\documentclass[reqno]{amsart}
\usepackage{hyperref}
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\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2015 (2015), No. 294, pp. 1--11.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2015 Texas State University.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2015/294\hfil Nonlinear Ritz approximation]
{Nonlinear Ritz approximation for Fredholm functionals}

\author[M. A. Abdul Hussain \hfil EJDE-2015/294\hfilneg]
{Mudhir A. Abdul Hussain}

\address{Mudhir A. Abdul Hussain
Department of Mathematics,
College of Education for Pure Sciences,
University of Basrah, Basrah,  Iraq}
\email{mud\_abd@yahoo.com}

\thanks{Submitted December 24, 2014. Published November 30, 2015.}
\subjclass[2010]{34K18, 34K10}
\keywords{Bifurcation theory; Lyapunov-Schmidt local method;
\hfill\break\indent nonlinear fourth-order differential equation}

\begin{abstract}
 In this article we use the modify Lyapunov-Schmidt
 reduction to find nonlinear Ritz approximation for a Fredholm
 functional. This functional corresponds to a nonlinear Fredholm 
 operator  defined by a nonlinear fourth-order differential equation.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{definition}[theorem]{Definition}
\allowdisplaybreaks


\section{Introduction}

  Many of the nonlinear problems that appear in Mathematics and 
Physics can be written in the  operator equation form
\begin{equation}
f(u,\lambda)=b, \quad  u\in O\subset X, \; b\in Y, \;
\lambda\in \mathbb{R}^n, \label{e1.1}
\end{equation}
where $f$ is a smooth Fredholm map of index zero and  $X$, $Y$ are Banach
spaces and $O$ is open subset of $X$. For these problems, the
method of reduction to finite dimensional equation,
\begin{equation}
\theta(\xi,\lambda)=\beta, \quad \xi\in M, \; \beta\in N,  \label{e1.2}
\end{equation}
can be used, where $M$ and $N$ are smooth finite dimensional manifolds.

A passage from  \eqref{e1.1} into  \eqref{e1.2} (variant local
scheme of Lyapunov -Schmidt) with the conditions that equation
\eqref{e1.2} has all the topological and analytical properties of
 \eqref{e1.1} (multiplicity, bifurcation
diagram, etc) can be found in \cite{l1,s1,s2,v1}.

Suppose that $f:\Omega \subset E\to F$   is a nonlinear
Fredholm map of index zero. A smooth map
$f:\Omega \subset E\to F$ has variational property,
if there exists a functional $V:\Omega \subset E\to R$ such that
$f=\operatorname{grad}_{H}V$ or equivalently,
$$
\frac{\partial V}{\partial u}(u,\lambda)h
=\langle f(u,\lambda),h\rangle_{H}, \quad \forall  u\in \Omega, \; h\in E,
 $$
 where $\langle \cdot,\cdot \rangle_{H}$ is the scalar product
in Hilbert space $H$. In this case, the solutions of equation
$f(u,\lambda)=0$  are the critical points of functional
$V(u,\lambda)$. Suppose that $f:E \to F$ is a smooth
Fredholm map of index zero, $E$, $F$  are Banach spaces and
$$
\frac{\partial V}{\partial u}(u,\lambda)h
=\langle f(u,\lambda),h\rangle_{H}, \quad h\in E.
$$
where $V$ is a smooth functional on $E$. Also it is assumed that
$E\subset F \subset H$, where $H $is a Hilbert space.
By using a method of finite dimensional reduction
(Local scheme of Lyapunov-Schmidt) the problem
$$
V(u,\lambda) \to \operatorname{extr}\quad  u\in E, \;
 \lambda \in \mathbb{R}^n
 $$
 can be reduced into equivalent problem
 $$
W(\xi,\lambda) \to \operatorname{extr} \quad \xi\in \mathbb{R}^n
$$
The function $W(\xi,\lambda)$  is called key function.

If $N=\operatorname{span}{\{e_1,\dots ,e_{n}}\}$ is a subspace of $E$, where
 $ e_1,\dots , e_{n}$ is an orthonormal set in $H$, then the key
function $W(\xi,\lambda)$ can be defined in the form of
$$
W(\xi,\lambda)=\inf_{u:\langle u,e_i \rangle
=\xi_i\, \forall  i} V(u,\lambda), \quad \xi=(\xi_1,\dots ,\xi_{n}).
$$
The function $W$ has all the topological and analytical properties
of the functional $V$ (multiplicity, bifurcation diagram, etc.)
\cite{s1}.
The study of bifurcation solutions of functional $V$ is
equivalent to the study of bifurcation solutions of key function.
If  $f$  has a variational property, then the equation
$$
\theta(\xi,\lambda)=\operatorname{grad}W(\xi,\lambda)=0
$$
is called bifurcation equation.

