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\frac{du}{dt} = f(u,\alpha), \quad u\in \ensuremath{\mathbf{R}}^{n},\alpha\in \ensuremath{\mathbf{R}}\quad f: \ensuremath{\mathbf{R}}^{n+1}\to \ensuremath{\mathbf{R}}^n\ .
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3662}%
$f(u,\alpha)=0$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath3481}%
\begin{displaymath}
F(x) = f(u,\alpha) = 0
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3664}%
$x=(u,\alpha)\in \ensuremath{\mathbf{R}}^{n+1}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3666}%
$v\in \ensuremath{\mathbf{R}}^{n+1}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

\stepcounter{subsection}
\stepcounter{subsection}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay12846}%
$\displaystyle \phi_1(u,\alpha)$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay12848}%
$\displaystyle \det\left(\begin{array}{c}
F_x\\
{v}^{T}\end{array}\right),$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay12849}%
$\displaystyle \phi_2(u,\alpha)$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay12851}%
$\displaystyle \left( \left[\begin{array}{cc}(2f_u(u,\alpha) \odot I_n)& w_1\\  v_1^T& d\end{array} \right] \: \backslash \left(\begin{array}{c}0\\  ...\\ 0\\  1\end{array} \right) \right) _{n+1},$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay12852}%
$\displaystyle \phi_3(u,\alpha)$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay12854}%
$\displaystyle v_{n+1},$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3670}%
$\odot$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3672}%
$v_1$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3674}%
$w_1$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3676}%
$\frac{n(n-1)}{2}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3678}%
$\phi_1=0$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3680}%
$\phi_2=0$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3682}%
$\phi_3=0, \ \phi_1\neq 0$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3686}%
$f_x$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath3527}%
\begin{displaymath}
S = \left(\begin{array}{ccc}
0 & - & -\\
- & 0 & -\\
1 & - & 0
\end{array}\right)
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath3533}%
\begin{displaymath}
a=\frac{1}{2} p^Tf_{uu}[q,q],

\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3688}%
$f_u q=f^T_u p =0, q^Tq =1, p^Tq =1$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3690}%
$a \neq 0.$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3692}%
$a$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3694}%
$\phi_3$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3696}%
$a=0$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath3542}%
\begin{displaymath}
l_1=\frac{1}{2}{\rm Re}\left\{p^T
\left(f_{uuu}[q,q,\bar{q}]-2f_{uu}[q,F^{-1}_uf_{uu}[q,\bar{q}]]+
f_{uu}[\bar{q},(2i\omega I_n-f_u)^{-1}f_{uu}[q,q]] \right)\right\},

\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3698}%
$f_uq=i\omega q,\  q^Tq =1,\ f^{T}_u p=-i\omega p,\ \bar{p}^Tq=1$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3700}%
$l_1<0$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3702}%
$l_1>0$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3704}%
$l_1=0$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

\stepcounter{subsubsection}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3706}%
$p\in \ensuremath{\mathbf{R}}^n$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3708}%
$\beta\in \ensuremath{\mathbf{R}}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath3569}%
\begin{displaymath}

\left\{ \begin{array}{rcl}
f(u,\alpha) + \beta p &=& 0 \\
f_u^T(u,\alpha)p &=& 0 \\
p^Tf_\alpha(u,\alpha) &=& 0 \\
p^Tp-1 &=& 0
\end{array}\right.
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3710}%
$\beta=0$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3712}%
$p: f_u^Tp=\mu p$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3714}%
$\mu$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3716}%
$(u,\alpha)$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3718}%
$(u,\alpha,0,p)$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3720}%
$f$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3722}%
$u$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3724}%
$\alpha$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

\stepcounter{subsection}
\stepcounter{subsection}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay12891}%
$\displaystyle x'$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay12893}%
$\displaystyle y - 2x + a e^{x}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay12894}%
$\displaystyle y'$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay12896}%
$\displaystyle x - 2y + a e^{y}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3953}%
$a=0.2$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3955}%
$(x,y,a)=(0,0,0)$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3963}%
$(x,y,a)\approx(3.0;3.0;0.15)$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3967}%
$\rightarrow$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlpictureA{tex2html_wrap13017}%
\includegraphics[scale=0.5]{ex/bratu3d.eps}%
\lthtmlpictureZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3979}%
$s$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3981}%
$s(6).$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3983}%
$p$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlpictureA{tex2html_wrap13024}%
\includegraphics[scale=0.5]{ex/bratuBP.eps}%
\lthtmlpictureZ
\lthtmlcheckvsize\clearpage}

\stepcounter{section}
\stepcounter{subsection}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay13028}%
$\displaystyle \frac{du}{dt} = f(u,\alpha)$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4635}%
$u \in \ensuremath{\mathbf{R}}^n$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4637}%
$\alpha \in \ensuremath{\mathbf{R}}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4639}%
$T$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay13033}%
$\displaystyle \left\{ \begin{array}{l}
\frac{du}{dt} = f(u,\alpha) \\
u(0) = u(T)
\end{array}\right.\ .$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay13036}%
$\displaystyle \left\{ \begin{array}{l}
\frac{du}{d\tau} = T\:f(u,\alpha) \\
u(0) = u(1)
\end{array} \right.\ .$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4643}%
$u(\tau)$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4645}%
$u(\tau+s)$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay13041}%
$\displaystyle \left\{ \begin{array}{rl}
\frac{du}{d\tau} - Tf(u,\alpha) & = 0 \\
u(0) - u(1) & = 0 \\
\int_0^1 \langle u(t),\dot u_{old}(t) \rangle dt & = 0
\end{array} \right.$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4649}%
$\dot u_{old}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

