6 蟾ｮ蛻�婿遞句ｼ上�Jacobi陦悟�縺ｨ縺昴�蝗ｺ譛牙､

縺薙�蝗ｺ螳夂せ $\mbox{\boldmath$\ x $}_0$ 縺ｮ菴咲嶌逧�ｧ雉ｪ(螳牙ｮ壽ｧ)縺ｯ�悟燕遽縺ｧ豎ゅａ縺� 蠑�(8)縺ｮ蛻晄悄蛟､縺ｫ蟇ｾ縺吶ｋJacobi陦悟��

$\begin{displaymath} \frac{\partial T(\mbox{\boldmath$ x $}_0)}{\partial \mbox{\b... ...l \varphi_2}{\partial y_0} (2\pi, x_0, y_0) \end{array}\right] \end{displaymath}$

(6.23)

縺ｮ蝗ｺ譛牙､縺ｧ迚ｹ蠕ｴ縺･縺代ｉ繧後ｋ��

縺薙�隱ｬ譏弱�蜑肴ｺ門ｙ繧偵＠繧医≧�主ｼ�(8)縺ｯ髮｢謨｣邉ｻ縺ｫ縺ｪ縺｣縺ｦ縺�ｋ�� 髮｢謨｣邉ｻ縺ｮ荳闊ｬ逧�↑譁ｹ遞句ｼ上��檎憾諷� $\mbox{\boldmath$\ x $} \in \mbox{\boldmath$R$}^n$ 縺ｫ髢｢縺励※��

$\begin{displaymath} \mbox{\boldmath$ x $}^{(k+1)} = \mbox{\boldmath$ f $}(\mbox{\boldmath$ x $}^{(k)}) \end{displaymath}$

(6.24)

縺ｨ縺�≧蟾ｮ蛻�婿遞句ｼ上〒荳弱∴繧峨ｌ繧具ｼ� $\mbox{\boldmath$\ f $}$ 縺ｮ蝗ｺ螳夂せ $\mbox{\boldmath$\ x $}_0$ 縺瑚ｦ九▽縺九▲縺溘→縺搾ｼ�

$\begin{displaymath} \mbox{\boldmath$ x $}_0 = \mbox{\boldmath$ f $}(\mbox{\boldmath$ x $}_0 ) \end{displaymath}$

(6.25)

縺ｨ縺ｪ繧九％縺ｨ縺後ｏ縺九ｋ��

縺ｨ縺薙ｍ縺ｧ�碁｣邯壽凾髢薙�閾ｪ蠕狗ｳｻ縺ｮ蝣ｴ蜷医ｒ蠕ｩ鄙偵☆繧九→[4]��

$\begin{displaymath} \frac{d\mbox{\boldmath$ x $}}{dt} = \mbox{\boldmath$ f $}(\mbox{\boldmath$ x $}) \end{displaymath}$

(6.26)

縺ｮ蟷ｳ陦｡轤ｹ繧� $\mbox{\boldmath$\ x $}^*$ 縺ｨ縺吶ｋ�� 蟷ｳ陦｡轤ｹ霑大ｍ $\mbox{\boldmath$\ x $}^*+\mbox{\boldmath$\ \xi $}$ 縺ｫ縺翫＞縺ｦ Taylor 螻暮幕繧貞ｮ溯｡後☆繧九→��

$\begin{displaymath} \frac{d(\mbox{\boldmath$ x $}^* + \mbox{\boldmath$ \xi $})}{... ... $}= \mbox{\boldmath$ x $}_0}\mbox{\boldmath$ \xi $} + \cdots \end{displaymath}$

(6.27)

縺薙�蠑上→蠑�(26)縺ｨ縺ｮ蟾ｮ繧貞叙繧九→�� $\mbox{\boldmath$\ \xi $}$ 縺ｫ髢｢縺吶ｋ螟牙�譁ｹ遞句ｼ�

$\begin{displaymath} \frac{d\mbox{\boldmath$ \xi $}}{dt} = \left. \frac{\partial... ...$}_0) \mbox{\boldmath$ \xi $} \equiv J \mbox{\boldmath$ \xi $} \end{displaymath}$

(6.28)