Now we formulate one of the most important theorem of bifurcation
analysis \cite{d1}.

\begin{theorem}[\cite{d1}] \label{thm1.1}
If a mapping $\tilde{f}(\cdot,\xi):E\cap
N^\bot\to F\cap N^\bot$ is proper and the condition
$\langle \frac{\partial f}{\partial x}(x)h,h \rangle >0$ is satisfied for
every $(x,h)$ in $E\times ((E\cap N^\bot) \setminus 0)$, then the
marginal mapping $\varphi:\xi\to \sum_{i=1}^{n}\xi_ie_i+h(\xi)$,
(where $h(\xi)$ is defined by equation $\tilde{f}(h,\xi)=0$),
establishes a one-to-one correspondence between critical points of
key function $W(\xi,\lambda)$ and critical points of the (given) functional
$V(u,\lambda)$.
 Moreover, the local singularity rings of the
corresponding functions at the points $\xi$ and $\varphi(\xi)$ are
isomorphic to each other and, if two simple critical points
correspond to each other, then their Morse indices are equal to each other.
\end{theorem}

\begin{definition}[\cite{d1}] \label{def1.1}\rm
The set of all $\lambda$ for which the function $W(\xi,\lambda)$ has
degenerate critical points is called Caustic and denoted by $\Sigma$.
$$
\Sigma={\{\lambda \in R:\frac{\partial W}{\partial \xi}=0,\;
 \frac{\partial^{2} W}{\partial \xi^{2}}=0}\}.
 $$
\end{definition}

It is well known that in the Lyapunov-Schmidt method, the space
$E$ is decomposed into two orthogonal subspaces
and then every element $u \in E$ can be written in the unique form
as a sum of two elements such that the solution of the equation
\eqref{e1.1} consists of the homogeneous solution and the particular
solution. Sapronov and his group \cite{d1,z1} used the complement
solution to find the function $W(\xi,\lambda)$ which denotes the
linear Ritz approximation of the functional $V(u,\lambda)$. The
study of boundary value problems by using Lyapunov-Schmidt
reduction can be found in \cite{a1,a2,a3,a4,a5,d1}. Most of the authors
that work  this way have studied the linear Ritz
approximation of Fredholm functional. A review for the finite
dimensional reduction can be found in
\cite{d1,s1,s2,s3,s4,z1}.
In \cite{a5} the author introduced an example to find nonlinear approximation
of bifurcation solutions of the  fourth-order differential equation
$$
\frac{d^4u}{dx^4} + \alpha \frac{d^2u}{dx^2} + \beta u +u^3 =0
$$

In \cite{a6} the author introduce a general method for finding
nonlinear Ritz approximation of Fredholm functionals. To the best
of our information the method is new. In this paper we find the
nonlinear Ritz approximation of a functional
$V(u,\lambda)$ which denotes the potential of the nonlinear
operator
$$
f(u,\lambda)=\frac{d^4u}{dx^4} + \lambda \frac{d^2u}{dx^2} +
u+u^2+ u^3.
$$

