INTRODUCTION

One of the most traditional versions of the Kuramoto–Sivashinsky equation (KSE) is dealt with in this paper [1–3]. The periodic boundary value problem (BVP) for this equation is studied. Several papers have been devoted to this problem [4–10]. The problem of local bifurcations has been considered. Most of these papers were based on reducing the BVP to a finite-dimensional dynamical system. As a rule, one version of the Galerkin method was used for such a reduction (see, for example, [5]). Thereafter, the resulting system of ordinary differential equations was analyzed, and local bifurcations for it were studied. Generally, such an analysis led to the establishment of conditions under which either Andronov–Hopf or Turing–Prigogine bifurcations occur, that is, the conditions were established under which cycles or spatially inhomogeneous solutions can be found for a finite-dimensional dynamical system.

An analysis of this problem without using the Galerkin method or other methods for reducing the problem to its finite-dimensional analog revealed the possibility of bifurcation of a two-dimensional local attractor the solutions on which are periodic functions of time, and all these solutions are unstable in the sense of Lyapunov’s classical definition. These results were obtained based on strict mathematically grounded methods for analyzing infinite-dimensional dynamical systems. The possibility of such a bifurcation was also noted earlier when studying other dynamical systems [11, 12]. The results presented below formed the basis for the authors' report at the conference “New Trends in Nonlinear Dynamics,” which took place from October 5 through October 7, 2017 [13].

1 STATEMENT OF THE MATHEMATICAL PROBLEM

We consider periodic BVP

$${{w}_{\tau }} + \alpha {{w}_{{\xi \xi \xi \xi }}} + \beta {{w}_{{\xi \xi }}} + 2\gamma w{{w}_{\xi }} = 0,$$
((1))
$$w(\tau ,\xi + 2H) = w(\tau ,\xi ),$$
((2))

where \(\alpha > 0,H > 0,\beta ,\gamma \in R.\) The differential equations in partial derivatives (1) is usually called the Kuramoto–Sivashinsky equation (see, for example, [1–3]). Changes

$$\xi = \frac{{xH}}{\pi },\,\,\tau = \frac{1}{\alpha }{{\left( {\frac{H}{\pi }} \right)}^{4}}t,\,\,w = \left( {\frac{\alpha }{\gamma }{{{\left( {\frac{\pi }{H}} \right)}}^{3}}} \right)u$$

make it possible to rewrite the BVP in a normalized form, thus reducing the number of parameters in the equation. From now on, we will study BVP

$${{u}_{t}} + {{u}_{{xxxx}}} + b{{u}_{{xx}}} + {{({{u}^{2}})}_{x}} = 0,$$
((3))
$$u(t,x + 2\pi ) = u(t,x),$$
((4))

where \(b = \tfrac{\beta }{\alpha }{{\left( {\tfrac{H}{\pi }} \right)}^{2}}.\) It is assumed in this equation that \({{({{u}^{2}})}_{x}} = 2u{{u}_{x}}.\)

We note certain properties typical of the solutions to BVPs (3) and (4).

First, \(u(t,x) = {\text{const}}\) is a solution to this problem. Second, if \(u(t,x)\) is its solution, then

$${{M}_{0}}(u) = \frac{1}{{2\pi }}\int\limits_{ - \pi }^\pi u(t,x)dx = c\,\,(c \in R).$$

Indeed, we have \(\frac{1}{{2\pi }}\int_{ - \pi }^\pi {{{u}_{{xxx}}}dx} = 0,\,\frac{1}{{2\pi }}\int_{ - \pi }^\pi {{{u}_{{xx}}}dx} = 0,\,\frac{1}{{2\pi }}\int_{ - \pi }^\pi {{{{({{u}^{2}})}}_{x}}dx} = 0.\)

Therefore, relations

$$\frac{1}{{2\pi }}\int\limits_{ - \pi }^\pi {{u}_{t}}dx = \frac{d}{{dt}}\frac{1}{{2\pi }}\int\limits_{ - \pi }^\pi udx = 0,\,\,\,{\text{hold, that is,}}\,\,\frac{1}{{2\pi }}\int\limits_{ - \pi }^\pi udx = c.$$

