\begin{flushright}
$BC+@n@6N4(B $BLu(B
\end{flushright}

\noindent
{\it Journal of Differential Equations} (1994), {\bf 110}, 289-321

\vspace{0.5cm}
\begin{center}
{\Large
{\bf $BJ?LL;0BNLdBj$K$*$1$k%"!<%N%k%I3H;6$H?6F02r(B \\
Arnold diffusion and oscillatory solutions in \\
the planar three-body problem} 
}

\vspace{0.3cm}
Zhihong Xia

\vspace{0.3cm}
Center for Dynamical Systems and Nonlinear Studies \\
Georgia Institute of Technology, Atlanta, Georgia 30332
\end{center}

\vspace{0.5cm}
\noindent
{\bf Abstract} 

$B$3$NO@J8$G$O%K%e!<%H%s;0BNLdBj$K4X$9$k0J2<$N7k2L$rF@$k!%(B(1) $BJ?LL;0BNLdBj$K$O%"!<%N%k%I3H;6$,B8:_$9$k!%$3$l$O%"!<%N%k%I(B(V.I.Arnold, 1964, {\it Dokl. Acad. Nauk SSSR} {\bf 156}, 9)$B$K$h$C$FM=A[$5$l$F$$$?!%(B(2) $B;0BNLdBj$K$O?6F02r$dJa3M$*$h$SC&=P(B(escape)$B2r$,$=$NB>$N%+%*%9E*$J2r$H$H$b$KB8:_$9$k!%(B(3) $B5<%"!<%N%k%I3H;6(B(pseudo Arnold difusion)$B$H8F$P$l$kFC<l$JLLGr$$8=>]$,;0BNLdBj$NL58B1s$N6a$/$G@8$8$k!%(B(4) $B%"!<%N%k%I3H;6$,B8:_$9$k$?$a$KJ?LL;0BNLdBj$OHs2D@QJ,$G$"$j!"CN$i$l$F$$$k$b$N0J30$K<B2r@OE*@QJ,$O$J$$!%(B

\vspace{0.5cm}
\noindent
{\bf 1. $B=x(B} 

$B%"!<%N%k%I3H;6$O9b<!85%O%_%k%H%s7O$KH<$&%+%*%98=>]$G$"$k!%:G=i$K%"!<%N%k%I3H;6$r8+$D$1$?$H$-!"%"!<%N%k%I(B[2]$B$O$3$N%+%*%98=>]$,%K%e!<%H%s(B$n$$BBNLdBj$KB8:_$9$k$@$m$&$HM=A[$7$?!%$3$NO@J8$G$O<g7k2L$N$R$H$D$H$7$F!"%"!<%N%k%I3H;6$,J?LL;0BNLdBj$KB8:_$9$k$3$H$r<($9$3$H$K$h$j!"$3$NM=A[$N@5$7$5$r<($9!%(B

$B%i%0%i%s%8%eB?MMBN8r:9M}O@(B(Weinstein[20], Arnold[4])$B$K$h$j!"%"!<%N%k%I3H;6$O<+M3EY$,(B2$B$h$jBg$-$$@8@.E*6a2D@QJ,%O%_%k%H%s7O$G@8$8$k!%$7$+$7$J$,$i!"(B$n$$BBNLdBj$N$h$&$JFCDj$N%O%_%k%H%s7O$,M?$($i$l$?$H$-$K$=$NB8:_$r<($9$N$K$$$/$D$+Bg$-$J:$Fq$,$"$k!%;v<B!"8=<B$NJ*M}7O$G%"!<%N%k%I3H;6$NB8:_$,<($5$l$?Nc$OHs>o$K>/$J$$!%%"!<%N%k%I(B[2]$B$d(BHolmes and Marsden[9]$B$K$h$kNc$O?6;R$r$D$J$2$?Hs>o$KFC<l$J%O%_%k%H%s4X?t$N>l9g$G$"$j!"5;=QE*$J>\:Y$J$7$K$3$N8=>]$r@bL@$9$k$?$a$K$D$/$i$l$?$b$N$G$"$k!%:G6a(BLim[10]$B$,12NO3X$K$*$$$F%"!<%N%k%I3H;6$N$$$/$D$+$NNc$r=P$7$?!%(B

