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諡｡蠑ｵBVP逋ｺ謖ｯ蝎ｨ縺ｫ縺翫￠繧区険蜍募●豁｢迴ｾ雎｡縺ｫ縺､縺�※
On Suppression of Oscillation in the Modified BVP Oscillator

荳顔伐蜩ｲ蜿ｲ	蟾昜ｸ� 蜊�
T. Ueta	H. Kawakami
(蠕ｳ蟲ｶ螟ｧ蟄ｦ)

**Figure 1:** 諡｡蠑ｵBVP逋ｺ謖ｯ蝎ｨ
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諡｡蠑ｵBVP逋ｺ謖ｯ蝎ｨ縺ｯ�悟ｼ帛ｼｵ逋ｺ謖ｯ繧偵→繧ゅ↑縺� 3髫手�蠕狗ｳｻ縺ｮ繧ｫ繧ｪ繧ｹ蜉帛ｭｦ邉ｻ縺ｧ縺ゅｋ[1]. 蝗ｳ1 縺ｫ蝗櫁ｷｯ蝗ｳ繧堤､ｺ縺呻ｼ� 邯ｲ縺後￠縺輔ｌ縺ｦ縺�ｋ驛ｨ蛻�′2谺｡蜈� BVP逋ｺ謖ｯ蝎ｨ縺ｧ縺ゅｊ�梧僑蠑ｵ縺励◆ , 縺ｮ繝繧､繝翫Α繧ｯ繧ｹ縺鯉ｼ後←縺�匱謖ｯ蝎ｨ蜈ｨ菴薙↓蠖ｱ髻ｿ縺吶ｋ縺九� 闊亥袖豺ｱ縺�撫鬘後〒縺ゅｋ�� [ $\mu$ F], [ $\Omega$ ], [mH] 縺ｫ驕ｸ縺ｳ��, 繧定ｧ｣謾ｾ髯､蜴ｻ縺励◆ 2谺｡蜈� BVP逋ｺ謖ｯ蝎ｨ縺ｯ閾ｪ蜉ｱ逋ｺ謖ｯ縺吶ｋ�� 縺昴�蜴溷屏縺ｯ蜴溽せ縺ｮ Hopf 蛻�ｲ舌〒縺ゅｋ縺薙→縺梧�繧峨°縺ｫ縺ｪ縺｣縺ｦ縺�ｋ[2]. [mH], [ $\Omega$ ] 縺ｨ縺励※�� 蠑帛ｼｵ謖ｯ蜍輔ｒ遉ｺ縺咏憾諷九↓險ｭ螳壹＠縺滂ｼ弱◎縺薙〒�� 繧貞､ｧ縺阪＞蛟､縺九ｉ蠕舌�↓貂帛ｰ代＆縺帙※謖ｯ蜍輔�螟牙喧繧定ｦｳ蟇溘＠縺滂ｼ� 縺悟､ｧ縺阪＞縺�■縺ｯ縺ｫ豬∝�縺吶ｋ髮ｻ豬√�蟆上＆縺擾ｼ�2谺｡蜈ィVP逋ｺ謖ｯ縺昴�繧ゅ�縺ｧ縺ゅｋ縺�(蝗ｳ2(a))�� 遯∝ｦ�[ $\Omega$ ] 莉倩ｿ代〒謖ｯ蜍輔′蛛懈ｭ｢縺吶ｋ(蝗ｳ2(b))�弱％縺ｮ縺ｨ縺搾ｼ� 莉悶↓螳牙ｮ壹↑蟷ｳ陦｡轤ｹ繧�Μ繝溘ャ繝医し繧､繧ｯ繝ｫ縺ｯ縺ｪ縺�ｼ� 縺昴�蠕鯉ｼ後＆繧峨↑繧�縺ｮ貂帛ｰ代↓繧医▲縺ｦ蜀阪�逋ｺ謖ｯ縺檎函縺倥ｋ(蝗ｳ2(c)(d))�� 莉･荳九〒縺ｯ縺薙ｌ繧峨�迴ｾ雎｡繧呈焚逅�噪縺ｫ隗｣譫舌☆繧具ｼ�

Figure 2: 螳滓ｸｬ蛟､��

縺ｨ $r_1\cdot i_1$ 繧呈ｸｬ螳壹＠縺滂ｼ� (a)

[k $\Omega$ ].