縺悟ｾ励ｉ繧後ｋ�弱％縺ｮ $J(n \times n)$ 縺ｮ蝗ｺ譛牙､ $\mu$ 縺ｮ蛟､縺ｫ繧医▲縺ｦ蟷ｳ陦｡轤ｹ縺ｮ螳牙ｮ壽ｧ繧貞愛蛻･縺ｧ縺阪◆縺薙→繧呈昴＞縺�縺昴≧�� 縺､縺ｾ繧奇ｼ� $\mbox{\boldmath$\ \xi $}$ 縺ｫ縺､縺�※縺ｮ荳闊ｬ隗｣縺ｯ谺｡蠑上〒陦ｨ縺輔ｌ繧具ｼ�

$\begin{displaymath} \mbox{\boldmath$ \xi $} = \sum^n_{k=1}c_k e^{\mu_k t}\mbox{\boldmath$ \xi $}_k \end{displaymath}$

(6.29)

縺薙％縺ｧ

縺ｯ螳壽焚�� $\mbox{\boldmath$\ \xi $}_k$ 縺ｯ蝗ｺ譛峨�繧ｯ繝医Ν繧堤､ｺ縺励※縺�ｋ�� 縺薙�蠑上〒萓九∴縺ｰ�悟崋譛牙､縺ｮ蜈ｨ縺ｦ縺� 雋�縺ｮ螳滄Κ繧呈戟縺｣縺ｦ縺�ｋ縺ｨ�後◎縺ｮ蟷ｳ陦｡轤ｹ縺悟ｮ牙ｮ壹↑縺ｮ縺ｯ荳逶ｮ迸ｭ辟ｶ縺ｧ縺ゅｍ縺�� 縺励◆縺後▲縺ｦ騾｣邯壽凾髢楢�蠕狗ｳｻ縺ｫ縺翫＞縺ｦ�後≠繧句ｹｳ陦｡轤ｹ縺ｧ縺ｮ Jocobi陦悟�縺ｮ蝗ｺ譛牙､縺ｯ�後◎縺ｮ蟷ｳ陦｡轤ｹ縺ｮ螳牙ｮ壽ｧ縺ｮ謖�焚縺ｨ縺ｪ縺｣縺ｦ縺�◆�� 縺薙�蝗ｺ譛牙､縺ｨ蜷梧ｧ倥↑�碁屬謨｣邉ｻ縺ｫ縺翫￠繧区欠讓吶�菴輔〒縺ゅｍ縺�°��

蜈�↓謌ｻ縺｣縺ｦ�碁屬謨｣邉ｻ繧ｷ繧ｹ繝�Β縺ｮ蝗ｺ螳夂せ縺ｯ�碁｣邯夂ｳｻ縺ｮ蟷ｳ陦｡轤ｹ縺ｫ蟇ｾ蠢懊☆繧九→閠�∴繧峨ｌ繧具ｼ� 縺吶↑繧上■�悟ｼ�(25)縺ｯ騾｣邯夂ｳｻ(3)縺ｮ��

$\begin{displaymath} \mbox{\boldmath$ f $}(\mbox{\boldmath$ x $}) = 0 \end{displaymath}$

(6.30)

縺ｫ蟇ｾ蠢懊＠縺ｦ縺�ｋ�弱→繧ゅ↓螳壼ｸｸ迥ｶ諷九↓關ｽ縺｡逹縺�※縺�ｋ縺ｨ諤昴∴縺ｰ繧医＞��

蠑�(25)縺ｫ縺､縺�※荳谺｡縺ｮ螟牙�譁ｹ遞句ｼ上ｒ縺溘※繧医≧�� 縺�∪�悟崋螳夂せ霑大ｍ縺ｫ縺､縺�※ $\mbox{\boldmath$\ x $}^{(k)}=\mbox{\boldmath$\ x $}_0 + \mbox{\boldmath$\ \xi $}^{(k)}$ 縺ｨ縺励※

$\begin{displaymath} \mbox{\boldmath$ x $}^{(k+1)} = \mbox{\boldmath$ x $}_0 + \m... ...f $}(\mbox{\boldmath$ x $}_0+ \mbox{\boldmath$ \xi $}^{(k)}). \end{displaymath}$

(6.31)

縺薙�蠑上↓蟇ｾ縺励※Taylor螻暮幕繧呈命縺吶→��

$\begin{displaymath} \mbox{\boldmath$ x $}^{(k+1)} = \mbox{\boldmath$ f $}(\mbox{... ...mbox{\boldmath$ x $}_0}\mbox{\boldmath$ \xi $}^{(k)} + \cdots \end{displaymath}$

(6.32)

