\documentclass[reqno]{amsart}
\usepackage{graphicx}
\usepackage{hyperref}

\AtBeginDocument{{\noindent\small
Sixth Mississippi State Conference on Differential Equations and
Computational Simulations,
{\em Electronic Journal of Differential Equations},
Conference 15 (2007),  pp. 153--158.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu  (login: ftp)}
\thanks{\copyright 2007 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document} \setcounter{page}{153}
\title[\hfilneg EJDE-2006/Conf/15\hfil Pitchfork bifurcation]
{On the direction of pitchfork bifurcation}

\author[X. Hou, P. Korman, Y. Li\hfil EJDE/Conf/15 \hfilneg]
{Xiaojie Hou, Philip Korman, Yi Li}  % in alphabetical order

\address{Xiaojie Hou \newline
Department of Mathematical Sciences \\
University of Cincinnati \\
Cincinnati, OH 45221-0025, USA}
\email{houx@email.uc.edu}

\address{Philip Korman \newline
Department of Mathematical Sciences \\
University of Cincinnati \\
Cincinnati, OH 45221-0025, USA}
\email{kormanp@math.uc.edu}

\address{Yi Li \newline
Department of Mathematics \\
Hunan Normal University\\
Changsha 410081, Hunan, China. \newline
University of Iowa \\
Iowa City Iowa 52242, USA}
\email{yi-li@uiowa.edu}

\thanks{Published February 28, 2007.}
\thanks{Supported by the Xiao-Xiang Grant from the Hunan Normal
 University, and by \hfill\break\indent
 grant 10471052 from the Natural Science Foundation of China}
\subjclass[2000]{34B15}
\keywords{Pitchfork bifurcation, multiplicity of solutions}

\begin{abstract}
 We present an algorithm for computing the direction of pitchfork
 bifurcation for two-point boundary value problems.
 The formula is rather involved, but its computational evaluation
 is quite feasible. As an application, we obtain a multiplicity result.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem*{example}{Example}

\section{Introduction}

We study  positive, negative, and sign-changing solutions for the
problem
\begin{equation}\label{1}
\begin{gathered}
u''(x)+f(u(x))-\lambda=0, \quad \text{for }-1<x<1, \\
u(-1)=u(1)=0.
\end{gathered}
\end{equation}
Our principal example will be $f(u)=u^{2k}$,
with an integer $k \geq 2$, although our result is considerably
more general. For $k=1$ this problem was exhaustively analyzed in
Scovel's Ph.D. thesis \cite{S}, and in  McKean and
Scovel \cite{MS}. They used explicit integration via elliptic
functions, which means that their method does not work for $k>1$.
Anuradha and  Shivaji \cite{AS} have studied a related
problem. Using the quadrature technique, they showed existence of
infinitely many points of bifurcation.   Korman \cite{K1} has
used bifurcation theory to approach the problem (\ref{1}), and in
particular the case of $f(u)=u^{2k}$, with  $k>1$. He was able to
generalize some, but not all, of the results of  McKean and
 Scovel \cite{MS}. One of the difficulties involved the
direction of  pitchfork bifurcation, which is the subject of the
present paper.

Let us briefly review part of what is known for this problem in
case of $f(u)=u^{2k}$, see \cite{K1} for more details. When $\lambda=0$,
the problem has a unique positive solution.
This solution continues for a while when $\lambda>0$. At a critical
$\lambda =\lambda _0$ the positive solution develops zero slope at the boundary,
i.e. $u'(-1)=u'(1)=0$, and a pitchfork bifurcation occurs at $\lambda =\lambda _0$.
Namely, we have a symmetric sign-changing solution for $\lambda >\lambda _0$,
and a parabola-like family of asymmetric solutions.
One of these solutions is negative near the $x=-1$ end and positive on
the rest of the interval $(-1,1)$, while the other one is negative
near $x=1$ end, see Figure $1$. The issue is: which way this
parabola-like curve of asymmetric solutions bifurcates, is it for
$\lambda >\lambda _0$, or toward decreasing $\lambda$?
(The numerical evidence of  Ramaswamy \cite{R}, and of  Korman \cite{K}
suggests that the pitchfork opens forward in $\lambda$.)