隗｣謾ｾ髯､蜴ｻ縺ｨ蜷悟柑譫懶ｼ守ｴ�6[kHz]. (b)

[ $\Omega$ ]遞句ｺｦ�梧険蜍募●豁｢�� (c)

[ $\Omega$ ]遞句ｺｦ�取険蜍募ｾｩ豢ｻ�檎ｴ� 34[kHz]. (d)

[ $\Omega$ ](

縺ｮ遏ｭ邨｡髯､蜴ｻ)�取険蟷�′髱槫ｸｸ縺ｫ螟ｧ縺阪＞��

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(a)	(b)
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(c)	(d)

**Figure 3:** - 蟷ｳ髱｢縺ｫ縺翫￠繧句�蟯仙峙��
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縺ｾ縺夲ｼ悟屓霍ｯ譁ｹ遞句ｼ上↓蟇ｾ縺玲ｭ｣隕丞喧繧呈命縺吶→谺｡縺ｮ3髫手�蠕狗ｳｻ繧貞ｾ励ｋ:

$\begin{displaymath} \dot{x} = -y - z + \epsilon ( x - x^3/3), \quad \dot{y} = x - k_1 y, \quad \alpha \dot{z}= x - k_2 z \end{displaymath}$

(1)

縺薙％縺ｧ��3谺｡迚ｹ諤ｧ縺ｧ霑台ｼｼ縺励◆髱樒ｷ壼ｽ｢繧ｳ繝ｳ繝繧ｯ繧ｿ繝ｳ繧ｹ縺ｯ�梧ｸｬ螳壹°繧� $\epsilon \approx 1.6$ 縺ｨ蜷悟ｮ壹＆繧後◆�� $\alpha = L_2/L_1$ 縺ｧ縺ゅｊ�御ｸ願ｨ倥�蝗櫁ｷｯ繝代Λ繝｡繝ｼ繧ｿ縺ｧ縺ｯ $\alpha = 0.1$ 縺ｨ縺ｪ繧具ｼ� 蠑�(1)縺ｮ謨ｰ蛟､險育ｮ励↓縺翫＞縺ｦ繧ゑｼ御ｸ願ｨ倥→蜷梧ｧ倥↑迴ｾ雎｡縺檎｢ｺ隱阪＆繧後◆�� 縺昴％縺ｧ�悟次轤ｹ縺ｫ縺､縺�※

蟷ｳ髱｢縺ｧ縺ｮHopf蛻�ｲ宣寔蜷医ｒ豎ゅａ縺滂ｼ� 蝗ｳ3繧貞盾辣ｧ�� 豁｣縺ｮ謚ｵ謚怜､ $r_1\ge 0$ , $r_2\ge 0$ 縺ｯ�� $k_1\ge 0$ , $k_2\ge 0$ 縺ｫ蟇ｾ蠢懊＠縺ｦ縺�ｋ�� $\alpha$ , $\epsilon$ 繧貞崋螳壹☆繧後��悟�蟯宣寔蜷医�

縺ｮ髢｢謨ｰ縺ｨ縺ｪ繧奇ｼ� $\xi(k_1,k_2) \pm \sqrt{\eta(k_1,k_2)}$ 縺ｮ蠖｢繧貞叙繧具ｼ主峙荳ｭ縺ｧ縺ｯ縺薙�隨ｦ蜿ｷ縺ｫ蜷医ｏ縺帙※ $H_{-}$ , $H_{+}$ 縺ｨ縺励◆�惹ｸ願ｨ倥�逋ｺ謖ｯ蛛懈ｭ｢縺ｯ縺薙ｌ繧我ｺ後▽縺ｮ Hopf 蛻�ｲ舌′髢｢荳弱＠縺ｦ縺�◆�� $H_{-}$ 縺ｯ $k_1 \approx 1.55$ 繧呈ｼｸ霑醍ｷ壹→縺励※謖√▽蜿梧峇邱壹〒縺ゅｊ�� 莉匁婿縺ｮ譖ｲ邱壹�

縺ｮ鬆伜沺縺ｧ迴ｾ繧後ｋ�� $H_{+}$ 縺ｯ��

繧貞､画峇轤ｹ縺ｨ縺吶ｋ繧上★縺九↑豁ｪ縺ｿ繧呈戟縺､縺鯉ｼ� 縺ｻ縺ｼ閾ｳ繧九→縺薙ｍ逶ｴ邱壹→縺ｪ繧具ｼ� $H_{-}$ 繧医ｊ荳企Κ縺ｯ菴弱＞蜻ｨ豕｢謨ｰ縺ｮ逋ｺ謖ｯ鬆伜沺�� $H_{+}$ 繧医ｊ荳矩Κ縺ｯ鬮倥＞蜻ｨ豕｢謨ｰ縺ｮ逋ｺ謖ｯ鬆伜沺縺ｨ縺ｪ繧具ｼ� Hopf蛻�ｲ仙ｾ後�繝ｪ繝溘ャ繝医し繧､繧ｯ繝ｫ縺ｮ隗貞捉豕｢謨ｰ繧抵ｼ悟峙3 荳ｭ縺ｫ謨ｰ蛟､(蜊倅ｽ阪�[deg/sec]) 縺ｨ縺励※險伜�縺励※縺�ｋ�惹ｸ｡蛻�ｲ宣寔蜷医�謖√▽ 蜻ｨ豕｢謨ｰ縺ｮ蟾ｮ縺悟､ｧ縺阪＞縺薙→縺悟�縺九ｋ�� 谺｡縺ｫ縺薙�蛻�ｲ仙峙縺ｮ $\alpha$ , $\epsilon$ 萓晏ｭ俶ｧ繧定ｪｿ譟ｻ縺励◆�� $\alpha$ 繧貞､牙喧縺輔○縺ｦ繧ょ�蟯仙峙縺ｫ縺ｯ縺ｻ縺ｩ繧薙←蠖ｱ髻ｿ縺後↑縺擾ｼ� 縺ｾ縺滂ｼ� $\epsilon$ 繧� $1.0 < \epsilon < 2.0$ 縺ｮ螳溽畑遽�峇縺ｧ螟牙喧縺輔○縺ｦ繧ゑｼ悟曙譖ｲ邱壹�貍ｸ霑醍ｷ壹′闍･蟷ｲ遘ｻ蜍輔☆繧九ｂ縺ｮ縺ｮ�� 菴咲嶌逧�↓縺ｯ蜷後§蛻�ｲ仙峙縺ｨ縺ｪ繧九％縺ｨ縺悟�縺九▲縺滂ｼ� 縺､縺ｾ繧奇ｼ悟峙1 縺ｮ蝗櫁ｷｯ讒区�縺ｧ縺ｯ�� 蛻�ｲ仙峙縺ｯ螳壽ｧ逧�↓蝗ｳ3縺ｮ蠖｢迥ｶ縺ｫ縺ｪ繧奇ｼ� 蝗櫁ｷｯ繝代Λ繝｡繝ｼ繧ｿ縺ｫ髱樔ｾ晏ｭ倥〒縺ゅｋ縺ｨ險縺｣縺ｦ繧医＞�� 蜴溽せ縺ｮ縺ｿ縺ｫ髢｢縺吶ｋ supercritical Hopf 蛻�ｲ舌〒�御ｸ縺､縺ｮ繝代Λ繝｡繝ｼ繧ｿ蛟､螟画峩縺ｫ繧医▲縺ｦ莠檎ｨｮ鬘槭�逡ｰ縺ｪ繧句捉豕｢謨ｰ縺ｮ逋ｺ謖ｯ縺悟ｾ励ｉ繧後ｋ蜉帛ｭｦ邉ｻ縺ｯ萓九′縺ｪ縺�→閠�∴繧峨ｌ繧具ｼ� 縺薙�遐皮ｩｶ縺ｯ�梧律譛ｬ蟄ｦ陦捺険闊井ｼ夂ｧ醍�碑ｲｻ闍･謇�(B)15760293 縺ｮ陬懷勧繧貞女縺代※縺�ｋ��

譁�鍵: [1]蜷画ｰｸ蜩ｲ蜩会ｼ梧�蠢懃ｾｩ蝪ｾ螟ｧ蟄ｦ蜊壼｣ｫ隲匁枚��1992. [2]蛹怜ｳｶ蜊壻ｹ具ｼ悟ｾｳ蟲ｶ螟ｧ蟄ｦ蜊壼｣ｫ隲匁枚, 1998.

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