As mentioned in the introduction, we study in this manuscript attractor bifurcation of the EFK equation under the periodic boundary condition. Then we have the following bifurcation theorem.

**Theorem 4.1** *For problem* (1.1),

*if* $2{a}_{2}^{2}+45\mu {a}_{3}+9\alpha {a}_{3}<0$ *is satisfied*,

*then the following assertions hold true*:

- (1)
*If* $\lambda \le \mu +\alpha $, *the steady state* $u=0$ *is locally asymptotically stable*.

- (2)
*If* $\lambda >\mu +\alpha $, *system* (1.1) *bifurcates from the trivial solution* $u=0$ *to an attractor* ${\mathrm{\Sigma}}_{\lambda}$.

- (3)
${\mathrm{\Sigma}}_{\lambda}$ *is homeomorphic to* ${S}^{1}$ *and consists of exactly one cycle of steady solutions of* (1.1).

- (4)
${\mathrm{\Sigma}}_{\lambda}$
*can be expressed as*
${\mathrm{\Sigma}}_{\lambda}=\{\tilde{x}cos(x+\theta )+o(|\tilde{x}|)|\theta \in \mathbb{R}\},$

*where* $\tilde{x}=\sqrt{\frac{4(16\mu +4\alpha -\lambda )(\mu +\alpha -\lambda )}{3{a}_{3}(16\mu +4\alpha -\lambda )+2{a}_{2}^{2}}}$ (${a}_{2}\ne 0$), *or* $\tilde{x}=\sqrt{\frac{4(\mu +\alpha -\lambda )}{3{a}_{3}}}$ (${a}_{2}=0$), *and* $\mu +\alpha <\lambda <\mu +\alpha +\u03f5$, *ϵ* *is sufficiently small*.

*Proof of Theorem 4.1* We shall prove Theorem 4.1 in four steps.

Step 1. In this step, we study the eigenvalue problem of the linearized equation of (3.2) and find the eigenvectors and the critical value of *λ*.

Consider the eigenvalue problem of the linear equation,

${L}_{\lambda}u=\beta u.$

(4.1)

It is not difficult to find that the eigenvalues and the normalized eigenvectors of (4.1) are

$\{\begin{array}{c}{\beta}_{2k-1}={\beta}_{2k}=\lambda -\mu {k}^{4}-\alpha {k}^{2},\phantom{\rule{1em}{0ex}}k=1,2,\dots ,\hfill \\ {e}_{2k-1}=\frac{sinkx}{\sqrt{\pi}},\phantom{\rule{2em}{0ex}}{e}_{2k}=\frac{coskx}{\sqrt{\pi}},\hfill \end{array}$

(4.2)

under condition that we get the principle of exchange of stabilities

$\begin{array}{c}{\beta}_{1}(\lambda )={\beta}_{2}(\lambda )\{\begin{array}{cc}<0,\hfill & \lambda <\mu +\alpha ,\hfill \\ =0,\hfill & \lambda =\mu +\alpha ,\hfill \\ >0,\hfill & \lambda >\mu +\alpha ,\hfill \end{array}\hfill \\ {\beta}_{j}(\mu +\alpha )<0,\phantom{\rule{1em}{0ex}}j\ge 3.\hfill \end{array}$

Step 2. We verify that for any $\lambda \in \mathbb{R}$, operator ${L}_{\lambda}+G$ satisfies conditions (2.2) and (2.3).

Thanks to the results in [9, 18, 19], we know that the operator ${L}_{\lambda}:{H}_{1}\to H$ is a sectorial operator which implies that condition (2.2) holds true.