\section{Modified Lyapunov-Schmidt reduction}

 Consider the nonlinear Fredholm operator of
index zero $f:E\to F$ defined by
\begin{equation}
f(u,\lambda)=0, \quad  \lambda \in \mathbb{R}^n , \;   u\in
\Omega\subset E  \label{e2.1}
\end{equation}
where $E$, $F$ are real Banach spaces and $\Omega$ is an open subset of
$E$. Assume that the operator $f$ has a variational property, i.e,
there exists a functional $V:\Omega \subset E\to R$ such
that $f=\operatorname{grad}_{H}V$ where $\Omega$ is a bounded domain. The
operator $f$ can be written as
$$
f(u,\lambda)=Au+Nu=0,
$$
where $A=\frac{\partial f}{\partial u}(u_0,\lambda)$ is a linear
continuous Fredholm operator,
$\frac{\partial f}{\partial u}(u_0,\lambda)$ the Frechet derivative
of the operator $f$ at the point $u_0$ and $N$ the nonlinear operator.
In this article we consider the operator $A$ as a differential operator.
By using Lyapunov-Schmidt reduction, the decomposition is obtained below
 $$
E=M \oplus M^{\bot}, \quad
F=\tilde{M} \oplus \tilde{M}^{\bot}
$$
 where $M=\ker A$ is the null
space of the operator $A$, $\dim M=\dim \tilde{M}=n$ and
$M^{\bot}$, $\tilde{M}^{\bot}$ are the orthogonal complements of the subspaces
$M$ and $\tilde{M}$ respectively. If $e_1, e_2, \dots  , e_{n}$
is an orthonormal set in $H$ such that
$Ae_i=\alpha_i(\lambda) e_i$, $ \alpha_i(\lambda)$ is continuous function,
$i=1,\dots ,n$, then every element $u\in E$ can be represented in the
unique form of
$$
u=w+v, \quad w=\sum_{i=1}^{n} \xi_i e_i\in M, \quad
M\bot v\in M^{\bot},  \; \xi_i=\langle u, e_i \rangle,
$$
where
$\langle \cdot, \cdot \rangle$ is the inner product in Hilbert
space $H$. There exist projections $p:E\to M$ and
$I-p:E\to M^\bot $ such that $w=pu$ and $(I-p)u=v$.
Similarly, there exist projections $Q:F\to \tilde{M}$ and
$I-Q:F\to \tilde{M}^\bot $ such that
\begin{equation}
f(u,\lambda)=Qf(u,\lambda)+(I-Q)f(u,\lambda)\quad \label{e2.2}
\end{equation}
or
$$
f(w+v,\lambda)=Qf(w+v,\lambda)+(I-Q)f(w+v,\lambda)
$$
It follows that
$$
Qf(w+v,\lambda)+(I-Q)f(w+v,\lambda)=0
$$
and hence the result becomes
\begin{gather*}
Qf(w+v,\lambda)=0,\\
(I-Q)f(w+v,\lambda)=0.
\end{gather*}
 The implicit function theorem implies that
$$
W(\xi,\delta)=V(\Phi(\xi,\delta),\delta), \quad \xi=(\xi_1,
\xi_2,\dots ,\xi_{n})^\top
$$
where $\deg W\geq 2$, then the linear
Ritz approximation of the functional $V$ is a function $W$ defined
by
\begin{equation}
W(\xi, \delta)=V\Big(\sum_{i=1}^n \xi_i
e_i,\delta\Big)=W_0(\xi)+W_1(\xi,\delta) \label{e2.3}
\end{equation}
where $W_0(\xi)$ is a homogenous
polynomial of order $n\geq 3$ such that $W_0(0)=0$ and
$W_1(\xi,\delta)$ is a polynomial function of degree less than $n$.
 Let $q_1,q_2,\dots ,q_{m}$ be the coefficients of the
quadratic terms of the function $W_1(\xi,\delta)$, then the
function $W_1(\xi,\delta)$ can be written in the form of
$$
W_1(\xi,\delta)=W_2(\xi,\delta)+\sum_{k=1}^m q_{k} \xi_{k}^2
$$
where $\deg W_2=d$,  $2< d < n$.