\stepcounter{subsection}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4651}%
$F(u,T,\alpha)$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4653}%
$[0,1]$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4655}%
$\times$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

\stepcounter{subsection}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4679}%
$1 \leq e1,e2 \leq nphase$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4681}%
$(e1,e2)$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4683}%
$1 \leq e2,e3 \leq nphase.$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4685}%
$(e2,e3)$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

\stepcounter{subsection}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4687}%
$(0.3;0.5;-0.1)$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4689}%
$(1,0.8)$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4691}%
$300.$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4693}%
$x1$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlfigureA{figure4220}%
\begin{figure}
\centering 
\centerline{\psfig{figure=ex/SelCyc1.eps,height=6cm}}
	

\end{figure}%
\lthtmlfigureZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4697}%
$t$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4705}%
$p=(1;0.8)$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4707}%
$ap=[2]$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4709}%
$20$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4711}%
$4.$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlfigureA{figure4228}%
\begin{figure}
\centering 
\centerline{\psfig{figure=ex/SelCyc2.eps,height=6cm}}
	

\end{figure}%
\lthtmlfigureZ
\lthtmlcheckvsize\clearpage}

\stepcounter{subsection}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay13086}%
$\displaystyle \left\{ \begin{array}{rl}
\dot v(\tau) - T f_u(u,\alpha)v(\tau) + G\varphi(\tau) & = 0 \\
v(0) + v(1) & = 0 \\
\int_0^1 \langle \psi(\tau),v(\tau)\rangle d\tau & = 1
\end{array} \right.$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4715}%
$\varphi$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4717}%
$\psi$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4719}%
$G \in \ensuremath{\mathbf{R}}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4723}%
$M_D$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4729}%
$h \in {\cal C}^1([0,1],{\mathrm{I\!R}}^n)$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath4731}%
\begin{displaymath}
Mh=
\left[
\begin{array}{c}
\dot{h}-Tf_x(x(t),\alpha)h \\
h(0)-h(1)
\end{array}
\right],
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath4733}%
\begin{displaymath}
M_D(h)_{dm}=
\left[
\begin{array}{c}
(\dot{h}-Tf_x(x(t),\alpha)h)_{dc} \\
h(0)-h(1)
\end{array}
\right],
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4735}%
$()_{dm}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4737}%
$()_{dc}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath4741}%
\begin{displaymath}
M_{Dbb}=
\left[
\begin{array}{ccc}
M_D & w_1 & w_2 \\
v_1^* & 0 & 0\\
v_2^* & 0 & 0
\end{array}
\right],
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4743}%
$M_{Dbb}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath4745}%
\begin{displaymath}
M_{Dbb}
\left[
\begin{array}{cc}
\psi_{11} & \psi_{12} \\
g_{BPC11} & g_{BPC12}  \\
g_{BPC21} & g_{BPC22}
\end{array}
\right]
= 
\left[
\begin{array}{cc}
0_{(Nm+1)n} & 0_{(Nm+1)n} \\
1 & 0 \\
0 & 1
\end{array}
\right],
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4747}%
$\psi_{11},\psi_{12}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4749}%
$(Nm+1)n$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4751}%
$g_{BPC11}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4753}%
$g_{BPC12}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4755}%
$g_{BPC21}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4757}%
$g_{BPC22}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay13110}%
$\displaystyle S = \left(\begin{array}{cccccccc}
0 & 0 & 0 & 0 & - & - & - & - \\
- & - & - & - & - & 0 & - & - \\
- & - & - & - & - & - & 0 & - \\
- & - & - & - & 1 & - & 1 & 0
\end{array}\right)$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4759}%
$4$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

\stepcounter{subsubsection}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay13119}%
$\displaystyle \left\{ \begin{array}{ll}
\frac{dx}{dt} - Tf(x,\alpha) +\beta p_1& = 0 \\
x(0) - x(1) +\beta p_2& = 0 \\
\int_0^1 \langle x(t),\dot x_{old}(t) \rangle dt+\beta p_3 & = 0 \\
G[x,T,\alpha] & = 0
\end{array} \right.$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4773}%
$[p_1^T p_2^T p_3]^T$%
\lthtmlinlinemathZ
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{\newpage\clearpage
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$[w_{01};w_{02};w_{03}]^T$%
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4777}%
$x,T,\alpha$%
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4779}%
$\beta$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4783}%
$(x,T,\alpha)$%
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4785}%
$(x,T,\alpha,0)$%
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4793}%
$v_{1st}$%
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{\newpage\clearpage
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$v_{1st}=\frac{v_1+v_2}{2}$%
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4799}%
$v_2$%
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\begin{displaymath}

\left\{\begin{array}{rcl}
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{\newpage\clearpage
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{\newpage\clearpage
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$\xi$%
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{\newpage\clearpage
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$a,b\in \mathbb{R}$%
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{\newpage\clearpage
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$O(\xi^3)$%
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\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
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$b$%
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{\newpage\clearpage
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$b\neq 0$%
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{\newpage\clearpage
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$b=0$%
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{\newpage\clearpage
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\begin{displaymath}

\left\{\begin{array}{rcl}
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{\newpage\clearpage
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$\tau \in [0,2T]$%
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{\newpage\clearpage
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$a,c \in \mathbb{R}$%
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4825}%
$2T$%
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4827}%
$O(\xi^4)$%
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\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4829}%
$c$%
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\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4831}%
$c<0$%
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\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
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$c>0$%
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4835}%
$c=0$%
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{\newpage\clearpage
\lthtmldisplayA{displaymath4358}%
\begin{displaymath}