In BVPs (3) and (4), we put

$$u(t,x) = c + {v}(t,x),\,\,\,c \in R,\,\,\,\frac{1}{{2\pi }}\int\limits_{ - \pi }^\pi {v}(t,x)dx = 0.$$

This change allows us to reduce BVPs (3) and (4) to auxiliary BVP

$${{{v}}_{t}} = A{v} - {{({{{v}}^{2}})}_{x}},$$
((5))
$${v}(t,x + 2\pi ) = {v}(t,x),{{M}_{0}}({v}) = 0.$$
((6))

In ВVPs (5) and (6), the linear differential operator (LDO) \(A = A(c)\) is defined by equality

$$A(c){v} = - {{{v}}_{{xxxx}}} - b{{{v}}_{{xx}}} - 2c{{{v}}_{x}}$$

and depends on the parameter \(c \in R.\) We emphasize that the value of \(c\) is arbitrary. LDO \(A(c),\) which is defined on sufficiently smooth functions \(p(x)\) satisfying conditions (6), is the generating operator of an analytic semigroup of linear bounded operators in Hilbert space \({{H}_{0}}:f(x) \in {{H}_{0}}\) if

$$(1)\,f(x + 2\pi ) = f(x),\,\,\,(2)\,f(x) \in {{L}_{2}}( - \pi ,\pi ),\,\,\,(3)\,{{M}_{0}}(f) = 0.$$

We let \({{H}_{k}}\) denote the Hilbert space consisting of \(2\pi \)-periodic functions \(f(x)\) having generalized derivatives up to order \(k\) that belong to space \({{L}_{2}}( - \pi ,\pi ).\) A norm in \({{H}_{k}}\) can be defined by equality

$$\left\| f \right\|_{{{{H}_{k}}}}^{2} = \left\| f \right\|_{{{{L}_{2}}}}^{2} + \left\| {f{\kern 1pt} {{'}}} \right\|_{{{{L}_{2}}}}^{2} + \ldots + \left\| {{{f}^{{(k)}}}} \right\|_{{{{L}_{2}}}}^{2},\,\,\,\left\| f \right\|_{{{{L}_{2}}}}^{2} = \int\limits_{ - \pi }^\pi {{f}^{2}}(x)dx.$$

Finally, \({{H}_{{k,0}}} \subset {{H}_{k}}\) and consists of functions \(f(x) \in {{H}_{k}}\) such that \({{M}_{0}}(f) = 0.\) If BVP (3), (4) is supplemented with initial condition

$$u(0,x) = f(x) \in {{H}_{{4,0}}},$$
((7))

combined (initial boundary value) problem (5)–(7) will be locally correctly solvable and its solutions form local semiflow

$$f(x) \to {{f}_{t}}(x) \to u(t,x) \in {{H}_{{4,0}}}\,\,\,{\text{for any}}\,\,\,t > 0.$$

Note that BVP (5), (6) has a zero-equilibrium point. In particular, the problems of the behavior of solutions to auxiliary BVP (5), (6) with the initial conditions \(f(x) \in Q(r) \subset {{H}_{{4,0}}}.\) as \(t \to \infty \) will be considered in this study. Here, \(Q(r)\) denotes a ball of radius \(r\) centered at zero of the phase space of solutions to BVP (5), (6) (that is, the ball centered at zero of the Hilbert space \({{H}_{{4,0}}})\)

2 LINEARIZED BOUNDARY VALUE PROBLEM

In this section, we consider a linearized version of BVP (5), (6), that is, BVP

$${{{v}}_{t}} = A{v},\,\,\,{v} = {v}(t,x),$$
((8))
$${v}(t,x + 2\pi ) = {v}(t,x),{{M}_{0}}({v}) = 0,$$
((9))

where LDO \(A = A(c)\) has been defined in the previous section.