It is easy to get the following inequality:

$\begin{array}{rcl}{\parallel G(u)\parallel}_{H}^{2}& =& {\int}_{0}^{2\pi}{|g(u)|}^{2}\phantom{\rule{0.2em}{0ex}}dx\\ \le & C{\int}_{0}^{2\pi}\left(\sum _{k=2}^{p}{|u|}^{2k}\right)\phantom{\rule{0.2em}{0ex}}dx\\ \le & C\sum _{k=2}^{p}{\parallel u\parallel}_{{L}^{2k}(0,2\pi )}^{2k}\\ \le & C\sum _{k=2}^{p}{\parallel u\parallel}_{{H}_{\frac{1}{2}}}^{2k},\end{array}$

which implies that

$G(u)=o({\parallel u\parallel}_{{H}_{\frac{1}{2}}})$, where

${H}_{\frac{1}{2}}=\{u\in {H}^{2}(0,2\pi )|{\int}_{0}^{2\pi}u\phantom{\rule{0.2em}{0ex}}dx=0,u(x+2\pi )=u(x)\},$

then condition (2.3) holds true.

Step 3. In this part, we prove the existence of attractor bifurcation and analyze the topological structure of the attractor ${\mathrm{\Sigma}}_{\lambda}$.

Let

${E}_{1}^{\lambda}={E}_{0}=span\{{e}_{1},{e}_{2}\}$,

${E}_{2}^{\lambda}={E}_{0}^{\mathrm{\perp}}$. Let Φ be the center manifold function, in the neighborhood of

$(u,\lambda )=(0,\mu +\alpha )$, we have

where $y={x}_{1}{e}_{1}+{x}_{2}{e}_{2}$.

Then the reduction equations of (3.2) are as follows:

$\{\begin{array}{c}\frac{d{x}_{1}}{dt}=(\lambda -\mu -\alpha ){x}_{1}+\u3008G(u),{e}_{1}\u3009,\hfill \\ \frac{d{x}_{2}}{dt}=(\lambda -\mu -\alpha ){x}_{2}+\u3008G(u),{e}_{2}\u3009.\hfill \end{array}$

(4.3)

To get the exact form of the reduction equations, we need to obtain the expression of $\u3008G(u),{e}_{1}\u3009$ and $\u3008G(u),{e}_{2}\u3009$.

Let

${G}_{2}:{H}_{1}\times {H}_{1}\to H$ and

${G}_{3}:{H}_{1}\times {H}_{1}\times {H}_{1}\to H$ be the bilinear and trilinear operators of

*G* respectively,

*i.e.*,

$\begin{array}{c}{G}_{2}({u}_{1},{u}_{2})={a}_{2}{u}_{1}{u}_{2},\hfill \\ {G}_{3}({u}_{1},{u}_{2},{u}_{3})={a}_{3}{u}_{1}{u}_{2}{u}_{3}.\hfill \end{array}$

Since

$\u3008{G}_{2}(y,y),{e}_{1}\u3009=\u3008{G}_{2}(y,y),{e}_{2}\u3009=0,$

the first order approximation of (4.3) does not work. Now, we shall find out the second order approximation of (4.3). And the most important of all is to obtain the approximation expression of the center manifold function.

By direct calculation, we have

$\u3008{G}_{2}(y,y),{e}_{k}\u3009=\{\begin{array}{cc}\frac{{a}_{2}}{\sqrt{\pi}}{x}_{1}{x}_{2},\hfill & k=3,\hfill \\ \frac{{a}_{2}}{2\sqrt{\pi}}{x}_{2}^{2}-\frac{{a}_{2}}{2\sqrt{\pi}}{x}_{1}^{2},\hfill & k=4,\hfill \\ 0,\hfill & k\ne 3,4.\hfill \end{array}$