\left\{\begin{array}{rcl}
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4841}%
$\tau$%
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{\newpage\clearpage
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$a \in \mathbb{R},d \in \mathbb{C}$%
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4847}%
$O(|\xi|^4)$%
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\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4849}%
$d$%
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4851}%
${\rm Re}~d<0$%
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4853}%
${\rm Re}~d>0$%
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4855}%
${\rm Re}~d=0$%
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\stepcounter{subsection}
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay13170}%
$\displaystyle \left\{\begin{array}{rcl}
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\dot z & = & -\alpha z -\beta y -x +x^2
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4857}%
$\alpha = 1$%
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4859}%
$\beta = 1$%
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\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4861}%
$245 \times 200$%
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\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4863}%
$243=(20 \times 4 +1)\times 3$%
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\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4867}%
$244-th$%
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4871}%
$245-th$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
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\includegraphics[scale=0.8]{ex/LCcurve.eps}%
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{\newpage\clearpage
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{\newpage\clearpage
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$fvector$%
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\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlfigureA{figure4482}%
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{\newpage\clearpage
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\begin{figure}
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\begin{displaymath}
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\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
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$(u,\alpha) \in {\bf R}^{n+2}$%
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\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6490}%
$g$%
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\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath5322}%
\begin{displaymath}
\left(
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\end{array}
\right)
\left(
\begin{array}{c}
v \\g
\end{array}
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0_n \\1
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\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6492}%
$w_{bor},v_{bor} \in \ensuremath{\mathbf{R}}^n$%
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\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6496}%
$f_u(u,\alpha)$%
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\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath6498}%
\begin{displaymath}
g_z = -w^T(f_u)_z v
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\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6500}%
$z$%
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\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath5340}%
\begin{displaymath}
\left(
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\left(
\begin{array}{c}
w \\
g
\end{array}
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=
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0_n \\
1
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\lthtmlcheckvsize\clearpage}

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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6502}%
$bp+3$%
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6504}%
$bp$%
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\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6506}%
$\phi_1=w^Tv$%
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\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath5368}%
\begin{displaymath}\phi_2(u,\alpha) =  
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\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
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$\phi_3=w^Tf_{uu}[v,v]$%
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6510}%
$\phi_i=w^Tf_{\beta_i}(u,\alpha)$%
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6512}%
$v, w$%
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\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6522}%
$\beta_i$%
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\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6526}%
$bp=0$%
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\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath5386}%
\begin{displaymath}
S = \left(\begin{array}{ccc}
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\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath6530}%
\begin{displaymath}
S = \left(\begin{array}{ccccc}
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- & - & 0 & - & -\\
- & - & - & 0 & -\\
- & - & - & - & 0
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\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6532}%
$(,bp)_{optional}$%
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\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6536}%
$x_{new}$%
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\lthtmlcheckvsize\clearpage}

\stepcounter{subsubsection}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6538}%
$v_{bor}$%
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\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6540}%
$w_{bor}$%
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\lthtmlcheckvsize\clearpage}

\stepcounter{subsubsection}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6550}%
$q_1, q_2, q_3, q_4, q_5, q_6, k$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6552}%
$q_2$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlpictureA{tex2html_wrap13250}%
\includegraphics[scale=0.6]{ex/testequilcataloscill.eps}%
\lthtmlpictureZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6560}%
$k$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlpictureA{tex2html_wrap13256}%
\includegraphics[scale=0.6]{ex/testLPcataloscill.eps}%
\lthtmlpictureZ
\lthtmlcheckvsize\clearpage}

\stepcounter{subsection}
\stepcounter{subsubsection}
{\newpage\clearpage
\lthtmldisplayA{displaymath5469}%
\begin{displaymath}
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\right.

\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6562}%
$u,\alpha,k,(i_1,j_1,i_2,j_2)\in\{1,2\}$%
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\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6564}%
$g=\left(
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\right)$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath5484}%
\begin{displaymath}
\left(
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\begin{array}{c}
V\\
G
\end{array}
\right)
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0_{n,2}\\I_2
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\right),

\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6566}%
$f_u$%
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\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6568}%
$\pm i\omega,\omega>0$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6570}%
$k=\omega^2$%
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\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6572}%
$V_{bor},W_{bor} \in \ensuremath{\mathbf{R}}^{n\times 2}$%
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\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6574}%
$i_1,j_1,i_2,j_2,V_{bor}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6576}%
$W_{bor}$%
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\lthtmlcheckvsize\clearpage}

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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6578}%
$\phi_1=k$%
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\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6580}%
$\phi_2=\det(f_u)$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6582}%
$\phi_3=\left( \left[\begin{array}{cc}(2f_u(u,\alpha) \odot I_n)& w_1\\v_1^T& d\end{array} \right] \: \backslash \left(\begin{array}{c}0\\...\\0\\1\end{array} \right) \right) _{n+1}$%
\lthtmlinlinemathZ
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6584}%
$\phi_4=l_1$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6590}%
$\frac{n(n-1)}{2} \times 2$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath5529}%
\begin{displaymath}
S = \left(\begin{array}{cccc}
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- & - & - & 0
\end{array}\right)
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

\stepcounter{subsubsection}
\stepcounter{subsubsection}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6596}%
$V_{bor}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6604}%
$i_1,j_1,i_2,j_2$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

\stepcounter{subsubsection}
{\newpage\clearpage
\lthtmlpictureA{tex2html_wrap13295}%
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\stepcounter{subsection}
\stepcounter{subsubsection}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay13299}%
$\displaystyle \left\{ \begin{array}{ll}
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u(0) - u(1) & = 0 \\
\int_0^1 \langle u(t),\dot u_{old}(t) \rangle dt & = 0 \\
G(u,T,\alpha) & = 0
\end{array} \right.$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6614}%
$G(u,T,\alpha) = 0$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6616}%
$G(u,T,\alpha)$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay13307}%
$\displaystyle \left\{ \begin{array}{ll}
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u(0) - u(1) & = 0 \\
\int_0^1 \langle u(t),\dot u_{old}(t) \rangle dt & = 0 \\
\dot v(\tau) - T f_u(u,\alpha)v(\tau) & = 0 \\
v(0) + v(1) & = 0 \\
\int_0^1 \langle v_{old}(\tau),v(\tau)\rangle d\tau -1 & = 0
\end{array} \right.$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