$B%"!<%N%k%I3H;6$NB8:_$r8!=P$9$k$?$a$N4pK\E*5;=Q$N$R$H$D$O%a%k%K%3%U$NJ}K!(B[12]$B$G$"$k(B($B%"!<%N%k%I(B[2], Holmes and Marsden[9], Robinson[17]$B;2>H(B)$B!%$7$+$7;0BNLdBj$OJ#;($G$"$k$N$G!"$J$K$h$j$b$^$:!"%a%k%K%3%U@QJ,$rDj<02=$9$k$3$H<+BN$,$[$H$s$IIT2DG=$G$"$j!"$I$3$+$i;O$a$F$h$$$N$+$o$+$i$J$$!%BhFs$K!"%a%k%K%3%U@QJ,$rI>2A$9$k$3$H$O!"EvA34|BT$5$l$k$h$&$K!"$?$H$(2DG=$G$"$k$K$;$h$*$=$m$7$/BgJQ$J$3$H$G$"$k!%$3$NO@J8$G$OJL$N$d$jJ}$r$9$k!%@)8B;0BNLdBj$N$"$k2#CGE*%[%b%/%j%K%C%/50F;$r$b$H$K9=@.$r9T$J$&!%$3$N$h$&$J%[%b%/%j%K%C%/50F;$NB8:_$O(BXia[21]$B$K$h$j<($5$l$F$$$k!%$3$NO@J8$G$O@)8B;0BNLdBj$+$i40A4$JJ?LL;0BNLdBj$X0\9T$9$k$H$-$K2#CGE*%[%b%/%j%K%C%/50F;$K$*$1$kGOD}<LA|$X$N@]F0$+$i%"!<%N%k%I3H;6$,@8$8$k$3$H$r<($9!%$3$N7k2L$O(B6$B@a$G=R$Y$k!%(B

$B$o$l$o$l$NBhFs$N<g7k2L$O;0BNLdBj$N?6F02r$K4X78$9$k!%%+%*%9NO3X$N=i4|$NNc$N(B1$B$D$,$$$o$f$k(BSitnikov$BLdBj$G$"$k!%(BSitnikov[19]$B$O(B$m_3=0$$B$J$k$3$NFC<l$J@)8B;0BNLdBj$K$*$$$F6/$/%+%*%9E*?6F0$r<($950F;$NB8:_$r<($7$?!%(BAlekseev[1]$B$O$3$NLdBj$r7OE}E*$KD4$Y!"7k2L$r(B$m_3>0$$B$X3HD%$7$?!%(BMcGehee[11]$B$N0BDjB?MMBN$N7k2L$rMQ$$$F(BMoser[13]$B$O(BSitnikov$BLdBj$r$b$C$H40A4$KCm0U?<$/D4$Y$?!%(B

$BIi%(%M%k%.!<$NJ?LL;0BNLdBj$N2r$,F1$8$h$&$J%+%*%9E*?6F0$r<($9$+$I$&$+$O<+A3$J5?Ld$G$"$k!%(BEaston and McGehee[8]$B$O4JC12=$7$?%b%G%kLdBj$rD4$Y!"$3$N%b%G%kLdBj$K?6F02r$,B8:_$9$k$3$H$r>ZL@$7$?!%?6F02r$O;0BNLdBj$NJ|J*(B-$BBJ1_2r$KL)@\$K4X78$7$F$$$k!%J|J*(B-$BBJ1_2r$K$*$$$FBh;0BN$O6K8BB.EY%<%m$GL58B1s$KF($2!";D$j$NFsBN$OBJ1_50F;$K6a$E$/$3$H$r;W$$$@$=$&!%(BEaston[6]$B$OJ|J*(B-$BBJ1_2rA4BN$N=89g$,%j%W%7%C%DB?MMBN$r$J$9$3$H$r<($7$?!%(BRobinson[16]$B$O$3$NLdBj$N8&5f$rB3$1!"$J$+$s$:$/!"$3$NB?MMBN$,<B$O<B2r@OE*$G$"$k$3$H$r<($7$?!%$5$i$K?J$s$@8&5f$,(BEaston[7]$B$K$h$C$F?tCME*$K9T$J$o$l$?!%(B

$B$3$NO@J8$G$O!"$3$NLdBj$r2r$/!%J?LL;0BNLdBj$N2r$,F1$8%+%*%9E*?6F0$rK\Ev$K<($9$3$H$r>ZL@$9$k!%$5$i$K!"%"!<%N%k%I3H;6$NB8:_$H7k$S$D$1$F!"3H;6E*?6F02r$NB8:_$r<($9!%$9$J$o$A50F;$O%+%*%9E*$K?6F0$9$k$@$1$G$J$/F1;~$K%+%*%9E*$K3H;6$9$k!%$3$l$r>ZL@$9$k$?$a$KL58B1s$K$*$1$kLLGr$$<LA|$rDj5A$7!"(BKAM$BM}O@$r;H$C$FL58B1s$K$*$1$k$3$N<LA|$NITJQ1_$NB8:_$r<($9!%?6F02r$O$3$l$i$NITJQ1_$N6a$/$N2r$r2r@O$9$k$3$H$K$h$C$FF@$i$l$k!%$3$N7k2L$O(B7$B@a$G=R$Y$k!%(B