According to the formula of Theorem 3.8 in [

15] (or Remark 4.1), the center manifold function Φ, in the neighborhood of

$(u,\lambda )=(0,\mu +\alpha )$, can be expressed as

$\begin{array}{rcl}\mathrm{\Phi}(y)& =& -\sum _{k=3}^{\mathrm{\infty}}{\beta}_{k}^{-1}\u3008{G}_{2}(y,y),{e}_{k}\u3009{e}_{k}+O\left(({|{\beta}_{1}|}^{2}+{|{\beta}_{2}|}^{2}){|y|}^{2}\right)+o\left({|y|}^{2}\right)\\ =& -{(\lambda -16\mu -4\alpha )}^{-1}\frac{{a}_{2}}{2\sqrt{\pi}}(2{x}_{1}{x}_{2}{e}_{3}+{x}_{2}^{2}{e}_{4}-{x}_{1}^{2}{e}_{4})\\ +O\left({|\lambda -\mu -\alpha |}^{2}({|{x}_{1}|}^{2}+{|{x}_{2}|}^{2})\right)+o({|{x}_{1}|}^{2}+{|{x}_{2}|}^{2}).\end{array}$

In the following, we calculate

$\u3008G(u),{e}_{j}\u3009$,

$j=1,2$.

$\begin{array}{rcl}\u3008G(u),{e}_{j}\u3009& =& \u3008{G}_{2}(y,\mathrm{\Phi}(y)),{e}_{j}\u3009+\u3008{G}_{2}(\mathrm{\Phi}(y),y),{e}_{j}\u3009+\u3008{G}_{3}(y,y,y),{e}_{j}\u3009\\ +O\left({|\lambda -\mu -\alpha |}^{2}({|{x}_{1}|}^{3}+{|{x}_{2}|}^{3})\right)+o({|{x}_{1}|}^{3}+{|{x}_{2}|}^{3}),\phantom{\rule{1em}{0ex}}j=1,2.\end{array}$

By direct calculation, we have

$\begin{array}{c}\u3008{G}_{2}(y,\mathrm{\Phi}(y)),{e}_{1}\u3009\hfill \\ \phantom{\rule{1em}{0ex}}=\u3008{G}_{2}(\mathrm{\Phi}(y),y),{e}_{1}\u3009\hfill \\ \phantom{\rule{1em}{0ex}}=-{(\lambda -16\mu -4\alpha )}^{-1}\frac{{a}_{2}^{2}}{4\pi}{x}_{1}^{3}-{(\lambda -16\mu -4\alpha )}^{-1}\frac{{a}_{2}^{2}}{4\pi}{x}_{1}{x}_{2}^{2}\hfill \\ \phantom{\rule{2em}{0ex}}+O\left({|\lambda -\mu -\alpha |}^{2}({|{x}_{1}|}^{3}+{|{x}_{2}|}^{3})\right)+o({|{x}_{1}|}^{3}+{|{x}_{2}|}^{3}),\hfill \\ \u3008{G}_{2}(y,\mathrm{\Phi}(y)),{e}_{2}\u3009\hfill \\ \phantom{\rule{1em}{0ex}}=\u3008{G}_{2}(\mathrm{\Phi}(y),y),{e}_{2}\u3009\hfill \\ \phantom{\rule{1em}{0ex}}=-{(\lambda -16\mu -4\alpha )}^{-1}\frac{{a}_{2}^{2}}{4\pi}{x}_{2}^{3}-{(\lambda -16\mu -4\alpha )}^{-1}\frac{{a}_{2}^{2}}{4\pi}{x}_{1}^{2}{x}_{2}\hfill \\ \phantom{\rule{2em}{0ex}}+O\left({|\lambda -\mu -\alpha |}^{2}({|{x}_{1}|}^{3}+{|{x}_{2}|}^{3})\right)+o({|{x}_{1}|}^{3}+{|{x}_{2}|}^{3}),\hfill \\ \u3008{G}_{3}(y,y,y),{e}_{1}\u3009=\frac{3{a}_{3}}{4\pi}{x}_{1}^{3}+\frac{3{a}_{3}}{4\pi}{x}_{1}{x}_{2}^{2},\hfill \\ \u3008{G}_{3}(y,y,y),{e}_{2}\u3009=\frac{3{a}_{3}}{4\pi}{x}_{2}^{3}+\frac{3{a}_{3}}{4\pi}{x}_{1}^{2}{x}_{2},\hfill \end{array}$

then we obtain the expression of

$\u3008G(u),{e}_{j}\u3009$,

$j=1,2$.