\stepcounter{subsubsection}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6622}%
$x-$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

\stepcounter{subsubsection}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6626}%
$\phi_1=(v_1^*)^T_{W_1}L_{C\times M}v_{1M}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6628}%
$\phi_2=(\psi_1^*)^T_{W_1}(F(u_{0,1}(t)))_C$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6630}%
$\phi_3=det\left(\left(M \odot M \right) - I_n \right)$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6632}%
$\phi_4=c$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6636}%
$M$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6642}%
$\psi_1$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6644}%
$\zeta \in {{\cal C}}^1([0,1],{\mathbb{R}^n})$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6646}%
$\zeta_M \in {\mathbb{R}^{(Nm+1)n}}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6648}%
$\zeta_C \in {\mathbb{R}^{Nmn}}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6650}%
$\zeta_W=\left[\begin{array}{c}\zeta_{W_1}\\\zeta_{W_2}\end{array}\right] \in {\mathbb{R}^{Nmn}}\times{\mathbb{R}^n}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6652}%
$\zeta_{W_1}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6654}%
$\zeta_{W_2}=\zeta(0)$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6656}%
$L_{C\times M}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6658}%
$\zeta_M$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6660}%
$\zeta_C$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6662}%
$\zeta_C=L_{C\times M}\zeta_M$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6664}%
$v_{1M}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath5642}%
\begin{displaymath}
\left[\begin{array}{c}D-TA(t)\\
\delta_0+\delta_1
\end{array}\right]_{disc}v_{1M}=0.

\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6668}%
$\sum_{i=1}^N\sum_{j=0}^m\sigma_j\langle(v_{1M})_{(i-1)m+j},(v_{1M})_{(i-1)m+j}\rangle\Delta_i=1$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6670}%
$\sigma_j$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6672}%
$\Delta_i$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath6674}%
\begin{displaymath}(v_{1W}^*)^T\left[\begin{array}{c}D-TA(t)\\\delta_0+\delta_1\end{array}\right]_{disc}=0.\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6676}%
$(v_1^*)_{W_1}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6678}%
$\sum_{i=1}^N\sum_{j=1}^m\left|((v_1^*)_{W_1})_{ij}\right|_1=1$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6680}%
$\int_{0}^{1} {\langle v_1^*(t),v_1(t)\rangle dt}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6682}%
$(v_1^*)^T_{W_1}L_{C\times M}v_{1M}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6684}%
$v^*_{1W}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6686}%
$\int_{0}^{1} {\langle v_1^*(t),v_1(t)\rangle dt}=\frac{1}{2}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6688}%
$\psi_{1W}^*$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath6690}%
\begin{displaymath}(\psi_{1W}^*)^T\left[\begin{array}{c}D-TA(t)\\\delta_0-\delta_1\end{array}\right]_{disc}=0\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6692}%
$\psi_{1W_1}^*$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6694}%
$\sum_{i=1}^N\sum_{j=1}^m\left|((\psi_1^*)_{W_1})_{ij}\right|_1=1$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6696}%
$\int_{0}^{1} {\langle \psi_1^*(t),F(u_{0,1}(t))\rangle dt}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6698}%
$(\psi_1^*)^T_{W_1}(F(u_{0,1}(t)))_C$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6700}%
$\psi^*_{1W}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6702}%
$\int_{0}^{1} {\langle \psi_1^*(t),F(u_{0,1}(t))\rangle dt}=1$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6704}%
$a_1$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6706}%
$(\psi^*_{W_1})^T(B(t,v_{1M},v_{1M}))_C$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6708}%
$(h_{2,1})_M$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath6710}%
\begin{displaymath}
\left\{\begin{array}{rcl}
(D-TA(t))_{C\times M}(h_{2,1})_M&=& (B(t;v_{1M},v_{1M}))_C + 2a_1(F(u_{0,1}(t)))_C,\ t \in [0,1] \\
(\delta(0)-\delta(1))(h_{2,1})_M&=&0,\\
(\psi^*_{W_1})^TL_{C\times M}(h_{2,1})_M&=&0.
\end{array}
\right.
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath6714}%
\begin{displaymath}
\begin{array}{c}
c=\frac{1}{3}((v^*_{1W_1})^T(C(t;v_{1M},\frac{1}{T}v_{1M},v_{1M})_C+3(B(t;v_{1M},(h_{2,1})_M)_C\\
-6\frac{a_1}{T}(v^*_{1W_1})^T(A(t)v_{1}(t))_C).
\end{array}
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath5732}%
\begin{displaymath}
S = \left(\begin{array}{ccc}
0 & - & - \\
- & 0 & -\\
1 & 1 & 0
\end{array}\right).
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

\stepcounter{subsubsection}
\stepcounter{subsubsection}
{\newpage\clearpage
\lthtmlpictureA{tex2html_wrap13368}%
\includegraphics[scale=0.8]{ex/PDbifurcation.eps}%
\lthtmlpictureZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6720}%
$246 \times 300$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6728}%
$\alpha, \beta$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6730}%
$245-$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6732}%
$246-$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

\stepcounter{subsection}
\stepcounter{subsubsection}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6734}%
$G=0$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay13381}%
$\displaystyle \left\{ \begin{array}{ll}
\frac{du}{dt} - Tf(u,\alpha) & = 0 \\
u(0) - u(1) & = 0 \\
\int_0^1 \langle u(t),\dot u_{old}(t) \rangle dt & = 0 \\
G[u,T,\alpha] & = 0
\end{array} \right.$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath5787}%
\begin{displaymath}
N^1\left(
\begin{array}{c}
        v\\
       S \\
       G
\end{array}
\right)= \left(\begin{array}{c}
0 \\
0\\
0\\
1
\end{array}\right).