The nonlinear Ritz approximation of the functional $V$ is a function $W$ defined
by
$$
W(\xi,\delta)=V\Big(\sum_{i=1}^n \xi_i e_i+\Phi(\sum_{i=1}^n \xi_i
e_i,\delta),\delta\Big)
$$
where $\Phi(w,\delta)=v(x,\xi,\delta)$, $v\in N^\bot$.
To determine the nonlinear Ritz approximation of the functional $V$,
Taylor's expansion of the functions $\mu_{k}(\xi)$ and
$v(x,\xi,\delta)$ is used by assuming the following:
\begin{gather*}
q_{k}=\hat{q}_{k}+\mu_{k}(\xi)=\hat{q}_{k}+\sum_{j=2}^r D_{k}^{(j)}(\xi),
\quad k=1,\dots ,m, \\
v(x,\xi,\delta)=\sum_{j=2}^r B^{(j)}(\xi).
\end{gather*}
where $D_{k}^{(j)}(\xi)$ and $B^{(j)}(\xi)$ are
homogenous polynomials of degree j with coefficients $\mu_{ki}$
and $v_{ji}(x,\delta)$ respectively,
$\xi=(\xi_1,\xi_2,\dots ,\xi_{n})$. Since
$$
Qf(u,\lambda)=\sum_{i=1}^n \langle f(u,\lambda), e_i \rangle
e_i=0
$$
it follows that
$$
\sum_{i=1}^n \langle Au+Nu, e_i \rangle e_i=0
$$
Hence
$$
\sum_{i=1}^n q_i\xi_i
e_i+\sum_{i=1}^n \langle Nu, e_i \rangle e_i=0, \ \ \ \
q_i=\alpha_i(\lambda)
$$
or
\begin{equation}
\sum_{i=1}^n q_i\xi_i e_i+\sum_{i=1}^n \Big[ \int_{\Omega} N(w+v) e_i \Big]
e_i=0. \label{e2.4}
\end{equation}
From \eqref{e2.2} it follows that
$$
(I-Q)f(u,\lambda)=f(u,\lambda)-Qf(u,\lambda)\,.
$$
From
$A(w+v)+N(w+v)=0$
it follows that
\begin{equation}
Av+N(w+v)+\sum_{i=1}^n q_i\xi_i
e_i=0 \label{e2.5}
\end{equation}
Substituting the values of $q_i$,  $\mu_i(\xi)$ and
$v(x,\xi,\delta)$ in \eqref{e2.4} and \eqref{e2.5} yields
\begin{gather}
\sum_{i=1}^n \Big[\hat{q}_i+\sum _{j=2}^r D_i^{(j)}(\xi) \Big] \xi_i
e_i+\sum_{i=1}^n \Big[ \int_{\Omega} N \Big( \sum_{i=1}^n
\xi_i e_i+\sum_{j=2}^r B^{(j)} (\xi) \Big) e_i \Big]
e_i=0,\label{e2.6}\\
A\Big( \sum_{j=2}^r B^{(j)}(\xi) \Big) +N \Big(\sum_{i=1}^n
\xi_i e_i+\sum_{j=2}^r B^{(j)} (\xi) \Big)+\sum_{i=1}^n
\Big(\hat{q}_i+\sum _{j=2}^r D_i^{(j)}(\xi) \Big)\xi_i
e_i=0. \label{e2.7}
\end{gather}
To determine the functions $v(x,\xi,\lambda)$ and $\mu_{k}(\xi)$ we
equate the coefficients of $\hat{\xi}=\xi_1 \xi_2 \dots
\xi_{n}$ in  \eqref{e2.6} to find the value of $\mu_{ki}$ and
after some calculations from \eqref{e2.7} it is obtained a linear
ordinary differential equation in the variable
$v_{ji}(x,\lambda)$. Solving the resulting equation one can find the
value of $v_{ji}(x,\lambda)$.