$\begin{array}{r}\u3008G(u),{e}_{1}\u3009=A{x}_{1}^{3}+A{x}_{1}{x}_{2}^{2}+O\left({|\lambda -\mu -\alpha |}^{2}({|{x}_{1}|}^{3}+{|{x}_{2}|}^{3})\right)+o({|{x}_{1}|}^{3}+{|{x}_{2}|}^{3}),\\ \u3008G(u),{e}_{2}\u3009=A{x}_{1}^{2}{x}_{2}+A{x}_{2}^{3}+O\left({|\lambda -\mu -\alpha |}^{2}({|{x}_{1}|}^{3}+{|{x}_{2}|}^{3})\right)+o({|{x}_{1}|}^{3}+{|{x}_{2}|}^{3}),\end{array}$

(4.4)

where $A=-{(\lambda -16\mu -4\alpha )}^{-1}\frac{{a}_{2}^{2}}{2\pi}+\frac{3{a}_{3}}{4\pi}$.

Putting (4.4) into (4.3), we have the reduction equations

$\{\begin{array}{c}\frac{d{x}_{1}}{dt}=(\lambda -\mu -\alpha ){x}_{1}+A{x}_{1}^{3}+A{x}_{1}{x}_{2}^{2}\hfill \\ \phantom{\frac{d{x}_{1}}{dt}=}+O({|\lambda -\mu -\alpha |}^{2}({|{x}_{1}|}^{3}+{|{x}_{2}|}^{3}))+o({|{x}_{1}|}^{3}+{|{x}_{2}|}^{3}),\hfill \\ \frac{d{x}_{2}}{dt}=(\lambda -\mu -\alpha ){x}_{2}+A{x}_{1}^{2}{x}_{2}+A{x}_{2}^{3}\hfill \\ \phantom{\frac{d{x}_{2}}{dt}=}+O({|\lambda -\mu -\alpha |}^{2}({|{x}_{1}|}^{3}+{|{x}_{2}|}^{3}))+o({|{x}_{1}|}^{3}+{|{x}_{2}|}^{3}).\hfill \end{array}$

(4.5)

For the case of $\lambda <\mu +\alpha $, it is obvious that $u=0$ is locally asymptotically stable. For the case of $\lambda =\mu +\alpha $, if $2{a}_{2}^{2}+(45\mu +9\alpha ){a}_{3}<0$, which implies that $A<0$, then $u=0$ is also locally asymptotically stable. Assertion (1) of Theorem 4.1 is proved.

Since the following equality holds true:

${x}_{1}(A{x}_{1}^{3}+A{x}_{1}{x}_{2}^{2})+{x}_{2}(A{x}_{1}^{2}{x}_{2}+A{x}_{2}^{3})=A{({x}_{1}^{2}+{x}_{2}^{2})}^{2},$

according to Theorems 2.1, 2.2 and Remark 2.1, we can conclude that if $\lambda >\mu +\alpha $, equation (1.1) bifurcates from $u=0$ to an attractor ${\mathrm{\Sigma}}_{\lambda}$, which is homeomorphic to ${S}^{1}$.

Step 4. In the last step, we show that the bifurcated attractor of (3.2) consists of a singularity cycle.