We can verify in a standard manner that BVP

$$ - {{p}^{{{\text{(IV)}}}}} - bp{{''}} - 2cp{{'}} = \lambda p,\,\,\,p(x + 2\pi ) = p(x),\,\,\,{{M}_{0}}(p) = 0$$

has nontrivial solutions

$$p(x) = {{p}_{n}}(x) = exp(inx),\,\,\,n = \pm 1, \pm 2, \ldots ,$$

if \({{\lambda }_{n}} = {{\lambda }_{n}}(c) = - {{n}^{4}} + b{{n}^{2}} + i\sigma n,n = \pm 1, \pm 2, \ldots ,\) and \(\sigma = - 2c.\)

We emphasize that \({{\tau }_{n}} = {\text{Re}}{{\lambda }_{n}}(c) = - {{n}^{4}} + b{{n}^{2}}\) and does not depend on \(c.\) In our case, quantity \(c\) plays the role of a parameter, and only \({\text{Im}}{{\lambda }_{n}}(c)\) depends on \(c\). We also note that \(\mathop {lim}\limits_{n \to \infty } {{\tau }_{n}} = - \infty .\) The family of eigenfunctions of the LDO under consideration \(exp(inx),n = \pm 1, \pm 2, \pm 3, \ldots \), forms a complete orthogonal system in separable Hilbert space \({{H}_{0}}.\) These remarks imply the assertion.

Lemma 1. The solutions to BVP (8), (9) are asymptotically stable if \(b < 1\) and unstable if \(b > 1.\) If \(b = 1,\) the solutions are stable.

The zero solution to nonlinear BVP (5), (6) is asymptotically stable for \(b < 1\), and it is unstable for \(b > 1.\) With \(b = 1\), the critical case concerning the stability of the zero solution to the BVP is realized. For this \(b\), we have \({{\lambda }_{ \pm }}(c) = \pm i\sigma ,\sigma = - 2c.\) This pair of eigenvalues is associated with eigenfunctions \(exp( \pm ix).\) For the other eigenvalues of LDO \(A(c)\), inequalities \({\text{Re}}{{\lambda }_{n}}(c) \leqslant - 12,n = \pm 2, \pm 3, \ldots ,\) hold.

3 THE MAIN RESULT

We consider nonlinear BVPs (3), (4) and (5), (6) with \(b = 1 + \gamma \varepsilon .\) In this section, it is convenient to assume that \(\gamma = \tfrac{1}{{12}}.\) This choice of \(\gamma \) is due to the convenience when stating the main result.

Theorem 1. There exists positive constant \({{\varepsilon }_{0}} > 0\) such that for all \(\varepsilon \in (0,{{\varepsilon }_{0}})\) BVP (5), (6) for any \(c \in R\) has unique stable limit cycle \(l(c,\varepsilon )\) belonging to sufficiently small neighborhood \(Q(r).\) All solutions with sufficiently small initial conditions approach \(l(c,\varepsilon )\) in the sense of the norm of the phase space with an exponential rate. For solutions forming this cycle, we have asymptotic formula

$${{{v}}_{p}}(t,x,c,\varepsilon ) = {{\varepsilon }^{{1/2}}}sin(x + \sigma t + {{\varphi }_{0}}) - \frac{\varepsilon }{{12}}sin(2x + 2\sigma t + 2{{\varphi }_{0}}) + \frac{{{{\varepsilon }^{{3/2}}}}}{{288}}sin(3x + 3\sigma t + 3{{\varphi }_{0}}) + o({{\varepsilon }^{{3/2}}}),$$
((10))

where \(\sigma = - 2c,{{\varphi }_{0}} \in R.\)

With \(c = 0\), we obtain not a cycle but a one-dimensional invariant manifold filled with spatially inhomogeneous equilibrium points of BVP (5), (6). In the case of general position \((c \ne 0)\), the Andronov–Hopf bifurcation theorem is applicable. When analyzing BVP (5), (6), we should take into account its characteristic feature lying in the fact that Eq. (5) depends on parameter \(c.\) This problem has a cycle for any \(c \ne 0\), and only the period depends on \(c\): \(T = T(c) = \tfrac{\pi }{{\left| c \right|}}.\) Theorem 1 can be proven in a rather standard manner. The derivation of (10), as well as the above-mentioned fact about the nature of the dependence on parameter \(c,\) require detailed explanation. This will be done in the next section (that is, in Section 4).