\section{Applications}

 In this section we introduced an example
to study the bifurcation of periodic solutions of the
nonlinear fourth-order differential equation
\begin{equation}
\frac{d^4u}{dx^4} + \lambda \frac{d^2u}{dx^2} +
u+u^2+ u^3=0, \label{e3.1}
\end{equation}
by finding the nonlinear Ritz approximation of the energy functional
$V(u,\lambda)$ given by
$$
V(u,\lambda)=\int_0^{2\pi} \Big(\frac{(u'')^2}{2}-\lambda
     \frac{(u^{^{/}})^2}{2}+\frac{u^2}{2}+\frac{u^3}{3}+\frac{u^4}{4}\Big) dx.
$$
To do this suppose that $f:E\to F$ is a nonlinear Fredholm
operator of index zero defined by
\begin{equation}
f(u,\lambda)=\frac{d^4u}{dx^4} + \lambda \frac{d^2u}{dx^2} +
u+u^2+ u^3 \label{e3.2}
\end{equation}
where $E=\Pi^{4}([0,2\pi],R)$ is the space of all periodic continuous
functions that have derivative of order at most four,
$F=\Pi_0([0,2\pi],R)$  is the space of all periodic continuous
functions, $u=u(x)$ and $x\in[0,2\pi]$. Since the operator $f$  is
variational, then there exists a functional $V$ such that $f$ is
the gradient of $V$, $i.e.$
$$
f(u,\lambda)=\operatorname{grad}_{H}V(u,\lambda)
$$
hence every solution of equation \eqref{e3.1} is a critical point of the
functional $V$ \cite{d1}.
Thus the study of the solutions of equation
\eqref{e3.1} is equivalent to the study of an extreme problem
$$
V(u,\lambda)\to \operatorname{extr}, \quad \ u\in E.
$$
Analysis of bifurcation can be found by using the local method of
Lyapunov-Schmidt, so by localizing the parameter
$$
\lambda=\lambda_1+\mu(\xi),\quad
\mu:\mathbb{R}\to\mathbb{R}\text{ is a continuous  function}
$$
the reduction leads to the function of one variable
$$
W(\xi,\delta)=\inf_{\langle u, e \rangle =\xi}
V(u,\delta).
$$
It is well known that in the reduction of
Lyapunov-Schmidt the function $W(\xi,\delta)$ is smooth. This
function has all the topological and analytical properties of
functional $V$  \cite{a6}. In particular, for small $\delta$ there is
one-to-one corresponding between the critical points of functional
$V$ and smooth function $W$, preserving the type of critical
points (multiplicity, bifurcation diagram, index Morse, etc.)
\cite{a6}. By using the scheme of Lyapunov-Schmidt, the linearized
equation corresponding to the equation \eqref{e3.1} is given by
$$
h'''+\lambda h''+ h=0, \quad h\in E
$$
Let $N=\ker(A)=\operatorname{span}{\{e}\}$,
$e=\sin(x)/\sqrt{\pi}$ and
$A=f_{u}(0,\lambda)=\frac{d^{4}}{dx^{4}}+\lambda
\frac{d^2}{dx^2}+1$,
then every element $u\in E$ can be written in
the  form
$$
u = w + v, \quad w=\xi e\in N, \quad
\xi\in\mathbb{R}, \; v\in \hat{E}=N^{\perp}\cap E.
$$
By the implicit function theorem, there
exists a smooth map $\Phi:N\to \hat{E}$  such that
$$
W(\xi,\delta)=V(\Phi(\xi,\delta),\delta),
$$
and then the linear Ritz approximation of the functional $V$
is a function $W$ given by
$$
W(\xi,\delta)=V(\xi  e,\delta)=\xi^4+q \xi^2.
$$
the nonlinear Ritz approximation of the functional $V$ is a
function $W$ given by
$$
W(\xi,\delta)=V(\xi  e+\Phi(\xi,\delta),\delta), \quad
v(x,\xi)=\Phi(\xi,\delta).
$$
We will apply the method in section 2 to find the nonlinear
Ritz approximation of the functional $V$. So from the Lyapunov-Schmidt
method we note that the space $E$ can be decomposed in direct sum of two
subspaces, $N$ and the orthogonal complement to $N$,
$$
E = N \oplus \hat{E}, \quad \hat{E}=N^\bot \cap E
= \{ v \in E:v \bot N\}.
$$
Similarly, the space $F$ decomposed in direct sum of two subspaces,
$N$ and orthogonal complement to $N$,
 $$
F = N \oplus \hat{F}, \quad \hat{F}=N^\bot \cap F = \{ v \in F:v \bot N\}.
$$
There exist projections $p:E\to N$  and $I-p:E\to \hat{E}$
such that $pu=w$ and $(I-p)u=v$, ($I$ is the identity operator).
Hence every vector $u\in E$  can be written in the form
$$
u = w + v, \quad w\in N, \quad N\perp v\in\hat{E}.
$$
Similarly, there exists projections
$Q:F\to N$  and $I-Q:F\to \hat{F}$ such that
\begin{equation}
f(u,\lambda)=Qf(u,\lambda)+(I-Q)f(u,\lambda)\label{e3.3}
\end{equation}
Accordingly, \eqref{e3.2} can be written in the form
\begin{gather*}
Qf(w+v,\lambda)=0, \\
(I-Q)f(w+v,\lambda)=0.
\end{gather*}
To determine the nonlinear Ritz approximation of the functional
$V$, the functions $v(x,\xi,\lambda)=O(\xi^{3})$ and
$\mu(\xi)=O(\xi^{2})$ must be found in the form of power series in
terms of $\xi$, as follows:
\begin{equation}
\begin{gathered}
v(x,\xi)=v_0(x) \xi^3+v_1(x) \xi^4+v_2(x) \xi^5+\dots ,\\
\mu(\xi)=\mu_0 \xi^2+\mu_1 \xi^3+\mu_2 \xi^4+\dots,
\end{gathered} \label{e3.4}
\end{equation}
and \eqref{e3.2} can be written in the form
$$
f(u,\lambda)=Au+Tu=0, \quad Tu=u^2+u^{3}.
$$
Since
$$
Qf(u,\lambda)=\langle f(u,\lambda),\sin(x) \rangle \sin(x) =0,
$$
 we have $\langle Au+Tu ,\sin(x) \rangle \sin(x) =0$
and hence
\begin{equation}
-\pi \xi \mu(\xi)+\int_0^{2 \pi}(v+\xi \sin(x))^{2}\sin(x) \,dx
+\int_0^{2 \pi} (v+\xi \sin(x))^3 \sin(x)\,dx=0.
 \label{e3.5}
\end{equation}
From \eqref{e3.3} and \eqref{e3.5} we have
\begin{equation}
v^{iv}+(\lambda_1+\mu(\xi))v''+v+(v+\xi \sin(x))^2+(v+\xi
\sin(x))^3-\xi\mu(\xi) \sin(x)=0 \label{e3.6}
\end{equation}
As a consequence
\begin{gather} \label{e3.7}
\begin{aligned}
&-\pi \xi \mu(\xi)+\int_0^{2 \pi} v^2
\sin(x) \,dx+2\xi\int_0^{2 \pi} v (\sin(x))^2 \,dx
+\frac{3\pi}{4}\xi^3\\
&+3\xi^2 \int_0^{2 \pi} v (\sin(x))^3 \,dx
+3\xi \int_0^{2 \pi} v^2 (\sin(x))^2 \,dx
+\int_0^{2 \pi} v^3 \sin(x) \,dx=0,
\end{aligned}\\
\begin{aligned}
&v^{iv}+(\lambda_1+\mu(\xi))v''+v+v^2+2 v
\xi \sin(x)+\xi^2 (\sin(x))^2\\
&+v^3+3 v^2 \xi \sin(x)+3 v \xi^2 (\sin(x))^2+\xi^3 (\sin(x))^3
-\xi \mu(\xi) \sin(x)=0.
\end{aligned} \nonumber
\end{gather}
 To determine the functions $v(x,\xi)$ and $\mu(\xi)$ first we substitute
\eqref{e3.4} in \eqref{e3.7} and then we find the
coefficients $\mu_0,\mu_1,\mu_2,v_0,v_1$ and $v_2$ by
equating the terms of $\xi$ as follows:
 Equating the coefficients of $\xi^3$ we have the following two equations,
\begin{equation}
\begin{gathered}
-\pi\mu_0+\frac{3\pi}{4}=0,\\
v_0^{(4)}+ \lambda_1 v_0''+ v_0+(\sin(x))^3- \mu_0 \sin(x)=0
\end{gathered} \label{e3.8}
\end{equation}
From the first equation in \eqref{e3.8} we have
$\mu_0=3/4$. Substituting this value in the second equation of \eqref{e3.8},
we have the linear differential equation
$$
v_0^{(4)}+ \lambda_1v_0''+v_0+ (\sin(x))^3-\frac{3}{4} \sin(x)=0,
$$
and then we have
\begin{equation} \label{e3.9}
v_0^{(4)}+ \lambda_1 v_0''+ v_0-\frac{1}{4} \sin(3x)=0. \\
\end{equation}
Then
$$
v_0(x)= \frac{\sin(3x)}{256}.
$$
Similarly, equating the coefficients of
$\xi^4$ we have
\begin{equation}
\begin{gathered}
-\pi\mu_1+2 \int _0^{2\pi} v_0(x) (\sin(x))^2 dx=0,\\
v_1^{(4)}+\lambda_1 v_1''+ v_1+2 v_0 \sin(x)-\mu_1=0.
\end{gathered} \label{e3.10}
\end{equation}
From the first equation in \eqref{e3.10} we have $\mu_1=0$.
Substituting this value in the second equation of \eqref{e3.10} we have
\begin{equation}
v_1^{(4)}+ \lambda_1 v_1''+ v_1+ \frac{\sin(x)\sin(3x)}{128}=0. \label{e3.11}
\end{equation}
Then
$$
v_1(x)= \frac{-1}{256}
\Big[\frac{\cos(2x)}{9}-\frac{\cos(4x)}{225}\Big].
$$
Equating
the coefficients of $\xi^5$ we have
\begin{equation}
\begin{aligned}
-\pi\mu_2+2 \int_0^{2\pi} v_1(x) (\sin(x))^2 dx
 +3 \int_0^{2\pi} v_0(x)(\sin(x))^3 dx=0, \\
v_2^{(4)}+ \lambda_1 v_2''+ v_2+ \mu_0
v_0''+ 2 v_1 \sin(x)+3 v_0 (\sin(x))^2-\mu_2 \sin(x)=0
\end{aligned} \label{e3.12}
\end{equation}
substituting the values of $v_0$
and $v_1$ in the first equation of \eqref{e3.12} and then solving this
equation we find that
$$
\mu_2=-\frac{23}{9216}.
$$
Also, substituting the values of $\mu_0, v_0$
and $v_1$ and  in the second equation of \eqref{e3.12} we have the
 linear differential equation
\begin{equation}
v_2^{(4)}+ \lambda_1 v_2''+ v_2+\frac{3}{1024}[7\sin(3x)+\sin(5x)]
-\frac{6 \sin(3x)-\sin(x)}{11520}=0 \label{e3.13}
\end{equation}
 Solving  \eqref{e3.13} we have
 $$
v_2(x)=\Big[\frac{21}{65536}-\frac{1}{122880}\Big] \sin(3x)
+\frac{1}{24}\Big[\frac{1}{8192}+\frac{1}{276480}\Big] \sin(5x).
$$
Now substituting the values of $\mu_0,\mu_1,\mu_2,v_0,v_1$ and
$v_2$ in \eqref{e3.4}  we have the bifurcation equation
\begin{equation} \label{e3.14}
\begin{gathered}
\begin{aligned}
u(x,\xi)&=\frac{\xi \sin(x)}{\sqrt{\pi}}+\frac{\xi^3}{256\pi\sqrt{\pi}}
\sin(3x)-\frac{\xi^4}{57600 \pi} \Big[25\cos(2x)-\cos(4x)\Big]\\
&\quad +\xi^5\Big(\Big[\frac{21}{65536\pi^2\sqrt{\pi}}
 -\frac{1}{122880\pi\sqrt{\pi}} \Big] \sin(3x)\\
&\quad +\frac{1}{24}\Big[\frac{1}{8192\pi^2\sqrt{\pi}}
 +\frac{1}{276480\pi\sqrt{\pi}}\Big]
\sin(5x)\Big)+O(\xi^7)
\end{aligned}\\
\lambda=\lambda_1+\frac{3}{4}\xi^2 -\frac{23}{9216}\xi^4 +O(\xi^6)
\end{gathered}
\end{equation}
From the above result we deduced the following theorem.