$BE,Ev$J:BI87OJQ49$K$h$C$FJ*M}6u4V$NL58B1s(B($BL58B1s$K$*$$$FBh;0BN$NB.EY%<%m!"(B$S^3$$B$KF1Aj(B)$B$rM-8BE@$K;}$C$F$/$k$3$H$,$G$-!"(B$S^3$$B$HF10l;k$5$l$kL58B1s$N2r$K$D$$$F8l$k$3$H$,$G$-$k!%;0BNLdBj$N@-<A$+$i$7$F(B$S^3$$B>e$N2r$O$9$Y$F<~4|2r$G$"$C$F(B$S^3$$B>e$K$OL5M}?tN.$l$NITJQ%H!<%i%9$O$J$$!%(B$S^3$$B$N0BDjB?MMBN$HIT0BDjB?MMBN$OJ#;($K8r$o$k$1$l$I$b!"%"!<%N%k%I3H;6$O$J$$!%$7$+$7$J$,$i!"(B$S^3$$B>e$N%X%F%m%/%j%K%C%/@\B3$rMQ$$$F>e$G=R$Y$?$*$b$7$m$$<LA|$rDj5A$G$-$k!%$3$N<LA|$O40A4(B(exact)$B%7%s%W%l%/%F%#%C%/<LA|$G$"$j(BKAM$BM}O@$,E,MQ$G$-$F$3$N<LA|$KBP$7$F(B$S^3$$B>e$NITJQ%H!<%i%9$NB8:_$,8@$($k!%ITJQ%H!<%i%9$NBgItJ,$N0BDjB?MMBN$HIT0BDjB?MMBN$,2#CGE*$K8r$o$k$3$H$,$o$+$k!%$3$N2#CGE*8r:9$+$i(B$S^3$$B$N6a$/$G3+@^$9$kJ#;($JNO3X$,F@$i$l!"%"!<%N%k%I3H;6$K;w$?Hs>o$K$*$b$7$m$$8=>]$,@8$8$k!%$3$N8=>]$r5<%"!<%N%k%I3H;6(B(pseudo Arnold difusion)$B$H8F$V!%$3$l$O%"!<%N%k%I3H;6$,=LB`$7$?$b$N$G$"$j!"<~4|2r$N=89g$N$_$,F@$i$l!"=`<~4|2r$O$J$$!%$$$/$D$+$N<e$$>r7o$N$b$H$G!"K-IY$J50F;9=B$$bF@$i$l!"Hs>o$K%+%*%9E*$J;EJ}$G50F;$,A+0\%H!<%i%9$N4V$r$&$m$D$-2s$k!%$3$N>l9g!"A+0\%H!<%i%9$O%"!<%N%k%I3H;6$K$*$1$k=`<~4|2r$G$O$J$/!"<~4|2r$N$"$k<o$N=89g$N%X%F%m%/%j%K%C%/@\B3$N=89g$G$"$k!%$3$N7k2L$O(B7$B@a$G=R$Y$k!%(B

$B;0BNLdBj$K$b$C$H@QJ,$,B8:_$9$k$+$I$&$+!";0BNLdBj$O@QJ,2DG=$+!"$H$$$&LdBj$OE7BNNO3X$N8&5f$K$*$$$FHs>o$K=EMW$JLr3d$r2L$7$F$-$?!%%"!<%N%k%I3H;6$,B8:_$9$k$3$H$ND>@\$N5"7k$O;0BNLdBj$,Hs2D@QJ,$G$"$C$F!"CN$i$l$F$$$k$b$N0J30$K4X?tE*$KFHN)$J<B2r@OE*@QJ,$,$J$$$3$H$G$"$k!%(B

\vspace{0.5cm}
\noindent
{\bf 2. $BJ?LL;0BNLdBj(B}

$m_1, m_2$$B$*$h$S(B$m_3$$B$O%K%e!<%H%s$N=ENOK!B'$N$b$H$G8GDjJ?LLFb$rF0$$$F$$$k(B3$B<AE@$N<ANL$G$"$k$H$9$k!%=E?4$O86E@$K8GDj$5$l$F$$$k$H$7!"(B${\bf r_1},{\bf r_2},{\bf r_3}$$B$O$=$l$>$l(B$m_1, m_2$$B$*$h$S(B$m_3$$B$N0LCV$rI=$o$9$H$9$k!%7O$NA4%(%M%k%.!<(B$h$$B$OIi$G$"$k$H$7!"3Q1?F0NL(B$C$$B$O8GDj$9$k!%(B

$B$$$/$i$+=$@5$r2C$($k$,!"(BRobinson[16]$B$*$h$S(BEaston[6]$B$K=>$C$FJQ?t$rJQ$(!"J}Dx<0$rK>$^$7$$7A$K;}$C$F$$$/!%>\:Y$O(B[6]$B$K>y$k!%$^$:%d%3%SJQ?t$rF3F~$9$k!%(B${\bf q} \in {\bf R}^2$$B$r(B$m_1$$B$+$i(B$m_2$$B$X$N%Y%/%H%k$H$7!"(B${\bf Q} \in {\bf R}^2$$B$rO"@1(B$m_1$$B$H(B$m_2$$B$N=E?4$+$i(B$m_3$$B$X$N%Y%/%H%k$H$9$k!%$3$N$H$-<!$rF@$k!%(B
$$ \begin{array}{l}
 {\bf q} = {\bf r_2} - {\bf r_1}, \\
 {\bf Q} = \left( {\bf r_3} - 
\frac{\displaystyle m_1{\bf r_1} + m_2{\bf r_2}}
     {\displaystyle m_1 + m_2} \right) = M \mu^{-1} {\bf r_3}.
   \end{array}
$$
$B$3$3$G(B$M=m_1+m_2+m_3$$B$*$h$S(B$\mu = m_1+m_2$$B$G$"$k!%(B

