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Next: 4.1 Hamilton $B7O(B Up: $B!V12;e$N:n$kNO3X7O!W$NJY6/$N=`Hw(B Previous: 3 $B?tCM%7%_%e%l!<%7%g%s$r$7$F$_$h$&(B

4 $B2,K\Bh(B1$B>O$rFI$`(B

$B<+M3EY(B $ n$ $B$N>oHyJ,J}Dx<0$H$O!"(B

$\displaystyle \frac{\D x_j}{\D t}=f_j(x_1,\cdots,x_n)$   $\displaystyle \mbox{($j=1,2,\cdots,n$)}$

$B<+M3EY$NJ*M}E*Dj5A(B: $B$"$kBP>]J*$dBN7O$r7hDj$9$k$?$a$KI,MW$J:BI8$d%Q%i%a!<(B $B%?!<$N8D?t$N$3$H(B

$B4v2?3XE*$J@bL@(B: $B@\%Y%/%H%k!"%Y%/%H%k>l!"NO3X7O!"Aj6u4V!"J?9UE@(B

$B!VM-8B;~4V$GGzH/$9$k!W!"!VGzH/;~9o!W$H$$$&!#(B

$BGzH/$NNc(B

$\displaystyle \frac{\D x}{\D t}=x^2,\quad x(0)=1
$

$\displaystyle x(t)=\frac{1}{1-t}$   $\displaystyle \mbox{($t\in(-\infty,1)$)}$$\displaystyle .
$

$\displaystyle \frac{\D x}{\Dt}=x^2+p(t)x+q(t)
$

$BFC$K(B $ p(t)=0$, $ q(t)=1$ $B$N$H$-(B $ x(t)=\tan(t-t_0)$ ($ t_0$ $BG$0UDj?t(B).

$ M$ $B$NE@(B $ a$ $B$rDL$k50F;$H$O!"(B $B=i4|>r7o(B $ x(0)=a$ $B$rK~$?$9(B $ \frac{\D x}{\D t}=X(x)$ $B$N(B $B2r6J@~(B $ \{x(t); t_{\rm min}<t<t_{\rm max}\}$ $B$N$3$H!#(B $ O(a)$ $B$GI=$o$9!#(B

$\displaystyle \exists T>0$   s.t.$\displaystyle \quad \forall t\quad
x(t)=x(t+T)
$

$B$,@.$jN)$D$H$-!"BP1~$9$k50F;$r(B$BJD50F;(B$B$^$?$O(B$B<~4|50F;(B$B$H8F$V!#(B

$ M$ $B$NE@(B $ a$ $B$rDL$k50F;$K4X$7$F(B $ t_{\rm max}=\infty$$B$H2>Dj$9$k!#(B $B$3$N$H$-!"(B$ y\in M$ $B$,(B $ a$ $B$rDL$k50F;$N(B $ \omega$ $B6K8BE@(B$B$H$O!"(B $B$"$k;~9o$NNs(B $ t_n$ ( $ n=1,2,\cdots$) $B$G!"(B $ n\to\infty$ $B$N$H$-!"(B $ t_n\to\infty$ $B$+$D(B $ x(t_n)\to y$ $B$J$k$b$N$,B8:_$9$k$3$H$r8@$&!#(B

$BNc$O!)!*(B $B:$$C$F$$$?(B (12/3)$B!#(B

($BB?J,!"6K8BJD50F;$K4,$-IU$$$F$$$/50F;$N(B $ \omega$ $B6K8B=89g$O(B $B6K8BJD50F;$=$N$b$N(B)

$ \omega$ $B6K8BE@$NA4BN$r!!(B$ \omega$ $B6K8B=89g(B$B$H8F$V!#(B


\begin{jtheorem}[Poincar\'e-Bendixson]\upshape
$BAj6u4V(B $M$$B$,(B $2$\ $B<!855eLL$G$




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Next: 4.1 Hamilton $B7O(B Up: $B!V12;e$N:n$kNO3X7O!W$NJY6/$N=`Hw(B Previous: 3 $B?tCM%7%_%e%l!<%7%g%s$r$7$F$_$h$&(B
Masashi Katsurada
$BJ?@.(B19$BG/(B12$B7n(B29$BF|(B