\begin{figure}[ht]
\begin{center}
\setlength{\unitlength}{0.8pt}
\begin{picture}(383,144)(-31,-60)
\thicklines
\put(143,0){\makebox(0,0)[l]{$x$}}
\put(57,86){\makebox(0,0)[1]{$u$}}
\qbezier(-25,0)(0,-37)(25,0)

\put(0,10){\makebox(0,0)[1]{$\xi$}}
\put(23,10){\makebox(0,0)[1]{$\theta$}}

\put(75,10){\makebox(0,0)[1]{$\eta$}}
\put(41,-18){\makebox(0,0)[1]{$w$}}
\put(41,50){\makebox(0,0)[1]{$v$}}

\qbezier(25,0)(75,100)(125,0)

\thinlines
\put(-31, 0){\vector(1,0){170}}
\put(50,-60){\vector(0,1){140}}
\put(0,-2){\line(0,1){4}}

\put(25,-2){\line(0,1){4}}
\put(75,-2){\line(0,1){4}}
\put(48,-18){\line(1,0){4}}
\put(48,50){\line(1,0){4}}

 \put(160, 0){\vector(1,0){169}}
\put(332,0){\makebox(0,0)[l]{$x$}}
\put(252,86){\makebox(0,0)[1]{$u$}}
\put(246,-60){\vector(0,1){140}}
\thicklines
\qbezier(170,0)(220,100)(270,0)
\qbezier(270,0)(295,-37)(320,0)
%\put(50,-100){\makebox(0,0)[l]{{\bf Figure 1:Two types of asymmetric solutions}}}
\end{picture}
\end{center}
\caption{Two types of asymmetric solutions}
\end{figure}


The following elementary example shows why this question is non-trivial.
 The solution set of the equation
\[
x( a x^2-b \lambda)=0
\]
exhibits a pitchfork in $(\lambda,x)$ plane, in  a neighborhood of the
point $(\lambda=0,x=0)$, for any non-zero constants $a$ and $b$.
If $a/b>0$, the pitchfork opens to the right, and if $a/b<0$ to the left.
The same behavior holds in case of more general equations
\[
f(x,\lambda) \equiv x( a x^2-b \lambda)+ \dots=0,
\]
where $\dots$ stands for higher order terms at $(0,0)$ (e.g.
$\lambda ^2x^2$, $x^4$, etc). To obtain the ratio $a/b$, governing the
direction of the pitchfork, one calculates
\[
a/b=-\frac{f_{xxx}(0,0)}{6f_{\lambda x}(0,0)} \,.
\]
If one tries the same approach for the equation (\ref{1}),
one needs to differentiate that equation three times. The resulting
equations has  a number of terms, which seems impossible to handle.

We approach the problem by using direct integration. Our algorithm
involves integrals that cannot be explicitly evaluated, but their
computational evaluation is quite feasible,  both in case $f(u)=u^{2k}$,
and for more general nonlinearities.

\section{The direction of bifurcation}

Let us consider the symmetry breaking solution, which is negative near the $x=-1$ end, and positive near $x=1$. Let us denote by $\xi$ the point of negative minimum, by $\eta$ the point of positive maximum, and by $\theta$ the root of $u(x)$.
We also denote $w=u(\xi)<0$ and $v=u(\eta)>0$, the minimum and maximum values respectively, see Figure $1$. Clearly $\xi=\xi(\lambda)$, $\eta=\eta (\lambda)$, but since solutions of autonomous equations are symmetric with respect to their extremal points, we have
\begin{equation}
\label{2}
\eta-\xi=1, \quad \text{for all }\lambda .
\end{equation}
(The points $\xi$ and $\eta$ are midpoints of the intervals $(-1,\theta)$ and
$(\theta,1)$ respectively.) Assume that the symmetry breaking solution
bifurcates at $\lambda=\lambda _0$. Then $w=0$ at  $\lambda=\lambda _0$. At other $\lambda$'s,
 $\lambda=\lambda(w)$, $w <0$. If we can show that $\frac{d \lambda}{dw}(0)<0$ ($>0$)
then the pitchfork opens forward (backward). (Observe that the function
$\lambda(w)$ is defined only for $w \leq 0$. Hence $\lambda(w)$ will have a point
of minimum at $w=0$, provided that $\lambda '(0) <0$.)

