect to argument *y* contains an arbitrary set of harmonics. We present one of the variants of strict statements about relations between the solutions of (106) and the solutions of the boundary value problem (79), (80).

**Theorem 14.** *Let Equation* (106) *have an R-periodic with respect to argument y solution ξ*(*τ*, *y*)*. Then, for ε* → 0*, the function*

$$\begin{split} N(t, \mathbf{x}, \varepsilon) &= 1 + \varepsilon^{1/4} \Big[ \tilde{\xi} (\varepsilon^{1/2} t, (\varepsilon^{-1/4} + \theta\_{\mathsf{R}}) \mathbf{x} + (\varepsilon^{-1/4} h\_{2} + \varepsilon^{1/4} \Delta) \mathbf{t}) \times \\ &\quad \exp(i[h\_{2} \varepsilon^{-1/4} \mathbf{c}\_{0} + \omega(\mathbf{c}\_{0})] \mathbf{t} + i[\varepsilon^{-1/2} \mathbf{c}\_{0} + \theta\_{\mathsf{0}}] \mathbf{x}) + \tilde{\xi} (\varepsilon^{1/2} t, (\varepsilon^{-1/4} + \theta\_{\mathsf{R}}) \mathbf{x} + \mathbf{x}) \\ &\quad (\varepsilon^{-1/4} h\_{2} + \varepsilon^{1/4} \Delta) \mathbf{t}) \exp(-i[h\_{2} \varepsilon^{-1/4} \mathbf{c}\_{0} + \omega(\mathbf{c}\_{0})] \mathbf{t} - i[\varepsilon^{-1/2} \mathbf{c}\_{0} + \theta\_{\mathsf{0}}] \mathbf{x})) + \\ &\quad \varepsilon^{1/2} u\_{2}(t, \mathbf{r}, \mathbf{y}, \mathbf{x}) \end{split}$$

*satisfies the boundary value problem* (79)*,* (80) *up to <sup>O</sup>*(*ε*3/4)*, where the expression <sup>θ</sup><sup>R</sup>* = *<sup>θ</sup>R*(*ε*) ∈ [0, 2*π*/*R*) *complements the summand ε*−1/4 *to an integer multiple of* 2*π*/*R.*