\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath5796}%
\begin{displaymath}
 N^1 =
\left[
\begin{array}{ccc}
D-Tf_u(u(t),\alpha) &~~~ -f(u(t),\alpha) &~~~w_{01}\\
\delta_1-\delta_0 &~~~ 0 &~~~ w_{02}\\
\mbox{\em Int}_{f(u(\cdot),\alpha)} &~~~ 0 &~~~ w_{03} \\
\mbox{\em Int}_{v_{01}} &~~~ v_{02} &~~~ 0
\end{array}
\right]

\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6746}%
$v_{01},w_{01}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6748}%
$w_{02}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6750}%
$v_{02}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6752}%
$w_{03}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6754}%
$N^1$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

\stepcounter{subsubsection}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6756}%
$bp+5$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6760}%
$\psi_i=w^Tf_{\beta_i}(u,\alpha),i\in (1,...,bp)$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6762}%
$\psi_{bp+1}=(\varphi_1^*)^T_{W_1}L_{C\times M}v_{1M}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6764}%
$\psi_{bp+2}= b$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6766}%
$\phi_{bp+3}=det\left( M + I_n \right)$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6768}%
$\psi_{bp+4}=det\left(\left( M2 \odot M2 \right) - I_n \right)$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6770}%
$\psi_i$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6778}%
$\psi_{bp+1}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath5849}%
\begin{displaymath}
\left[\begin{array}{c}D-TA(t)\\
\delta_0-\delta_1\\
\int_0^1{F(u_{0,1}(t))L_{C\times M}}
\end{array}\right]_{disc}v_{1M}=
\left[\begin{array}{c}TF(u_{0,1}(t))\\
0\\
0
\end{array}\right]_{disc}.

\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath6782}%
\begin{displaymath}(\varphi_{1W}^*)^T\left[\begin{array}{c}D-TA(t)\\\delta_0-\delta_1\end{array}\right]_{disc}=0.\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6784}%
$(\varphi_1^*)_{W_1}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6786}%
$\sum_{i=1}^N\sum_{j=1}^m\left|((v_1^*)_{W_1})_{(i-1)m+j}\right|_1=1$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6788}%
$\int_{0}^{1} {\langle \varphi_1^*(t),v_1(t)\rangle dt}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6790}%
$(\varphi_1^*)^T_{W_1}L_{C\times M}v_{1M}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6792}%
$\varphi^*_{1W}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6794}%
$\int_{0}^{1} {\langle \varphi_1^*(t),v_1(t)\rangle dt}=1$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath6798}%
\begin{displaymath}b = \frac{1}{2}((\varphi_{1W_1}^*)^T((B(t;v_{1M},v_{1M})_C+2(A(t)v_{1}(t))_C)).\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6802}%
$M2$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6804}%
$(n-2) \times (n-2)$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6806}%
$n \times n$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath6814}%
\begin{displaymath}
S = \left(\begin{array}{ccccc}
 0 & - & - & - & -\\
 - & 0 & - & - & -\\
 - & - & 0 & - & -\\
 - & - & - & 0 & -\\
 - & - & - & 1 & 0
\end{array}\right).
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

\stepcounter{subsubsection}
\stepcounter{subsubsection}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay13429}%
$\displaystyle \left\{\begin{array}{rcl}
\dot v&=&y-0.5(v+0.5)-2w(v+0.7)-m_{\infty}(v-1)\\
\dot w&=&1.15(w_{\infty}-w)\tau
\end{array}\right.$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6818}%
$m_{\infty}(v)=(1+\tanh((v+0.01)/0.15))/2$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6820}%
$w_{\infty}(v)=(1+\tanh((v-z)/0.145))/2$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6822}%
$\tau=\cosh((v-0.1)/0.29).$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6832}%
$w_{\infty}(z)=\frac{1}{2}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6834}%
$y=0.110472$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6836}%
$z=0.1$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6838}%
$(0.04722,0.32564)$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlpictureA{tex2html_wrap13446}%
\includegraphics[scale=0.6]{ex/LPLC1.eps}%
\lthtmlpictureZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6844}%
$y=0.075659$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6846}%
$y=-0.020727$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6848}%
$y=0.083257$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6850}%
$y=0.033207$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6852}%
$l_1=8.234573$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6858}%
$N=30$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6860}%
$m=4$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6862}%
$y=0.084569$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlpictureA{tex2html_wrap13460}%
\includegraphics[scale=0.7]{ex/LPLC2.eps}%
\lthtmlpictureZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlpictureA{tex2html_wrap13466}%
\includegraphics[scale=0.8]{ex/Figure22.eps}%
\lthtmlpictureZ
\lthtmlcheckvsize\clearpage}

\stepcounter{subsection}
\stepcounter{subsubsection}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay13472}%
$\displaystyle \left\{ \begin{array}{ll}
\frac{dx}{dt} - Tf(x,\alpha) & = 0 \\
x(0) - x(1) & = 0 \\
\int_0^1 \langle x(t),\dot x_{old}(t) \rangle dt & = 0 \\
G[x,T,\alpha,\kappa] & = 0
\end{array} \right.$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath6872}%
\begin{displaymath}
G=\left(\begin{array}{cc}
G^{11}&G^{12}\\
G^{21}&G^{22}
\end{array}\right)\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath5982}%
\begin{displaymath}
N^3\left(
\begin{array}{cc}
        v^1&v^2\\
       G^{11}&G^{12} \\
       G^{21}&G^{22}
\end{array}
\right)= \left(\begin{array}{cc}
0&~~0 \\
0&~~0\\
1&~~0\\
0&~~1
\end{array}\right).