縺ｨ縺ｪ繧具ｼ�2谺｡莉･髯阪ｒ辟｡隕悶＠�悟ｼ�(25)縺ｨ縺ｮ蟾ｮ繧貞叙繧九→�� 邱壼ｽ｢蟾ｮ蛻�婿遞句ｼ擾ｼ�

$\begin{displaymath} \mbox{\boldmath$ \xi $}^{(k+1)} = \left. \frac{\partial \mb... ...{\boldmath$ \xi $}^{(k)} \equiv J\mbox{\boldmath$ \xi $}^{(k)} \end{displaymath}$

(6.33)

縺悟ｾ励ｉ繧後ｋ�弱％縺薙〒

縺ｯ $(n \times n)$ 縺ｮ Jocobi陦悟�縺ｨ縺ｪ繧具ｼ�

縺輔※��

$\begin{displaymath} ( J - \mu I)\mbox{\boldmath$ \xi $}^{(0)} = 0 \end{displaymath}$

(6.34)

縺梧�繧顔ｫ九▽繧ｼ繝ｭ縺ｧ縺ｪ縺�ｧ｣ $\mbox{\boldmath$\ \xi $}^{(0)}$ 縺悟ｭ伜惠縺吶ｋ縺ｨ縺搾ｼ� $\mu$ 繧貞ｷｮ蛻�婿遞句ｼ�(33)縺ｮ蝗ｺ譛牙､縺ｨ縺�≧�� 縺ｾ縺滂ｼ� $\mbox{\boldmath$\ \xi $}^{(0)}$ 繧抵ｼ悟崋譛牙､ $\mu$ 縺ｫ蟇ｾ蠢懊☆繧句崋譛峨�繧ｯ繝医Ν縺ｨ縺�≧�� 蝗ｺ譛牙､縺ｯ迚ｹ諤ｧ譁ｹ遞句ｼ擾ｼ�

$\begin{displaymath} \det \bigl[J - \mu I \bigr] = 0 \end{displaymath}$

(6.35)

縺ｧ豎ゅａ繧峨ｌ�悟崋譛峨�繧ｯ繝医Ν縺ｯ蠑�(34)繧堤峩謗･險育ｮ励☆繧九→ 豎ゅａ繧峨ｌ繧具ｼ�

蝗ｺ譛峨�繧ｯ繝医Ν繧貞�逋ｺ轤ｹ縺ｨ縺励※�梧婿遞句ｼ�(33)縺ｧ谺｡縲�→逕滓� 縺輔ｌ縺ｦ繧�￥繝吶け繝医Ν蛻励��

$\begin{displaymath} \mbox{\boldmath$ \xi $}^{(1)} = \mu \mbox{\boldmath$ \xi $}^... ...{\boldmath$ \xi $}^{(k)} = \mu^k \mbox{\boldmath$ \xi $}^{(0)} \end{displaymath}$

(6.36)

縺ｨ縺ｪ繧具ｼ弱％縺ｮ蟾ｮ蛻�婿遞句ｼ上�荳闊ｬ隗｣縺ｯ��

$\begin{displaymath} \mbox{\boldmath$ \xi $}^{(k)} = \sum^n_{i=1} c_i \mu^k_i \mbox{\boldmath$ \xi $}^{(0,i)} \end{displaymath}$

(6.37)

縺ｧ陦ｨ縺吶％縺ｨ縺後〒縺阪ｋ�弱％縺薙〒

縺ｯ螳壽焚�� $\mbox{\boldmath$\ \xi $}^{(0,i)}$ 縺ｯ蝗ｺ譛牙､ $\mu_i$ 縺ｫ蟇ｾ蠢懊＠縺溷崋譛峨�繧ｯ繝医Ν縺ｧ縺ゅｋ��

蠑�(37)縺ｯ縺ゅｋ螟ｧ縺阪＆縺ｮ蝗ｺ譛峨�繧ｯ繝医Ν縺ｫ�� 縺ゅｋ繧ｹ繧ｫ繝ｩ蛟､ $\mu_i$ 繧�蝗槭°縺代※縺�￥縺薙→縺ｫ縺ｪ繧具ｼ� 縺昴％縺ｧ�後�繧ｯ繝医Ν $\mbox{\boldmath$\ \xi $}^{(k)}$ 縺ｮ髟ｷ縺輔′逋ｺ謨｣縺励↑縺�◆繧√↓縺ｯ

$\begin{displaymath} \left\vert \mu_i \right\vert < 1 \end{displaymath}$

(6.38)