Since the even function space is an invariant subspace of

${L}_{\lambda}+G$ defined by (3.1), we shall consider the bifurcation problem in the even function space and prove that system (1.1) bifurcates from

$(u,\lambda )=(0,\mu +\alpha )$ to two steady solutions. For any function

*v* in the even function space can be expressed as follows:

$v=\sum _{k\ge 1}{x}_{2k}{e}_{2k},$

by the Lyapunov-Schmidt reduction method used in Step 3, we can deduce that the reduction equation of (1.1) is as follows:

$\frac{d{x}_{2}}{dt}=(\lambda -\mu -\alpha ){x}_{2}+A{x}_{2}^{3}+O\left({|\lambda -\mu -\alpha |}^{2}{|{x}_{2}|}^{3}\right)+o\left({|{x}_{2}|}^{3}\right),$

(4.6)

which implies that (1.1) bifurcates from $(u,\lambda )=(0,\mu +\alpha )$ to two steady solutions ${V}_{\lambda}^{\pm}(x,t)=\pm \sqrt{\frac{4(16\mu +4\alpha -\lambda )(\mu +\alpha -\lambda )}{3{a}_{3}(16\mu +4\alpha -\lambda )+2{a}_{2}^{2}}}cosx+\text{h.o.t.}$ in the space of even functions.

Since the solutions of (2.1) are translation invariant,

${V}_{\lambda}^{+}(x,t)\to {V}_{\lambda}^{+}(x+\theta ,t),\phantom{\rule{1em}{0ex}}\mathrm{\forall}\theta \in \mathbb{R},$

the set

$T=\{{V}_{\lambda}^{+}(x+\theta ,t)|\theta \in \mathbb{R}\}$

represents ${S}^{1}$ in ${H}_{1}$, which implies that ${\sum}_{\lambda}$ consists of exactly one circle of steady solutions of (1.1). This completes the proof of Theorem 4.1. □

**Remark 4.1** Suppose that

$\{{e}_{i}\}$, the generalized eigenvectors of

${L}_{\lambda}$, form a basis of

*H* with the dual basis

$\{{e}_{i}^{\ast}\}$ such that

${\u3008{e}_{i},{e}_{j}^{\ast}\u3009}_{H}\{\begin{array}{cc}=0,\hfill & \text{if}i\ne j,\hfill \\ \ne 0,\hfill & \text{if}i=j.\hfill \end{array}$

We have

$\begin{array}{c}v=x+y\in {E}_{1}^{\lambda}\oplus {E}_{2}^{\lambda},\hfill \\ x=\sum _{i=1}^{m}{x}_{i}{e}_{i}\in {E}_{1}^{\lambda},\hfill \\ y=\sum _{i=m+1}^{\mathrm{\infty}}{x}_{i}{e}_{i}\in {E}_{2}^{\lambda}.\hfill \end{array}$

Then near

$\lambda ={\lambda}_{0}$, the center manifold function

$\varphi (x,\lambda )$ in Theorem 3.8 in [

15] can be expressed as follows:

$\varphi (x,\lambda )=\sum _{j=m+1}^{\mathrm{\infty}}{\varphi}_{j}(x,\lambda ){e}_{j}+O(|Re\beta (\lambda )|\cdot {\parallel x\parallel}^{k})+o\left({\parallel x\parallel}^{k}\right),$

(4.7)

where

${\varphi}_{j}(x,\lambda )=-\frac{1}{{\beta}_{j}(\lambda )}{\u3008{G}_{k}(x,\dots ,x),{e}_{j}^{\ast}\u3009}_{H}.$

**Remark 4.2** If $g(s)$ in (1.1) is not a polynomial but a ${C}^{\omega}$ with Taylor’s expansion in $s=0$ as $g(s)={\sum}_{k=2}^{\mathrm{\infty}}{a}_{k}{s}^{k}$; if $2{a}_{2}^{2}+45\mu {a}_{3}+9\alpha {a}_{3}<0$ is satisfied, then the conclusions of Theorem 4.1 also hold true.

**Remark 4.3** If the higher order terms ${\sum}_{k=4}^{p}{a}_{k}{u}^{k}$ in $g(u)$ are omitted, from the proof of Theorem 4.1, it is easy to see that the conclusions of Theorem 4.1 also hold true.