Obviously, equality

$${{u}_{p}}(t,x,c,{{\varphi }_{0}},\varepsilon ) = c + {{{v}}_{p}}(t,x,c,{{\varphi }_{0}},\varepsilon )$$
((11))

defines a two-parameter family of periodic solutions to BVP (3), (4) if \(b = 1 + \tfrac{\varepsilon }{{12}}.\) The two-parameter family of solutions (11) forms two-dimensional invariant manifold \({{V}_{2}}(\varepsilon ),\) which geometrically is the direct product of cycle \(l(c,\varepsilon )\) and a line. All other solutions from its small neighborhood approach \({{V}_{2}}(\varepsilon )\) at an exponential rate with an exponent that does not depend on \(c\) (recall that \({\text{Re}}{{\lambda }_{n}}(c)\) does not depend on \(c\)). Therefore, \({{V}_{2}}(\varepsilon )\) is a local attractor for solutions to BVP (3), (4).

On the other hand, all solutions \({{u}_{p}}(t,x,c,\varepsilon ) \in {{V}_{2}}(\varepsilon )\) are individually unstable in the norm of space \({{H}_{4}}.\) To see this, we consider two distinct solutions from the family of periodic solutions (11), that is, \({{u}_{p}}(t,x,{{c}_{1}},{{\varphi }_{1}},\varepsilon )\) and \({{u}_{p}}(t,x,{{c}_{2}},{{\varphi }_{2}},\varepsilon )({{c}_{1}} \ne {{c}_{2}})\), and single out the leading parts in the asymptotic representations for these two solutions:

$$\begin{array}{*{20}{c}} {{{u}_{p}}(t,x,{{c}_{1}},{{\varphi }_{1}},\varepsilon ) = {{w}_{p}}(t,x,{{c}_{1}},{{\varphi }_{1}},\varepsilon ) + o(\varepsilon ),\,\,\,{{u}_{p}}(t,x,{{c}_{2}},{{\varphi }_{2}},\varepsilon ) = {{w}_{p}}(t,x,{{c}_{2}},{{\varphi }_{2}},\varepsilon ) + o(\varepsilon ),} \\ {{{w}_{1}}(t,x) = {{w}_{p}}(t,x,{{c}_{1}},{{\varphi }_{1}},\varepsilon ) = {{c}_{1}} + {{\varepsilon }^{{1/2}}}sin(x + {{\sigma }_{1}}t + {{\varphi }_{1}}),\,\,\,{{\sigma }_{1}} = - 2{{c}_{1}},} \\ {{{w}_{2}}(t,x) = {{w}_{p}}(t,x,{{c}_{2}},{{\varphi }_{2}},\varepsilon ) = {{c}_{2}} + {{\varepsilon }^{{1/2}}}sin(x + {{\sigma }_{2}}t + {{\varphi }_{2}}),\,\,\,{{\sigma }_{2}} = - 2{{c}_{2}}.} \end{array}$$

Put \(\Delta w = {{w}_{1}} - {{w}_{2}}.\) Then, we have

$$\left\| {\Delta w} \right\|_{{{{H}_{4}}}}^{2} = \left\| {\Delta w} \right\|_{{{{L}_{2}}}}^{2} + \left\| {{{{(\Delta w)}}_{x}}} \right\|_{{{{L}_{2}}}}^{2} + \left\| {{{{(\Delta w)}}_{{xx}}}} \right\|_{{{{L}_{2}}}}^{2} + \left\| {{{{(\Delta w)}}_{{xxx}}}} \right\|_{{{{L}_{2}}}}^{2} + \left\| {{{{(\Delta w)}}_{{xxxx}}}} \right\|_{{{{L}_{2}}}}^{2}.$$