繧定ｦ√☆繧九％縺ｨ縺後ｏ縺九ｋ��

縺輔※�後ｂ縺ｨ縺ｮPoincaré蜀吝ワ縺ｫ繧医▲縺ｦ谺｡縲�→逕滓�縺輔ｌ縺ｦ繧�￥繝吶け繝医Ν蛻励�� 蠑�(36)縺ｫ莨ｼ縺溷ｽ｢縺ｮ蠑上↓縺ｪ縺｣縺ｦ縺�ｋ縺ｨ豌励′縺､縺上□繧阪≧:

$\begin{displaymath} \mbox{\boldmath$ x $}^{(1)} = T(\mbox{\boldmath$ x $}^{(0)})... ... x $}^{(k+1)} = T(\mbox{\boldmath$ x $}^{(k)}),\enskip \cdots \end{displaymath}$

(6.39)

縺励◆縺後▲縺ｦ�悟ｼ�(6)縺ｮ迚ｹ諤ｧ譁ｹ遞句ｼ�

$\begin{displaymath} \det \left[ \frac{\partial T(\mbox{\boldmath$ x $}_0)} {\partial \mbox{\boldmath$ x $}_0} - \mu I \right] = 0 \end{displaymath}$

(6.40)

縺ｮ蝗ｺ譛牙､�� $\mu_1,\mu_2$ 縺ｫ繧医▲縺ｦ�悟崋螳夂せ $\mbox{\boldmath$\ x $}_0$ 縺ｮ螳牙ｮ壽ｧ縺瑚ｭｰ隲悶〒縺阪ｋ�� 2谺｡蜈�鴨蟄ｦ邉ｻ縺ｫ縺翫￠繧句崋螳夂せ縺ｮ蛻�｡槭��後％縺薙〒縺ｯ邁｡蜊倥↓ 谺｡陦ｨ縺ｫ縺ｾ縺ｨ繧√ｋ��

**Table 1:** 蝗ｺ螳夂せ縺ｮ蛻�｡�
蜷咲ｧｰ	譚｡莉ｶ	險伜捷
螳悟�螳牙ｮ壼崋螳夂せ	$\vert\mu_1\vert < 1,\, \vert\mu_2\vert <1$	S
螳悟�荳榊ｮ牙ｮ壼崋螳夂せ	$\vert\mu_1\vert > 1,\, \vert\mu_2\vert >1$	U
豁｣荳榊ｮ牙ｮ壼崋螳夂せ	$0 < \mu_1 < 1 < \mu_2$	D
騾�ｸ榊ｮ牙ｮ壼崋螳夂せ	$\mu_1 < -1 < \mu_2 < 0$	I

荳縺､縺ｮ蝗ｺ螳夂せ縺ｫ縺ｯ2縺､縺ｮ譁ｹ蜷代�蝗ｺ譛峨�繧ｯ繝医Ν縺悟ｯｾ蠢懊☆繧具ｼ惹ｾ九∴縺ｰ�梧ｭ｣荳榊ｮ牙ｮ� 蝗ｺ螳夂せ縺ｪ繧画ｵ∝��梧ｵ∝�縺ｮ2譁ｹ蜷代�繝吶け繝医Ν縺ｫ縺ｪ縺｣縺ｦ縺�ｋ�弱％繧後ｒ髷榊梛轤ｹ(saddle point) 縺ｨ縺�≧�弱∪縺滂ｼ碁�ｸ榊ｮ牙ｮ壼崋螳夂せ縺ｯ�悟�蜒上�蠎ｦ縺ｫ蜿榊ｯｾ譁ｹ蜷代↓轤ｹ蛻励′蜿崎ｻ｢縺励※ 縺�￥(繝槭う繝翫せ縺ｮ蛟､繧堤ｹｰ繧願ｿ斐＠縺九￠繧九°繧�)髷榊梛轤ｹ縺ｫ縺ｪ縺｣縺ｦ縺�ｋ��

縺輔※�御ｻ･荳翫�莠亥ｙ遏･隴倥ｒ繧ゅ→縺ｫ�悟崋螳夂せ繧剃ｻｻ諢上�繝代Λ繝｡繝ｼ繧ｿ縺ｫ縺翫＞縺ｦ豎ゅａ繧九％縺ｨ縺後〒縺阪ｋ繧医≧縺ｫ縺ｪ縺｣縺滂ｼ�

User & 2017-09-07

6 蟾ｮ蛻�婿遞句ｼ上�Jacobi陦悟�縺ｨ縺昴�蝗ｺ譛牙､

6 蟾ｮ蛻�婿遞句ｼ上�Jacobi陦悟�縺ｨ縺昴�蝗ｺ譛牙､