\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6874}%
$v^1$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6876}%
$v^2$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6878}%
$G^{11},G^{12},G^{21}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6880}%
$G^{22}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath5999}%
\begin{displaymath}
 N^3 =
\left[
\begin{array}{ccc}
D-Tf_x(x(\cdot),\alpha) &~~~ w_{11} &~~~w_{12}\\
\delta_0-2\kappa\delta_1+\delta_2 &~~~ w_{21} &~~~ w_{22}\\
\mbox{\em Int}_{v_{01}} &~~~ 0 &~~~ 0 \\
\mbox{\em Int}_{v_{02}} &~~~ 0 &~~~ 0
\end{array}
\right]

\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6882}%
$v_{01},v_{02},w_{11},w_{12}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6884}%
$w_{21}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6886}%
$w_{22}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6888}%
$N^3$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6890}%
$\kappa$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6892}%
$[0\ 2]$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

\stepcounter{subsubsection}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6896}%
$\psi_1=\kappa-1$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6898}%
$\psi_2=\kappa+1$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6900}%
$\psi_3=\kappa-1/2$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6902}%
$\psi_4=\kappa$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6904}%
$\psi_5=(v_1^*)^T_{W_1}L_{C\times M}v_{1M}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6906}%
$\psi_6=Re(d)$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6908}%
$\psi_7=det\left( M + I_n \right)$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6910}%
$\psi_8=det\left(\left( M2 \odot M2 \right) - I_n \right)$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath6051}%
\begin{displaymath}

\left[\begin{array}{c}D-TA(t)+i\theta I\\
\delta_0-\delta_1
\end{array}\right]_Dv_{1M}=0.
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6916}%
$\sum_{i=1}^N\sum_{j=0}^m\sigma_j\langle(v_{1M})_{ij},(v_{1M})_{ij}\rangle t_i=1$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath6920}%
\begin{displaymath}(v_{1W}^*)^H\left[\begin{array}{c}D-TA(t)+i\theta \\\delta_0+\delta_1\end{array}\right]_{disc}=0.\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6932}%
$\int_{0}^{1} {\langle v_1^*(t),v_1(t)\rangle dt}=1$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6934}%
$\varphi_{1W}^*$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath6936}%
\begin{displaymath}(\varphi_{1W}^*)^T\left[\begin{array}{c}D-TA(t)\\\delta_0-\delta_1\end{array}\right]_{disc}=0\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6938}%
$\varphi_{1W_1}^*$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6940}%
$\sum_{i=1}^N\sum_{j=1}^m\left|((\varphi_1^*)_{W_1})_{ij}\right|_1=1$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6942}%
$\int_{0}^{1} {\langle \varphi_1^*(t),F(u_{0,1}(t))\rangle dt}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6944}%
$(\varphi_1^*)^T_{W_1}(F(u_{0,1}(t)))_C$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6948}%
$\int_{0}^{1} {\langle \varphi_1^*(t),F(u_{0,1}(t))\rangle dt}=1$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6950}%
$h_{20,1M}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath6952}%
\begin{displaymath}
\left[\begin{array}{c}D-TA(t)+2i\theta I\\
\delta_0-\delta_1
\end{array}\right]_{disc}h_{20,1M}=
\left[\begin{array}{c}B(t,v_{1M}(t),v_{1M}(t))\\
0
\end{array}\right].
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6956}%
$(\varphi^*_{W_1})^T(B(t,v_{1M},\overline{v}_{1M}))_C$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6958}%
$(h_{11,1})_M$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath6960}%
\begin{displaymath}
\left[\begin{array}{c}
(D-TA(t))_{C\times M}\\
\delta(0)-\delta(1)\\
(\varphi^*_{W_1})^TL_{C\times M}
\end{array}\right](h_{11,1})_M=
\left[\begin{array}{c}
B(t;v_{1M},\overline{v}_{1M}))_C - a_1(F(u_{0,1}(t)))_C\\
0\\
0
\end{array}
\right]
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath6964}%
\begin{displaymath}
\begin{array}{c}
d= \frac{1}{2}((v_{1W_1}^*)^T,(B(t;h_{11,1M},v_{1M}))_C+(B(t;h_{20,1M},\overline{v}_{1M}))_C+\frac{1}{T}(C(t;v_{1M},v_{1M},\overline{v}_{1M}))_C) \\
-\frac{a_1}{T}(v_{1W_1}^*)^T(A(t)v_{1}(t))_C+ \frac{ia_1\theta}{T^2}.
\end{array}
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath6158}%
\begin{displaymath}
S = \left(\begin{array}{cccccc}
0 & - & - & - & - & -\\
- & 0 & - & - & - & -\\
- & - & 0 & - & - & -\\
- & - & - & 0 & - & -\\
- & - & - & - & 0 & -\\
- & - & - & - & 1 & 0 
\end{array}\right).
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

\stepcounter{subsubsection}
\stepcounter{subsubsection}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6976}%
$x,y$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6980}%
$\beta,\gamma,r,a_3,b_3,\nu$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath6183}%
\begin{displaymath}
\left\{\begin{array}{rcl}
\dot x&=&(-(\beta+\nu)x+\beta y-a_3x^3+b_3(y-x)^3)/r\\
\dot y&=&\beta x-(\beta+\gamma)y-z-b_3(y-x)^3\\
\dot z&=&y
\end{array}\right.