Direct calculations show that

$$\left\| {\Delta w} \right\|_{{{{L}_{2}}}}^{2} = 2\pi {{({{c}_{1}} - {{c}_{2}})}^{2}} + 4\varepsilon \pi si{{n}^{2}}\left( {\frac{{({{\sigma }_{1}} - {{\sigma }_{2}})t + {{\varphi }_{2}} - {{\varphi }_{1}}}}{2}} \right).$$

Calculations for other terms are similar; finally, we arrive at

$$\left\| {\Delta w} \right\|_{{{{H}_{4}}}}^{2} = 2\pi {{({{c}_{1}} - {{c}_{2}})}^{2}} + 20\varepsilon \pi si{{n}^{2}}\left( {({{c}_{2}} - {{c}_{1}})t + \frac{{{{\varphi }_{2}} - {{\varphi }_{1}}}}{2}} \right).$$

Without loss of generality, we can assume that \({{c}_{2}} > {{c}_{1}}({{c}_{2}} - {{c}_{1}} > 0).\) We put

$${{t}_{k}} = \frac{{\left( {\tfrac{\pi }{2} + \pi k - \tfrac{{{{\varphi }_{2}} - {{\varphi }_{1}}}}{2}} \right)}}{{{{c}_{2}} - {{c}_{1}}}},\,\,\,\mathop {lim}\limits_{k \to \infty } {{t}_{k}} = \infty .$$

With \({{t}_{k}}\) chosen in this way, inequality

$$\left\| {\Delta w} \right\|_{{{{H}_{4}}}}^{2} \geqslant 20\varepsilon \pi \,\,\,{\text{or}}\,\,{{\left\| w \right\|}_{{{{H}_{4}}}}} \geqslant 2\sqrt {5\pi \varepsilon } $$

holds. Therefore, for sufficiently small \(\varepsilon (\varepsilon \in (0,{{\varepsilon }_{0}}))\), inequality

$${{\left\| {{{u}_{p}}({{t}_{k}},x,{{c}_{1}},{{\varphi }_{1}},\varepsilon ) - {{u}_{p}}({{t}_{k}},x,{{c}_{2}},{{\varphi }_{2}},\varepsilon )} \right\|}_{{{{H}_{4}}}}} \geqslant 2\sqrt {5\pi \varepsilon } $$

is valid. On the other hand, with sufficiently small \(\Delta c = {{c}_{2}} - {{c}_{1}}\) and \(\Delta \varphi = {{\varphi }_{1}} - {{\varphi }_{2}}\), we have

$${{\left\| {{{u}_{p}}({{t}_{k}},x,{{c}_{1}},{{\varphi }_{1}},\varepsilon ) - {{u}_{p}}({{t}_{k}},x,{{c}_{2}},{{\varphi }_{2}},\varepsilon )} \right\|}_{{{{H}_{4}}}}} < \delta ,$$

where \(\delta \) is an arbitrarily small positive constant. The last remark proves that any solution from family (11) is unstable.

4 AUXILIARY BOUNDARY VALUE PROBLEM

In this section, we consider another auxiliary BVP

$${{w}_{t}} = B(\varepsilon )w - {{({{w}^{2}})}_{y}},\,\,\,w = w(t,y),$$
((12))
$$w(t,0) = w(t,\pi ) = {{w}_{{yy}}}(t,0) = {{w}_{{yy}}}(t,\pi ) = 0,$$
((13))

which is also of independent interest. Here, \(y \in [0,\pi ]\) and the LDO is \(B(\varepsilon )g(y) = - {{g}^{{{\text{(IV)}}}}}(y) - \left( {1 + \tfrac{\varepsilon }{{12}}} \right)g{{''}}(y),\) whose domain of definition contains sufficiently smooth functions \(g(y)\) satisfying boundary conditions