Clearly,
\begin{equation} \label{3}
\frac{1}{2} {u'}^2(x)+F(u(x))-\lambda u(x)=c,
\end{equation}
where $F(u)=\int_0^u f(t) \, dt$, and $c$ is a constant.
Evaluating the formula (\ref{3}) at $x=\xi$, and then at $x=\eta$,
we have $c=F(w)-\lambda w=F(v)-\lambda v$, which implies
\begin{equation} \label{4}
\lambda=\frac{F(v)-F(w)}{v-w}.
\end{equation}
Also from (\ref{3}) we have on the interval $(\xi,\eta)$
(where $\frac{du}{dx}>0$)
\[
\frac{du}{dx}=\sqrt{2} \sqrt{F(v)-F(u)-\lambda (v-u)}.
\]
Integrating over the unit interval $(\xi,\eta)$ (see (\ref{2})),
\begin{equation}
\label{5}
\int_w^v \frac{du}{\sqrt{F(v)-F(u)-\lambda (v-u)}} =\sqrt{2}.
\end{equation}

In formulas (\ref{4}) and (\ref{5}) we regard $v$ and $\lambda$ as
functions of $w$, i.e. $v=v(w)$, $\lambda =\lambda(w)$, with $w \leq 0$.
We have $\lambda(0)=\lambda _0$, and $v(0) \equiv v_0=u_0(0)$,
where $(\lambda _0,u_0(x))$ is the point of  pitchfork bifurcation.
Let us calculate $\lambda _0$ and $v_0$ for our $f(u)=u^{2k}$.
Setting $w=0$ in (\ref{4}), we have
\begin{equation}
\label{6}
\lambda _0=\frac{F(v_0)}{v_0}=\frac{v_0^{2k}}{2k+1}.
\end{equation}
Setting $w=0$ in (\ref{5}), and using (\ref{4}) and \eqref{6}
\begin{equation} \label{7}
\int_0^{v_0} \frac{du}{\sqrt{\frac{v_0^{2k}}{2k+1}  u
 -  \frac{1}{2k+1} u^{2k+1}  }}=\sqrt{2}.
\end{equation}
We evaluate the integral in (\ref{7}) by making a substitution $u=v_0z$,
\begin{align*}
 \int_0^{v_0} \frac{du}{\sqrt{\frac{v_0^{2k}}{2k+1}  u
 -  \frac{1}{2k+1} u^{2k+1}  }}
&=v_0^{-k+1/2} \sqrt{2k+1}
 \int_0^1 \frac{dz}{\sqrt{z-z^{2k+1}}}\\
&\equiv v_0^{-k+1/2} \sqrt{2k+1} \, J(k),
\end{align*}
where, using {\em Mathematica}, we express in terms of the standard
gamma function,
\[
J(k) \equiv \int_0^1 \frac{dz}{\sqrt{z-z^{2k+1}}}
=\frac{2 \sqrt{\pi}  \Gamma(1+\frac{1}{4k})}{\Gamma(\frac{1}{2}+\frac{1}{4k})}.
\]
Returning to (\ref{7}), we have
\begin{equation}
\label{8}
v_0=\Big[\frac{(2k+1)J^2(k)}{2} \Big]^{1/(2 k-1)}.
\end{equation}
We calculate $v_0$ from this formula, and then use \eqref{6} to
calculate $\lambda _0$.
We now turn to the calculation of $\frac{d \lambda}{dw}(0)$.
In the formula (\ref{4}) we multiply through by $v-w$, and differentiate
with respect to $w$,
\[
\lambda '(w) \left(v(w)-w \right)+\lambda(w)(v'(w)-1)=f(v)v'(w)-f(w).
\]
We now set $w=0$. Then $\lambda =\lambda _0$, and $v=v_0$. Since $f(0)=0$, we obtain
\begin{equation}
\label{9}
\lambda '(0)=\frac{1}{v_0} \left[ \lambda _0 +v'(0) \left(f(v_0)-\lambda _0 \right) \right].
\end{equation}
In order to find $v'(0)$, we plug (\ref{4}) into (\ref{5}), obtaining
\begin{equation}
\label{10}
\int_w^v \frac{du}{\sqrt{F(v)-F(u)-\frac{F(v)-F(w)}{v-w} (v-u)}} =\sqrt{2}.
\end{equation}
The integral in (\ref{10}) is improper at both end-points.
To regularize it, we use the substitution
\begin{equation}
\label{*}
u=\frac{1}{2} (v-w) \sin \theta+\frac{w}{2}+\frac{v}{2}.
\end{equation}
 Then (\ref{10}) takes the form
\begin{equation} \label{11}
G(v,w) \equiv \int_{-\pi/2}^{\pi/2}  H(v,w,\theta) \, d \theta=\sqrt{2},
\end{equation}
where
\[
H(v,w,\theta)=\frac{1}{2} \frac{(v-w)^{3/2} \cos \theta}{  \sqrt{ \left(F(v)-F(u) \right)(v-w) -\left( F(v)-F(w) \right)(v-u)}},
\]
and $u$ is given by (\ref{*}).
{\em Mathematica} seems unable to evaluate exactly the integral in (\ref{11}) for general $k$, however it easily evaluates a very accurate numerical approximation for any particular $k$. We now differentiate (\ref{11}) with respect to $w$
\[
G_v(v,w)v'(w)+G_w(v,w)=0,
\]
where $G_v=\int_{-\pi/2}^{\pi/2}  H_v(v,w,\theta) \, d \theta$,
 and $G_w=\int_{-\pi/2}^{\pi/2}  H_w(v,w,\theta) \, d \theta$.
 We now set $w=0$, $v=v_0$, and solve for $v'(0)$,
\begin{equation} \label{12}
v'(0)=-\frac{ G_w(v_0,0)}{G_v(v_0,0)}.
\end{equation}
After calculating $v'(0)$ from (\ref{12}), we are able to calculate
$\lambda '(0)$ from (\ref{9}).