${\bf p} \in {\bf R}^2$$B$*$h$S(B${\bf P} \in {\bf R}^2$$B$r(B${\bf q}$$B$*$h$S(B${\bf Q}$$B$KBP1~$9$k1?F0NL$H$9$k!%BP1~$9$k1?F0J}Dx<0$O(B
$$ \begin{array}{ll}
 \dot{\bf Q} = k_1^{-1} {\bf P}, & 
   \dot{\bf P} = -Mk_1 |{\bf Q}|^{-3} {\bf Q} + O(|{\bf Q}|^{-3}), \\
 \dot{\bf q} = k_2^{-1} {\bf p}, & 
  \dot{\bf p} = -\mu k_2 |{\bf q}|^{-3} {\bf q} + O(|{\bf Q}|^{-3}), \\
   \end{array}             \eqno(1)
$$
$B$H$J$k!%$3$3$G(B$k_1= m_3 \mu M^{-1}$$B$*$h$S(B$k_2 = m_1m_2\mu^{-1}$$B$G$"$k!%(B

$B$o$l$o$l$,$H$/$K6=L#$r;}$C$F$$$k$N$O$$$o$f$k(B\underline{$BJ|J*(B}$B50F;$G$"$k!%(B$\omega$($B$^$?$O(B$\alpha$)\underline{$BJ|J*(B}$B50F;$H$O!"(B$t$$B$,L58B$K8~$+$&$H$-(B($B$^$?$OIi$NL58B1s$K8~$+$&$H$-(B)${\bf q}$$B$,M-3&$G!"(B${\bf Q}$$B$,HsM-3&$G!"(B${\bf P}$$B$,%<%m$K8~$+$&50F;$G$"$k!%$@$+$i!"Bh;0BN(B$m_3$$B$,6K8BB.EY%<%m$GN%$l$F$$$-!"1?F0$OL58B$K6a$E$/!%(B$|{\bf Q}|= \infty$$B$rM-8BCM$K$9$k$?$a$K!"(B$\alpha=M\mu^{-1}$$B$H$7$F<\EYJQ49(B$|{\bf Q}|=\alpha x^{-2}$$B$r;H$&!%$^$?J#AGJ?LL$N3QJQ?t(B$s$$B$r(B${\bf Q} = \alpha x^{-2}s$$B$GDj5A$9$k!%1?F0NL$O<!$N$h$&$KF07B$*$h$S3QEY@.J,$KJ,2r$5$l$k!%(B
$$
  {\bf P} = k_1 \alpha ys + k_1 \alpha x^2 \rho is.
$$
$B$3$3$G(B$is$$B$OC10L%Y%/%H%k(B$s$$B$KD>8r$9$kC10L%Y%/%H%k$NJ#AG?tI=<($G$"$j!"(B$\rho$$B$O(B$m_3$$B$N3Q1?F0NL$r(B$m_3$$B$G<\EYJQ49$7$?$b$N$G$"$k!%J}Dx<0(B(1)$B$O=E?4$N$^$o$j$N2sE>$N$b$H$GITJQ$G$"$k!%$3$NBP>N@-$r8rE@$N>C5n$K$h$C$F<h$j=|$-!"J}Dx<0$r(B$s$$B$+$iFHN)$K$9$k$?$a$K!"(B
$$ \begin{array}{lcl}
  {\bf q} = zs & $B$*$h$S(B & {\bf p} = k_2 \mu^{-1} ws,
   \end{array} 
$$
$B$H$*$$$F(B${\bf q}$$B$*$h$S(B${\bf p}$$B$r(B$z$$B$*$h$S(B$w$($B$H$b$KJ#AG?t(B)$B$GCV$-49$($k!%$3$3$GJ#AG?tI=<($*$h$SJ#AG?t$N>h;;$rMQ$$$?!%$9$k$H1?F0J}Dx<0$O<!$N$h$&$K$J$k!%(B
$$ \begin{array}{l}
 \dot{x} = \frac{1}{2} x^3 y, \\
 \dot{y} = -\mu x^4 + O(x^6), \\
 \dot{z} = \mu^{-1} w + O(x^4), \\
 \dot{w} = -\mu^2 |z|^{-3}z + O(x^4), \\
 \dot{s} = x^4 \rho is, \\
 \dot{\rho} = O(x^6).
   \end{array}             \eqno(2)
$$
$B3Q1?F0NL@QJ,$H%(%M%k%.!<@QJ,$O<!$NDL$j$G$"$k!%(B
$$ \begin{array}{l}
 C = m_3 \rho + \frac{\displaystyle k_2^2} {\displaystyle 2\mu i} 
   (z \overline{w} - w \overline{z}), \\ 
 h = \frac{\displaystyle 1}{\displaystyle 2} m_3 y^2 + 
\frac{\displaystyle 1}{\displaystyle 2} k_2^3 \mu^{-2} w \overline{w} + 
m_1m_2|z|^{-1} + O(x^2).
   \end{array} 
$$
$BJ}Dx<0(B(2)$B$N(B$s$$B$K4X$9$k<0$OMn$H$;$k!%(B$s$$B$OB>$N<0$K$O=P$F$3$J$$$+$i$G$"$k!%(B$m_3 \neq 0$$B$J$i(B$\rho$$B$K4X$9$k<0$bMn$H$;$k!%(B$\rho$$B$O3Q1?F0NL@QJ,(B$C$$B$r;H$C$F$[$+$NJQ?t$+$i7hDj$G$-$k$+$i$G$"$k!%$7$+$7(B$m_3=0$$B$N$H$-!"$9$J$o$A@)8B;0BNLdBj$N>l9g$K$O$3$N4JC1$J$d$jJ}$O;H$($:!"(B$\rho$$B$K4X$9$k<0$OMn$H$;$J$$!%(B