\begin{theorem} \label{thm3.1}
The key function of the functional $V$ has the form
\begin{equation}
\begin{aligned}
\hat{W}(\xi,\delta)
&=U(\xi,\delta)+O(|\xi|^{20})+O(|\xi|^{20})O(|\delta|)\\
&=c_1\xi^{20}+c_2\xi^{18}+c_3\xi^{16}+c_4
\xi^{14}+c_5\xi^{12}+\alpha_1\xi^{10}+\alpha_2\xi^{8}\\
&\quad + \alpha_3\xi^{6}+c_6 \xi^{4}+\alpha_4\xi^{2}
 +O(|\xi|^{20})+O(|\xi|^{20})O(|\delta|),
\end{aligned}\label{e3.15}
\end{equation}
where
\begin{gather*}
c_1=0.11742\times 10^{19}, \quad
c_2=0.52310\times10^{21}, \quad c_3=0.47769\times10^{23},\\
c_4=0.23733\times10^{27},\quad c_5=-0.41660\times10^{29},\\
\alpha_1=-(0.63868\times 10^{31}+0.99142\times 10^{29}\lambda_1),\\
\alpha_2=-(0.11220\times 10^{32}\lambda_1+0.94599\times 10^{32}),\\
\alpha_3=0.31596\times 10^{35}-0.17154\times 10^{33}\lambda_1,\\\
c_6=-0.77749\times 10^{37},\quad
\alpha_4=-(0.24658\times 10^{38}+0.12329\times 10^{38}\lambda_1)
\end{gather*}
\end{theorem}