$$g(0) = g(\pi ) = g{{''}}(0) = g{{''}}(\pi ) = 0.$$

The spectrum of LDO \(B(\varepsilon )\) consists of a countable set of eigenvalues \({{\lambda }_{n}}(\varepsilon ) = - {{n}^{4}} + \left( {1 + \tfrac{\varepsilon }{{12}}} \right){{n}^{2}},\) which are associated with eigenfunctions \({{e}_{n}}(y) = sinny.\) All eigenvalues of this LDO are real and of multiplicity one. In particular, \({{\lambda }_{1}}(\varepsilon ) = \tfrac{\varepsilon }{{12}},\) that is, \({{\lambda }_{1}}(0) = 0.\) Nonlinear BVP (12), (13) has a zero equilibrium point for which a case close to the critical case of the simple zero eigenvalue is realized. In a neighborhood of the zero solution to BVP (12), (13), there exists one-dimensional invariant manifold \({{M}_{1}}(\varepsilon ).\) Other solutions in the vicinity of the zero solution to BVP (12), (13), approach \({{M}_{1}}(\varepsilon )\) with time at an exponential rate. The dynamics of solutions on \({{M}_{1}}(\varepsilon )\) is governed by scalar first-order equation (the normal form (NF))

$$\dot {z} = \varepsilon \Psi (z,\varepsilon ) = \varepsilon \psi (z) + o(\varepsilon ),\,\,\,z = z(t).$$
((14))

It is advisable to seek solutions on \({{M}_{1}}(\varepsilon )\) in form

$$w(t,y,\varepsilon ) = {{\varepsilon }^{{1/2}}}{{w}_{1}}(z,y) + \varepsilon {{w}_{2}}(z,y) + {{\varepsilon }^{{3/2}}}{{w}_{3}}(z,y) + o({{\varepsilon }^{{3/2}}}).$$
((15))

Here, \(z = z(t)\) are solutions to NF (14), \({{w}_{1}}(z,y) = zsiny,\) and functions \({{w}_{k}}(y,z),k = 2,3,4, \ldots \), belong to the following class of functions: \(f(z,y) \in W\) if

(1) for a fixed \(z\)\(f(z,y) \in W_{2}^{4}\left[ {0,\pi } \right]\).

(2) The function has continuous partial derivatives with respect to \(z\) and \({{f}_{k}}(0,y) = 0\).

(3) The function satisfies hinge support boundary conditions (13).

(4) Equalities \(\frac{2}{\pi }\int_0^\pi {f(z,y)sinydy} = 0\) hold.

Upon substituting sum (15) into BVP (12), (13) and equating the terms at the same powers of \(\varepsilon (\varepsilon ,{{\varepsilon }^{{3/2}}}, \ldots )\), we obtain inhomogeneous BVPs to determine the terms of (15):

$${{B}_{0}}{{w}_{2}} = {{\Phi }_{2}}(z,y),\,\,\,{{w}_{2}}(z,0) = {{w}_{2}}(z,\pi ) = {{w}_{{2yy}}}(z,0) = {{w}_{{2yy}}}(z,\pi ) = 0,$$
((16))
$${{B}_{0}}{{w}_{3}} = {{\Phi }_{3}}(z,y),\,\,\,{{w}_{3}}(z,0) = {{w}_{3}}(z,\pi ) = {{w}_{{3yy}}}(z,0) = {{w}_{{3yy}}}(z,\pi ) = 0,$$
((17))

where the LDO is \({{B}_{0}}{{w}_{k}} = - {{w}_{{yyyy}}} - {{w}_{{yy}}},k = 2,3.\) The derivatives of \(z(t)\) should be calculated using Eq. (14), that is, the NF. Therefore, we have

$${{\Phi }_{2}}(z,y) = 2{{(w_{1}^{2})}_{y}} = {{z}^{2}}sin2y,\,\,\,{{\Phi }_{3}}(z,y) = 2{{({{w}_{1}}{{w}_{2}})}_{y}} + \frac{1}{{12}}{{({{w}_{1}})}_{{yy}}} + \psi (z)siny.$$

An analysis of the solvability of BVP (16), (17) has shown that

$${{w}_{2}} = - \frac{1}{{12}}{{z}^{2}}sin2y,\,\,\,{{w}_{3}} = \frac{1}{{288}}{{z}^{3}}sin3y,\,\,\,\psi (z) = \frac{1}{{12}}(z - {{z}^{3}}).$$

We now consider a truncated version of NF (14), namely, ordinary differential equation

$$\dot {z} = \frac{\varepsilon }{{12}}(z - {{z}^{3}}),$$

which has three equilibrium points: \({{S}_{0}}:z = 0;\,\,{{S}_{ + }}:z = 1;\,\,{{S}_{ - }}:z = - 1.\) Zero equilibrium point \({{S}_{0}}\) is unstable, while nontrivial equilibrium points \({{S}_{ + }}\) and \({{S}_{ - }}\) are asymptotically stable. The following assertion is true.