$B1?F0J}Dx<0$OO"@1(B$m_1$$B$H(B$m_2$$B$,(B$z=0$$B$G>WFM$r5/$3$9$H$3$m$KFC0[@-$r;}$D!%(BLevi-Civita$B$N@5B'2=$rMQ$$$F$3$NFC0[@-$r<h$j=|$1$k!%(B
$$ \begin{array}{l}
z = 2 \mu \xi^2, \ \ \ \ w = \mu h^{-1} \eta \overline{\xi}^{-1}, \\
K = 4(\xi \overline{\xi} + \eta \overline{\eta}) \mu \xi \overline{\xi},
   \end{array} 
$$
$B$H$*$-!"%Y%/%H%k>l$K(B$K$$B$r3]$1$F;~4V$N<\EY$rJQ$($k!%$9$k$H1?F0J}Dx<0$O<!$N$h$&$K$J$k!%(B
$$ \begin{array}{l}
 \dot{x} = -\frac{1}{2} K x^3 y, \\
 \dot{y} = -K \mu x^4 + O(x^6), \\
 \dot{\xi} = \eta + O(x^4), \\
 \dot{\eta} = - \xi + O(x^4,y^2), \\
 \dot{\rho} = O(x^6).
   \end{array}             \eqno(3)
$$
$B%(%M%k%.!<$*$h$S3Q1?F0NL@QJ,$O<!$N$h$&$K$J$k!%(B
$$ \begin{array}{l}
 h = H = \frac{\displaystyle 1}{\displaystyle 2} m_3 y^2 - 
k_2^3(\xi \overline{\xi} + \eta \overline{\eta}) + O(x^2), \\
 C = m_3 \rho + \frac{\displaystyle k_2^2 \mu} {\displaystyle hi} 
   (\xi \overline{\eta} - \eta \overline{\xi}). 
   \end{array} 
$$

$B$3$l$i$NJ}Dx<0$O(B$x=0$$B$K<+A3$K3HD%$5$l$F$$$k!%$3$NE@$G(B$H=h<0$$B$G$"$j!"$^$?3Q1?F0NL@QJ,$K$h$j3F(B$y$$B$KBP$7$F;0<!855e$,Dj5A$5$l$k!%3HD%$5$l$?J}Dx<0$G(B$x=0$$B$H$7(B$y$$B$r8GDj$9$k$H!"N.$l$O(B$S^3$$B>e$N(BHopf$BN.$G$"$k!%$3$l$i$NJ}Dx<0$G(B$\omega$$BJ|J*(B($B$^$?$O(B$\alpha$$BJ|J*(B)$B50F;$O!"(B$t$$B$,L58B$K(B($B$^$?$OIi$NL58B$K(B)$B8~$+$C$?$H$-$K(B$x(t)$$B$H(B$y(t)$$B$,$H$b$K%<%m$K9T$/$h$&$J50F;$G$"$k!%$@$+$i(B$\omega$$BJ|J*(B($B$^$?$O(B$\alpha$$BJ|J*(B)$B50F;$N=89g$O(B$x=0$$B$*$h$S(B$y=0$$B$GDj5A$5$l$kITJQ(B3$B<!855e(B$S^3$$B$N0BDj!J$^$?$OIT0BDj!KB?MMBN(B$W^s(S^3)$($B$^$?$O(B$W^u(S^3)$)$B$K@53N$KBP1~$9$k!%J*M}6u4V$O(B$x>0$$B$KBP1~$9$k$3$H$r;XE&$7$F$*$/!%(B