\begin{example} \rm
 Let $k=2$, i.e. $f(u)=u^4$. Using {\em Mathematica},
we calculate $\lambda_0 \simeq 6.454$, $v_0 \simeq 2.383$,
$v'(0) \simeq -0.542$, and $\lambda '(0) \simeq -3.160$.
Conclusion: we have a pitchfork bifurcation at $\lambda_0 \simeq 6.454$,
with the pitchfork facing forward in $\lambda$.
\end{example}

One can verify that $\lambda '(0)<0$ for larger $k$ too
(the values of  $\lambda '(0)<0$ increase with $k$, and $\lambda '(0) \simeq -2.003$
at $k=720$), although at $k=721$ (and larger $k$) our program runs into
a problem: {\em Mathematica} is unable to calculate the integral
for $G_w(v_0,0)$ to the  accuracy it desires. When we had replaced
{\em Mathematica}'s {\bf NIntegrate} command by a ``home-made" numerical
integration routine, the program worked for larger $k$ too, and the
results were similar. However, we  state the next result conservatively.

\begin{theorem}\label{thm:3}
Consider the problem
\begin{equation} \label{13}
u''(x)+u^{2k}(x)-\lambda=0, \quad \text{for $-1<x<1$}, \; u(-1)=u(1)=0,
\end{equation}
with $1 \leq k \leq 720$. Compute $\lambda _0=\lambda _0(k)$ by using the
formulas \eqref{8} and \eqref{6}.
Then there is a negative $\bar \lambda=\bar \lambda(k)<0$, so that
the problem \eqref{13} has exactly two positive solutions for
$\bar \lambda <\lambda < 0$, it has exactly one positive and one negative
solution on $(0,\lambda _0)$. Moreover, there is a $\lambda _1 (k)>\lambda _0$,
so that the problem \eqref{13} has  four solutions on $(\lambda _0,\lambda _1)$,
one negative (and symmetric), one sign-changing and symmetric (with $u(0)>0$),
and two asymmetric solutions (see Figure 2).
\end{theorem}