\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6982}%
$x=y=z=0$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6984}%
$\beta=0.5$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6986}%
$\gamma=-0.6$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6988}%
$r=0.6$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6990}%
$a_3=0.32858$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6992}%
$b_3=0.93358$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6994}%
$\nu=-0.9$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline6996}%
$\nu$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline7000}%
$\nu=-0.58934$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath7002}%
\begin{displaymath}
\dot z=y+\epsilon  
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline7006}%
$\epsilon=0.001$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline7010}%
$x=0.00125$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline7012}%
$y=-0.001$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline7014}%
$z=0.00052502$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline7022}%
$\nu=-0.58928$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline7024}%
$x=0.0056037$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline7028}%
$z=0.0027021$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline7030}%
$N=25$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlpictureA{tex2html_wrap13566}%
\includegraphics[scale=0.7]{ex/Figure23.eps}%
\lthtmlpictureZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline7036}%
$\nu=-0.59575$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline7038}%
$\nu,\epsilon$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline7042}%
$\nu=-0.591612,$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline7044}%
$\epsilon=-0.000006\approx 0,$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline7046}%
$0.822406.$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlpictureA{tex2html_wrap13576}%
\includegraphics[scale=0.8]{ex/Figure24.eps}%
\lthtmlpictureZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline7048}%
$\beta,\nu$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline7050}%
$xns2$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline7052}%
$607 \times 50$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlpictureA{tex2html_wrap13584}%
\includegraphics[scale=0.8]{ex/Figure25.eps}%
\lthtmlpictureZ
\lthtmlcheckvsize\clearpage}

\stepcounter{section}
\stepcounter{subsection}
\stepcounter{subsubsection}
{\newpage\clearpage
\lthtmldisplayA{displaymath7643}%
\begin{displaymath}
\left\{
\begin{array}{ccl}
f(u,\alpha) & = & 0, \\
g_1(u,\alpha) & = & 0,\\
g_2(u,\alpha) & = & 0,
\end{array}
\right.

\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline7910}%
$g_1$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline7912}%
$g_2$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath7651}%
\begin{displaymath}
N^4\left(
\begin{array}{cc}
v_{11}&v_{21}\\
v_{12}&v_{22}\\
g_1&g_2
\end{array}\right)=
\left(
\begin{array}{cc}
0_n& 0_n\\
1  & 0\\
0  & 1
\end{array}
\right).

\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline7914}%
$v_{11}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline7916}%
$v_{21}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline7918}%
$v_{12},v_{22},g_1$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath7922}%
\begin{displaymath}N^4=
\left[\begin{array}{ccc}
f_u(u,\alpha)&f_{\beta}(u,\alpha)&w_{01}\\
v_0^{11T}&v_0^{12T}&0\\
v_0^{21T}&v_0^{22T}&0
\end{array}
\right]
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline7924}%
$v_0^{11},v_0^{21},w_{01}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline7926}%
$v_0^{12},v_0^{22}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline7928}%
$N^4$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

\stepcounter{subsubsection}
\stepcounter{subsubsection}
\stepcounter{subsubsection}
{\newpage\clearpage
\lthtmldisplayA{displaymath7698}%
\begin{displaymath}

\left\{
\begin{array}{rcl}
\dot x&=& \alpha_3-(1+\lambda)x+\lambda\alpha_1/(1+\lambda\alpha_2*e^{-\alpha_4x/(1+x)})
\end{array}\right.
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline7932}%
$\alpha_1=10-9*\beta+\gamma,\alpha_2=10-9*\beta$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline7934}%
$\alpha_3=-0.9+0.4*\beta$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline7936}%
$\lambda,\beta,\gamma,\alpha_4,$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline7938}%
$x=-0.9$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline7940}%
$(0,0,0,3)$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline7942}%
$\lambda$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlpictureA{tex2html_wrap13621}%
\includegraphics[scale=0.6]{ex/LPBP.eps}%
\lthtmlpictureZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline7956}%
$\lambda,\beta$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline7958}%
$\gamma$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline7962}%
$\alpha_4$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlpictureA{tex2html_wrap13631}%
\includegraphics[scale=0.8]{ex/BP.eps}%
\lthtmlpictureZ
\lthtmlcheckvsize\clearpage}

\stepcounter{subsection}
\stepcounter{subsubsection}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline7968}%
$G_1=0$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline7970}%
$G_2=0$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline7972}%
$G_1$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline7974}%
$G_2$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay13639}%
$\displaystyle \left\{ \begin{array}{ll}
\frac{dx}{dt} - Tf(x,\alpha) & = 0 \\
x(0) - x(1) & = 0 \\
\int_0^1 \langle x(t),\dot x_{old}(t) \rangle dt & = 0 \\
G[x,T,\alpha] & = 0
\end{array} \right.$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath7976}%
\begin{displaymath}
G=\left(\begin{array}{c}
G_{1}\\
G_{2}
\end{array}\right)\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath7746}%
\begin{displaymath}
N\left(
\begin{array}{cc}
        v_1&v_2\\
        G_1&G_2
\end{array}
\right)= \left(\begin{array}{cc}
0&0 \\
0&0\\
0&0\\
1&0\\
0&1
\end{array}\right).