$S^3$$B$O@55,AP6JE*(B(normally hyperbolic)$B$G$J$$$3$H$KCm0U$7$h$&!%=LB`$N860x$OJ}Dx<0(B(3)$B$N(B$x$$B$H(B$y$$B$N<0$NCf$N0x;R(B$x^3$$B$G$"$k!%$@$+$iI8=`E*0BDjB?MMBNDjM}$O$3$NLdBj$KE,MQ$G$-$:!"=89g(B$W^s(S^3)$$B$*$h$S(B$W^u(S^3)$$B$,3j$i$+$JB?MMBN$N9=B$$r;}$D$+$I$&$+L@$i$+$G$J$$!%$7$+$7(BRobinson$B$NF@$?DjM}$+$i!"$3$l$i$,<B:]3j$i$+$JB?MMBN$G$"$k$3$H$,$$$($k!%>\$7$/$O<!$NDL$j$G$"$k!%(B

\vspace{0.3cm}
\noindent
{\bf $BDjM}(B 2.1}(Robinson[16]). (a) $BJ}Dx<0(B(3)$B$N(B$\{ x \geq 0 \}$$B$K$*$1$k(B$S^3$$B$N0BDj$*$h$SIT0BDjB?MMBN$O(B($x=0$$B$K6-3&$r;}$D(B)$C^{\infty}$$BB?MMBN$G$"$k!%(B
$B$=$N>e(B$W^s(S^3) \cap \{ x>0 \}$$B$H(B$W^u(S^3) \cap \{ x>0 \}$$B$O<B2r@OE*$G$"$k!%(B

(b) $\pi: S^3 \rightarrow S^3/ \sim \cong S^2$$B$O50F;$r>&6u4V(B$S^3/ \sim \cong S^2$$B$NE@$K<M1F$9$k(BHopf$B<LA|$G$"$k$H$7!"(B$\alpha^*:W^u(S^3) \rightarrow S^2$$B$O(B$W^u(S^3)$$B$N3FE@$r(B$S^3$$BFb$N$=$N(B$\alpha$$B6K8B$K<L$94X?t$H>e$GDj5A$7$?(B$\pi$$B$N9g@.$H$9$k!%F1MM$K<LA|(B$\omega^*:W^s(S^3) \rightarrow S^2$$B$rDj5A$9$k!%$3$N$H$-(B$\alpha^*$$B$H(B$\omega^*$$B$O(B$C^{\infty}$$B4X?t(B($B$+$D(B$\{ x > 0 \}$$B$G<B2r@OE*(B)$B$G$"$k!%(B

\vspace{0.3cm}
$B$3$NDjM}$OJ?6QK!$*$h$S%0%i%UJQ49(B(graph transform)$B$r;H$C$F>ZL@$5$l$k!%>\:Y$O(BRobinson[16]$B$r;2>H$5$l$?$$!%(B

\vspace{0.5cm}
\noindent
{\bf 3. $BA+0\:?$H%"!<%N%k%I3H;6(B}

$B%"!<%N%k%I3H;6$O9b<!85$N%O%_%k%H%s7O$KH<$&Hs>o$KJ#;($J%+%*%9E*NO3X8=>]$G$"$k!%$3$N8=>]$NGX8e$K$"$k4v2?3X$H5!9=$r8+$k$?$a$K<!$N7A$N6a2D@QJ,%O%_%k%H%s7O$r9M$($h$&!%(B
$$
  H_{\varepsilon}(q,p,x,y) = F(q,p) + G(x,y) + 
\varepsilon H^1(q,p,x,y,t).           \eqno(4)
$$
$B$3$3$G(B$(q,p,x,y)$$B$O(B$2(n+1)$$B<!85%7%s%W%l%/%F%#%C%/B?MMBN(B$M$$B>e$N@5=`:BI8$G$"$k!%(B$q$$B$H(B$p$$B$O<B?t$G(B$x=(x_1,x_2,\ldots,x_n)$$B$*$h$S(B$y=(y_1,y_2,\ldots,y_n)$$B$G$"$k!%(B$\varepsilon$$B$OHy>.%Q%i%a!<%?!"(B$H^1$$B$O(B$t$$B$K4X$7$F(B$2\pi$$B<~4|$G$"$k!%$5$i$K:nMQ(B-$B3Q:BI8(B$(\theta_1,\theta_2,\ldots,\theta_n,I_1,I_2,\ldots,I_n)$$B$r8+$D$1$k$3$H$,$G$-$F<0(B(4)$B$,0J2<$N7A$K$J$k$H2>Dj$9$k!%(B
$$ \begin{array}{l}
  H_0 = F(q,p) + G(x,y) = 
        \displaystyle F(q,p) + \sum_{i=1}^{n} G_i(I_i), \\
  H_{\varepsilon} = H_0(q,p,I) + \varepsilon H^1(q,p,\theta,I,t).
   \end{array}            \eqno(5)
$$
$B$3$3$G(B$H^1$$B$O(B$\theta_1,\theta_2,\ldots,\theta_n$$B$*$h$S(B$t$$B$K4X$7$F(B$2\pi$$B<~4|$G$"$k!%$^$?Hs=LB`>r7o(B
$$
\Omega_i(I_i) = \frac{\displaystyle \partial G_i(I_i)}
                     {\displaystyle \partial I_i} \neq 0,
$$
$B$,Aj6u4V$N$"$kNN0h$G@.$jN)$D$3$H$b2>Dj$9$k!%(B