\begin{proof}
It is well known that at $\lambda =0$ there exists a unique
 positive solution. This solution is known to be non-degenerate,
so that we can continue it for small $\lambda >0$. Setting $u(x)=\mu v(x)$,
with $\mu$ determined by the relation $\mu ^{2k-1}=\lambda$,
we convert the problem \eqref{13} into
\begin{equation}
\label{14}
v''(x)+\lambda(v^{2k}(x)-1)=0, \quad \text{for }-1<x<1,\; v(-1)=v(1)=0.
\end{equation}
With the parameter now in front of the nonlinearity, the results
of \cite{KLO} apply. They imply that we can always continue both positive
and sign-changing  solutions of (\ref{14}) (and hence of \eqref{13},
 and that the curve of positive solutions does not turn for $\lambda >0$
(for $g(v) \equiv v^{2k}-1$, we have $vg'(v)>g(v)$ for all $v>0$).
By Korman \cite{K}  this curve of positive solutions cannot be
continued for all $\lambda >0$ (the function $g(v) = v^{2k}-1$ has
no ``stable" roots, i.e. roots where derivative is negative).
By the Sturm's comparison theorem, it is easy to see that positive
solutions cannot become unbounded at a finite $\lambda$. Hence, solutions
on this curve must eventually stop being positive, and the only way this
can happen is that $u'(\pm 1)=0$ at some $ \lambda _0$ (in view of the
symmetry of positive solutions). By P. Korman \cite{K1} a pitchfork
bifurcation occurs at $ \lambda _0$, and by the result of the present paper,
 the pitchfork faces forward in $\lambda$.
\end{proof}


\begin{figure}[ht]
\begin{center}
\includegraphics[width=0.7\textwidth]{fig2}
\end{center}
\caption{Pitchfork bifurcation}
\end{figure}


\noindent {\bf Remarks}
\begin{enumerate}
\item
The bifurcation diagram for the Theorem \ref{thm:3} is given in Figure 2,
where we draw $u'(-1)$ as a function of $\lambda$. In that figure solid lines
denote positive and negative solutions, the dashed line denotes
sign-changing symmetric solutions, and the doted lines stand for the
symmetry breaking solutions.
\item
Our result applies to more general $f(u)$. If one is only interested in
local direction of pitchfork bifurcation, one can consider any
differentiable $f(u)$, with $f(0)=0$.
\item
Our computations constitute a {\em proof} that the pitchfork opens forward,
rather than a numerical simulation. We have computed the integrals
in (\ref{12}) by using a sophisticated adaptive routine of {\em Mathematica}.
We had $\lambda '(0) <-2$ for all $1 \leq k \leq 720$. Even assuming a
$50 \%$ relative error, $\lambda '(0)$ is still negative. {\em Mathematica}'s
relative error is much less than that, and in fact our program quit at
$k=721$, when it could not deliver high accuracy. If someone desires an
absolute assurance, one can do error analysis of the integration method,
together with computations in exact arithmetics. This would be very
time consuming, but straightforward.

\item
It was shown in P. Korman \cite{K1} that the problem \eqref{13}
has infinitely many points of pitchfork bifurcation. It follows
from  our result that they all face forward.
\end{enumerate}


\begin{thebibliography}{00}

\bibitem{AS} V. Anuradha and R. Shivaji,
\emph{Existence of infinitely many non-trivial bifurcation points},
 Results in Math., {\bf 22}, 641-650 (1992).

\bibitem{K} P. Korman,
\emph{Curves of sign-changing solutions for semilinear equations},
 Nonlinear Anal. TMA, {\bf 51}, no. 5,  801-820 (2002).

\bibitem{K1} P. Korman,
\emph{The global solution set for a class of semilinear problems},
J. Math. Anal. Appl., {\bf 226},  no. 1, 101-120 (1998).

\bibitem{KLO} P. Korman, Y. Li and T. Ouyang,
\emph{Exact multiplicity results for boundary value
      problems with nonlinearities generalizing cubic},
Proc. Royal Soc. Edinburgh,  Ser. A. {\bf 126A}, 599-616 (1996).

\bibitem{MS} H. P. McKean and J.C. Scovel,
\emph{Geometry of some simple nonlinear differential operators},
Ann. Scuola Norm. Sup. Pisa Cl. Sci.  (4) {\bf 13}, no. 2, 299-346, (1986).

\bibitem{R} M. Ramaswamy,
\emph{On the global set of solutions of a nonlinear ODE:
 Theoretical and numerical description},  J. Differential Equations,
 {\bf 65}, 1-48 (1986).

\bibitem{S} J. C. Scovel,
\emph{Geometry of Some Nonlinear Differential Operators},
 Ph. D. thesis, Courant Institute, NYU (1983).

\end{thebibliography}
\end{document}