\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath7755}%
\begin{displaymath}
 N =
\left[
\begin{array}{cccc}
D-Tf_x(x(\cdot),\alpha) &~~~ -f(x(\cdot),\alpha)&~~~ -Tf_{\beta}(x(\cdot),\alpha) &~~~w_{01}\\
\delta_1-\delta_0 &~~~ 0&~~~ 0 &~~~ w_{02}\\
\mbox{\em Int}_{\dot x_{old}(\cdot)} &~~~ 0 &~~~ 0 &~~~ w_{03} \\
v_{11} &~~~ v_{12}&~~~ v_{13} &~~~ 0\\
v_{21} &~~~ v_{22}&~~~ v_{23} &~~~ 0
\end{array}
\right]

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\dot{z}&=y
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{\newpage\clearpage
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$Y_S$%
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$Y_U$%
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$Q(0)$%
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{\newpage\clearpage
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$m$%
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{\newpage\clearpage
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$A=f_x(x_0,\alpha)$%
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{\newpage\clearpage
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$T_{ij}$%
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{\newpage\clearpage
\lthtmldisplayA{displaymath8200}%
\begin{displaymath} 

\left[ \begin{array} {cc}T_{11}\ \ T_{12}\\T_{21}\ \ T_{22}\end{array}\right] =\ Q(0)^T\:A\:Q(0).
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline8442}%
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline8444}%
$m \times m$%
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\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline8446}%
$T_{22}$%
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline8448}%
$(n-m) \times (n-m)$%
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\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath8212}%
\begin{displaymath} 

\begin{array}{l}
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\lthtmlinlinemathA{tex2html_wrap_inline8456}%
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline8460}%
$Q_U(0)$%
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\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline8462}%
$Q_U^\bot(s)$%
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\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath8464}%
\begin{displaymath}
Q_U^\bot(s) = Q_U(0) \left[ \begin{array}{c} -Y_{U}^T\\I \end{array} \right].
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{\newpage\clearpage
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$Q_{1U}$%
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\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
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\begin{displaymath} 

\begin{array}{l}
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\end{displaymath}%
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline8470}%
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline8472}%
$x(1)$%
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\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
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\begin{displaymath} 

\begin{array}{l}
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\end{displaymath}%
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{\newpage\clearpage
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$x(t)$%
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{\newpage\clearpage
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$\epsilon_0$%
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{\newpage\clearpage
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$\epsilon_1$%
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{\newpage\clearpage
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$Q$%
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{\newpage\clearpage
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{\newpage\clearpage
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{\newpage\clearpage
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$ns$%
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{\newpage\clearpage
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{\newpage\clearpage
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$p_1^u$%
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{\newpage\clearpage
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$A^T(x_0,\alpha_0)$%
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{\newpage\clearpage
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$A(x_0,\alpha_0)$%
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{\newpage\clearpage
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$(x_0,\alpha_0)$%
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{\newpage\clearpage
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$\textstyle \ \ \ \ $%
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{\newpage\clearpage
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline8578}%
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{\newpage\clearpage
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$\phi \in {\cal C}^1([0,1],{\mathbb{R}}^n)$%
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{\newpage\clearpage
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$L_u$%
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{\newpage\clearpage
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$c\phi(t)$%
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{\newpage\clearpage
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$c \in {\mathbb{R}}$%
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline8594}%
$Q_U$%
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline8598}%
$Q_S$%
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\stepcounter{subsection}
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{\newpage\clearpage
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{\newpage\clearpage
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\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline8610}%
$t \rightarrow -\infty$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline8612}%
$u(t) 
\rightarrow u_1$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline8614}%
$t \rightarrow \infty.$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

\appendix
\stepcounter{section}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline8774}%
$(x,y)$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline8776}%
$x^2+y^2=1$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath8768}%
\begin{displaymath}
F(x,y) = x^2+y^2-1
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline8867}%
$(x,y)=(1,0)$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlpictureA{tex2html_wrap13913}%
\includegraphics[scale=0.5]{ex/circle.eps}%
\lthtmlpictureZ
\lthtmlcheckvsize\clearpage}

\stepcounter{section}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline9055}%
$X(x,t),Y(x,t), A(x,t)$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline9057}%
$B(x,t)$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline9063}%
$x\in [0,1]$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline9065}%
$L$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath8987}%
\begin{displaymath}

\begin{array}{ccl}
\ensuremath{\frac{\partial X}{\partial t}} &=& \frac{D_x}{L^2}\ensuremath{\frac{\partial^2X}{\partial x^2}} + A - (B-1)X + X^2Y \\
\ensuremath{\frac{\partial Y}{\partial t}} &=& \frac{D_y}{L^2}\ensuremath{\frac{\partial^2Y}{\partial x^2}} + BX - X^2Y
\end{array}
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline9071}%
$x\in[0,1],\ t\geq 0$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline9073}%
$D_x,D_y$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline9079}%
$x=0$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline9081}%
$x=1$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath9001}%
\begin{displaymath}

\left\{ \begin{array}{l}
X(0,t) = X(1,t) = A \\
Y(0,t) = Y(1,t) = \frac{B}{A}
\end{array}\right.
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline9083}%
$X(x)$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline9085}%
$Y(x)$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath9009}%
\begin{displaymath}

\left\{ \begin{array}{ccl}
X(x) &=& A + 2\sin(\pi x) \\
Y(x) &=& \frac{B}{A} - \frac{1}{2}\sin(\pi x)
\end{array}\right.
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline9089}%
$A=2$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline9091}%
$B=4.6$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline9093}%
$D_x=0.0016$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline9095}%
$D_y=0.08$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline9097}%
$L=0.06$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline9101}%
$\ensuremath{\frac{\partial^2f}{\partial x^2}} = \frac{1}{h^2}\left(f_{i-1} - 2f_i + f_{i+1}\right)$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline9103}%
$h=\frac{1}{N+1}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline9113}%
$2N$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline9115}%
$6.5$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline9117}%
$7$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline9427}%
$A,B,D_x,D_y$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline9437}%
$0.06$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline9443}%
$x0$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlpictureA{tex2html_wrap14287}%
\includegraphics[scale=0.5]{ex/bruss2.eps}%
\lthtmlpictureZ
\lthtmlcheckvsize\clearpage}


\end{document}