$B%O%_%k%H%s4X?t(B(4)$B$*$h$S(B(5)$B$K$*$1$k@]F0$O<~4|E*$K;~4V$K0MB8$9$k$+$i!"7O$r%7%s%W%l%/%F%#%C%/<LA|$K5"Ce$5$;$k$N$,ET9g$h$$!%$3$N$?$a$K(B$\{ (q,p, \theta,I,t)|t=t_0 \in [0,2\pi) \}$$B$GDj5A$5$l$kBg0hE*2#CGLL(B$\Sigma^{t_0}$$B$r8GDj$9$k!%(B$\Pi: \Sigma^{t_0} \rightarrow \Sigma^{t_0}$$B$O(B$\Sigma^{t_0}$$B>e$KDj5A$5$l$?%]%"%s%+%l<LA|$H$9$k!%0J2<$N5DO@$O<LA|(B$\Pi$$B$*$h$S2#CGLL(B$\Sigma^{t_0}$$B$K8B$i$l$k!%(B

$\varepsilon=0$$B$J$i%O%_%k%H%s7O(B(5)$B$O40A42D@QJ,$G$"$k!%$3$3$G%O%_%k%H%s4X?t(B$F(q,p)$$B$,AP6J%5%I%kE@(B$(q_0,p_0)$$B$r7k$V%[%b%/%j%K%C%/50F;(B$\gamma=(\overline{q}(t),\overline{p}(t))$$B$r;}$D$H2>Dj$9$k!%$@$+$i(B$H_0$$B$KBP$9$k%O%_%k%H%s7O$O2#CGLL(B$\Sigma^{t_0}$$B>e$K<!<0$GM?$($i$l$kITJQ(B$n$$B<!85%H!<%i%9(B$T(h_1,h_2,\ldots,h_n)$$B$N(B$n$$B%Q%i%a!<%?B2$r;}$D!%(B
$$ \begin{array}{lll}
 I_i = h_i = $B0lDj(B, &  &  i = 1,2, \ldots, n \\
 t = t_0,           & \theta_i \in [0, 2\pi), & i = 1,2, \ldots, n \\
 q = q_0,      & p = p_0. & 
   \end{array}            \eqno(6)
$$
$B%H!<%i%9(B$T(h_1,h_2,\ldots,h_n)$$B$O$=$l<+?H$,(B$(n+1)$$B<!85%[%b%/%j%K%C%/B?MMBN(B
$$ \begin{array}{lll}
 I_i = h_i  &  i = 1,2, \ldots,n & \\
 t = t_0,   & \theta_i \in S^1  &  \\
 q = q(s),  &  p = \overline{p}(s), & s \in {\bf R}, 
   \end{array}           
$$
$B$K$h$C$F7k$P$l$F$$$k!%$3$N%[%b%/%j%K%C%/B?MMBN$O(B$n$$B<!85%H!<%i%9(B$T(h_1,h_2,\ldots,h_n)$$B$N0BDjB?MMBN$G$"$jIT0BDjB?MMBN$G$b$"$k!%(B

$\varepsilon \neq 0$$B$J$i7O$O$O$k$+$KJ#;($K$J$k!%$o$l$o$l$,6=L#$r;}$C$F$$$k$N$O>e$NITJQB?MMBN$HBP1~$9$k0BDj$*$h$SIT0BDjB?MMBN$,>.$5$J@]F0$N$b$H$G$I$&JQ$o$k$+$G$"$k!%$^$:CGLL(B$\Sigma^{t_0}$$B>e$G(B$p=p_0, q=q_0$$B$HCV$$$FF@$i$l$k(B$2n-2$$B<!85ITJQB?MMBN(B$N_0$$B$,@55,AP6JE*$G$"$k$3$H$KCm0U$9$k!%ITJQB?MMBNM}O@$K$h$j!"@55,AP6JITJQB?MMBN$O>.$5$J@]F0$N$b$H$G@8$-;D$k!%$@$+$i(B$\varepsilon \neq 0$$B$,>.$5$$$H$-!"(B$N_0$$B$O@]F0$r<u$1$F(B$\Sigma^{t_0}$$BFb$G6a$/$NITJQB?MMBN$?$H$($P(B$N_{\varepsilon}$$B$K$J$k!%4JC1$K$o$+$k$h$&$K!"(B$N_{\varepsilon}$$B>e$KDj5A$5$l$k%]%"%s%+%l<LA|$O%7%s%W%l%/%F%#%C%/$G$"$k!%$9$J$o$A!"(B$N_0$$B>e$GDj5A$5$l$?ITJQHs=LB`(B2-$B7A<0$,$"$k!%;v<B!"$3$N(B2-$B7A<0$O(B$\Sigma^{t_0}$$B>e$N@5=`(B2-$B7A<0$+$i(B$N_{\varepsilon}$$B$NJq4^<LA|$K$h$C$FM6F3$5$l$k!%Hs=LB`$J$N$OHs=LB`(B2-$B7A<0$+$i$N@]F0$@$+$i$G$"$k!%$3$3$G(BKAM(Kolmogorov-Arnold-Moser)$BM}O@$,ITJQB?MMBN(B$N_{\varepsilon}$$B$KE,MQ$G$-$k!%Hs=LB`>r7o(B$\Omega_i(I_i) = \partial G_i(I_i)/ \partial I_i \neq 0$$B$r2>Dj$9$k$H!"Hs6&LDITJQ%H!<%i%9(B$T(h_1,h_2, \ldots, h_n)$$B$NBgB??t$O>.$5$J@]F0$N$b$H$G@8$-;D$j!"@]F0$r<u$1$F6a$/$N%H!<%i%9(B$T_{\varepsilon}(h_1,h_2, \ldots, h_n)$$B$K$J$k!%(B