The prove of Theorem \ref{thm3.1} follows directly from the formula
$$
\hat{W}(\xi,\delta)=V(\xi  e+\Phi(\xi,\delta),\delta).
$$


We note that $c_1,c_2,c_3,c_4,c_5,c_6$ are constants
and $\alpha_1,\alpha_2,\alpha_3,\alpha_4$ are
parameters.The key function $\hat{W}(\xi,\delta)$ in Theorem \ref{thm3.1}
is the required nonlinear Ritz approximation of the functional
$V(\xi  e+\Phi(\xi,\delta),\delta)$. The geometry of the
bifurcation of critical points and the principal asymptotic of the
branches of bifurcating points for the function
$\hat{W}(\xi,\delta)$ are entirely determined by its principal
part $U(\xi,\delta)$. The function has all the topological and
analytical properties of functional $V$ , also the function have
19 critical points. The point $u(x)=\xi  e+v(x,\xi)$ is a
critical point of the functional $V(u,\lambda)$ if and only if the
point $\xi$ is a critical point of the function
$\hat{W}(\xi,\delta)$, (Theorem \ref{thm1.1}). This means that the
existence of the solutions of equation \eqref{e3.2} depend on the
existence of the critical points of the functional $V(u,\lambda)$
and then on the existence of the critical points of the function
$\hat{W}(\xi,\delta)$. From this notation we can find a nonlinear
approximation of the solutions of equation \eqref{e3.2} corresponding to
each critical point of the function $\hat{W}(\xi,\delta)$. To
avoid the singularities of the function $U(\xi,\delta)$ we must
find the caustic, so from definition \ref{def1.1} the caustic of the
function $U(\xi,\delta)$ is the set of all $\lambda_1$
satisfying the equation
\begin{equation}
\begin{aligned}
&(\lambda_1+6.168595401)(\lambda_1+6.168557117)(\lambda_1+1.997966599)\\
&\times (\lambda_1-7.420188558)(\lambda_1-7.420220242)
 (\lambda_1^2+10.79759396 \lambda_1+39.30282312) \\
&\times (\lambda_1^2+10.79751137 \lambda_1+39.30229176)
 (\lambda_1^2+6.369291114 \lambda_1+42.39826243) \\
&\times (\lambda_1^2+6.369244386 \lambda_1+42.39775611)
 (\lambda_1^2+4.002037322 \lambda_1+4.004078787) \\
&\times (\lambda_1^2-0.1636805324 \lambda_1+46.42060955)
 (\lambda_1^2-0.1636921686 \lambda_1+46.42102879) \\
&\times (\lambda_1^2-7.233265906 \lambda_1+50.57130598)
 (\lambda_1^2-7.233315078 \lambda_1+50.57170600)\\
&\times (\lambda_1^2-12.74882579 \lambda_1+53.82009714)
 (\lambda_1^2-12.74888749 \lambda_1+53.82053975)\\
&=0.
\end{aligned} \label{e3.16}
\end{equation}
The only real values satisfying the above
equation are
$$
\Sigma={\{-6.168595401,-6.168557117,-1.997966599,7.420188558,7.420220242}\}.
$$
Hence the caustic dividing the real lines into following six sets
\begin{gather*}
(-\infty,-6.168595401),\; (-6.168595401,-6.168557117), \\
(-6.168557117,-1.997966599),\; (-1.997966599,7.420188558),\\
(7.420188558,7.420220242),\;(7.420220242,\infty)
\end{gather*}
every set has a fixed number of nondegenerate critical points. The
spreading of real critical points of the function $U(\xi,\delta)$
is given below:

If $\lambda_1\in (-\infty,-6.168595401)$, then we have five
nondegenerate critical points (three minima and two maxima).

If $\lambda_1\in (-6.168595401,-6.168557117)$, then we have
five nondegenerate critical points (three minima and two maxima).

If $\lambda_1\in (-6.168557117,-1.997966599)$, then we have
five nondegenerate critical points (three minima and two maxima).

If $\lambda_1\in (-1.997966599,7.420188558)$, then we have
three nondegenerate critical points (two minima and one maximum).

If $\lambda_1\in (7.420188558,7.420220242)$, then we have three
nondegenerate critical points (two minima and one maximum).

If $\lambda_1\in (7.420220242,\infty)$, then we have three
nondegenerate critical points (two minima and one maximum).
\smallskip

To explain our results we have the following:
 We found that the linear Ritz approximation of the functional
$V(u,\lambda)$ is the function
$$
W(\xi,\delta)=\xi^{4}+q \xi^{2};
$$
the critical points of this function are degenerate when $q=0$, so
for every $q\neq0$ we have three nondegenerate critical points of
the function $W(\xi,\delta)$. Corresponding to each nondegenerate
critical point we have a linear approximation solution of
\eqref{e3.1} in the form of $w=\xi \sin(x)/\sqrt{\pi}$. These solutions
have only the two geometric representations shown in Figure \ref{fig1}.

\begin{figure}[htb]
\begin{center}
\includegraphics[width=0.7\textwidth]{fig1.png}
\end{center}
\caption{Graphs of the function $w=\xi \sin(x)/\sqrt{\pi}$.}
\label{fig1}
\end{figure}

\begin{figure}[htb]
\begin{center}
\includegraphics[width=0.7\textwidth]{fig2.png}
\end{center}
\caption{Graphs of the function \eqref{e3.14}.}
\label{fig2}
\end{figure}

In theorem \ref{thm3.1} we proved that the nonlinear Ritz approximation
of the functional $V(u,\lambda)$ is the function \eqref{e3.15}. All
critical points of this function are degenerate when $\lambda_1$
is a solution of \eqref{e3.16}, so for every
$\lambda_1\in R\backslash \Sigma$ we have only three or five nondegenerate
critical points. Corresponding to each nondegenerate critical point
we have nonlinear approximation solution of  \eqref{e3.1} in the
form of function \eqref{e3.14}. These solutions have the  four
geometric representations shown in Figure \ref{fig2}.





\subsection*{Acknowledgments}
I would like to thank the anonymous referee for the useful comments and
suggestions.


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\end{document}