Theorem 2. There exists \({{\varepsilon }_{0}} > 0\) such that for all \(\varepsilon \in (0,{{\varepsilon }_{0}})\) BVP (12), (13) has two asymptotically stable equilibrium points

$$\begin{array}{*{20}{c}} {{{S}_{ + }}:{{w}_{ + }}(y,\varepsilon ) = {{\varepsilon }^{{1/2}}}siny - {{q}_{1}}\varepsilon sin2y + {{q}_{2}}{{\varepsilon }^{{3/2}}}sin3y + o({{\varepsilon }^{{3/2}}}),} \\ {{{S}_{ - }}:{{w}_{ - }}(y,\varepsilon ) = - {{\varepsilon }^{{1/2}}}siny - {{q}_{1}}\varepsilon sin2y - {{q}_{2}}{{\varepsilon }^{{3/2}}}sin3y + o({{\varepsilon }^{{3/2}}}),} \end{array}$$

where \({{q}_{1}} = \tfrac{1}{{12}}\,{\text{and}}\,{{q}_{2}} = \tfrac{1}{{288}}.\)

Note that equality

$${{w}_{ - }}(y,\varepsilon ) = - {{w}_{ + }}(\pi - y,\varepsilon )$$

remains valid; that is, physically, we have the same equilibrium point but in another system of coordinates. Therefore, equilibrium point \({{S}_{ + }}\) is dealt with in the sequel.

We extend function \({{w}_{ + }}(y,\varepsilon )\) to interval \([ - \pi ,0]\) to be an odd function and then to the entire number line to be a periodic function with a period of \(2\pi .\) Resulting function \({{{v}}_{ + }}(y,\varepsilon )\) is an equilibrium point of BVP

$${{{v}}_{t}} + {{{v}}_{{yyyy}}} + \left( {1 + \frac{1}{{12}}\varepsilon } \right){{{v}}_{{yy}}} + {{({{{v}}^{2}})}_{y}} = 0,$$
((18))
$${v}(t,y + 2\pi ) = {v}(t,y),{{M}_{0}}({v}) = 0.$$
((19))

When verifying this assertion, we use the fact that Eq. (18) is invariant for odd functions. Indeed, if \({v}(t,y)\) is an odd function of variable \(y,\) then all terms in the left-hand side of Eq. (18) preserve this property.

We now consider family of functions \({{{v}}_{p}}(t,x,c,\varepsilon ) = {{{v}}_{ + }}(x + \sigma t + {{\varphi }_{0}},\varepsilon ),\) where \(\sigma = - 2c;\) for each \(c\) and \({{\varphi }_{0}}\), this family gives a periodic solution to auxiliary BVP (5), (6) with the preservation of stability of equilibrium point \({{S}_{ + }}\) in the sense that periodic solution \({{{v}}_{p}}(t,x,c,\varepsilon )\) is orbitally asymptotically stable. The last fragment of this section completes the proof of formula (10). We also note that equilibrium point \({{w}_{ - }}(y,\varepsilon )\) undoubtedly leads to the same family of periodic solutions. Equilibrium points \({{{v}}_{ + }}(y,\varepsilon )\) and \({{{v}}_{ - }}(y,z)\) of BVP (18), (19) are associated with different representatives of the family of periodic solutions (10). They differ in phase value.

In conclusion, we note that solution \({{{v}}_{p}}(t,x,c,\varepsilon )\) has a period of \(T = 2\pi /\left| c \right|\) if \(c \ne 0\) and quantity \(T\) assumes any value in \({{R}_{ + }}.\)