$B@8$-;D$C$F$$$kITJQ%H!<%i%9(B$T_{\varepsilon}(h_1,h_2,\ldots,h_n)$$B$r9M$($h$&!%(B$T_{\varepsilon}(h_1,h_2,\ldots,h_n)$$B$N0BDj$*$h$SIT0BDjB?MMBN$,B8:_$7!"(B$\varepsilon$$B$H$H$b$K3j$i$+$KJQ$o$k$3$H$O4JC1$K<($;$k!%(B$\varepsilon=0$$B$N$H$-$3$l$i(B2$B$D$NB?MMBN$O0lCW$9$k!%(B$\varepsilon \neq 0$$B$J$i!"0BDj$*$h$SIT0BDjB?MMBN$OE57?E*$K$OHs>o$KJ#;($J;EJ}$G8r$o$k!%%i%0%i%s%8%e8r:9M}O@(B(Arnold[4], Weinstein[20])$B$N<($9$H$3$m$K$h$l$P!"$3$l$i(B2$B$D$NB?MMBN$O8r$o$k$O$:$G$"$j!"$3$N8r:9$,2#CGE*$G$"$k$+$I$&$+$r<($9$N$K!"$H$-$K%a%k%K%3%U$NJ}K!(B[12,2,9]$B$,;H$($k!%(B

$B$"$kITJQ%H!<%i%9(B$T_{\varepsilon}(h_1,h_2,\ldots,h_n)$$B$N0BDj$*$h$SIT0BDjB?MMBN$,!"$"$kE@$G2#CGE*$K8r$o$C$F$$$k$H2>Dj$9$k!%$3$N$H$-(B$\lambda$$BJdBj$h$j!"0[$J$k8rE@$,L5?t$K$"$j!"J#;($JNO3X$,F@$i$l$k!%$3$l$OAP6JITF0E@$"$k$$$O<~4|E@$N0BDj$*$h$SIT0BDjB?MMBN$N2#CGE*8r:9$+$i@8$8$k%[%b%/%j%K%C%/Mm$_9g$$(B(tangle)$B$K;w$F$$$k!%<B:]$K$O$b$C$HJ#;($J!"$$$o$f$k%"!<%N%k%I3H;6$H8F$P$l$k8=>]$,$3$N>l9g@8$8$k!%(B

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$\varepsilon$$B$,==J,>.$5$1$l$P!"3FA+0\%H!<%i%9$N6a$/$GA+0\:?$r9=@.$G$-$k!%(BKAM$BM}O@$,<($9$H$3$m$K$h$l$P!"(B$\varepsilon \rightarrow 0$$B$N$H$-ITJQ%H!<%i%9$N=89g$OA4B,EY$r;}$D$h$&$K$J$k!%$7$?$,$C$F3FA+0\%H!<%i%9$NG$0U$K>.$5$J6aK5$KL58B$KB?$/$NA+0\%H!<%i%9$,B8:_$9$k!%$@$+$iA+0\%H!<%i%9$NNs$r=&$$>e$2$FA+0\:?$r9=@.$9$k$h$&$K$G$-$k!%$7$+$7!"$3$NA+0\:?$,$I$N$/$i$$1s$/$^$G9T$/$+$OL@$i$+$G$J$$!%$3$l$O@]F0$N5,LO$K0MB8$9$k$h$&$K;W$($k!%@]F0$,Bg$-$/$J$k$K$D$l$F:?$O$?$/$5$s$NCGJR$K2u$l$k$+$b$7$l$J$$!%6&LD$N6a$/$GITJQ%H!<%i%9$,$I$N$h$&$K2u$l$k$+$r$b$C$H>\$7$/M}2r$9$l$P!"A+0\:?$NGK2u$NGX8e$K@x$`J#;($JNO3X$,L@$i$+$K$J$k$+$b$7$l$J$$!%(B

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