periodic solution of van der pol differential equation
DESCRIPTION
Periodic solution of van der Pol differential equation. Seminar at Kitami Institute of Technology January 8, 2009, By Akio Arimoto ( 武蔵工業大学). van der Pol Equation. 概略. Van der Pol の微分方程式 がリミットサイクルをもつということ 2.微分方程式から導かれる積分方程式 (いくつかの基本公式). 3. 周期解の様子 (円、半径が決まってくる) 4.周期 . 簡単な例. 両辺に をかけると. - PowerPoint PPT PresentationTRANSCRIPT
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Periodic solution of van der Pol differential equation.
Seminar at Kitami Institute of Technology
January 8, 2009,By Akio Arimoto (武蔵工業大学)
![Page 2: Periodic solution of van der Pol differential equation](https://reader036.vdocuments.site/reader036/viewer/2022081513/5681315e550346895d97d513/html5/thumbnails/2.jpg)
van der Pol Equation
22 1 0w w w w
0 1
w w t dw t
wdt
![Page 3: Periodic solution of van der Pol differential equation](https://reader036.vdocuments.site/reader036/viewer/2022081513/5681315e550346895d97d513/html5/thumbnails/3.jpg)
概略
1. Van der Pol の微分方程式
がリミットサイクルをもつということ2.微分方程式から導かれる積分方程式(いくつかの基本公式)
22 1 0w w w w
![Page 4: Periodic solution of van der Pol differential equation](https://reader036.vdocuments.site/reader036/viewer/2022081513/5681315e550346895d97d513/html5/thumbnails/4.jpg)
3. 周期解の様子(円、半径が決まってくる)
4.周期
0
2 0
2 2
0lim , , 4w t w t
![Page 5: Periodic solution of van der Pol differential equation](https://reader036.vdocuments.site/reader036/viewer/2022081513/5681315e550346895d97d513/html5/thumbnails/5.jpg)
簡単な例
0w w dw
wdt
2
2
d ww
dt
0
![Page 6: Periodic solution of van der Pol differential equation](https://reader036.vdocuments.site/reader036/viewer/2022081513/5681315e550346895d97d513/html5/thumbnails/6.jpg)
両辺に をかけるとw
0ww ww
2 2 0w w c
相空間 で円をなす
,w w
![Page 7: Periodic solution of van der Pol differential equation](https://reader036.vdocuments.site/reader036/viewer/2022081513/5681315e550346895d97d513/html5/thumbnails/7.jpg)
両辺に を掛けるw
2 22 1 0ww w w ww
22 2 2
0
4 1t
w t w t w s w s ds c
Limit Cycle の存在の可能性
![Page 8: Periodic solution of van der Pol differential equation](https://reader036.vdocuments.site/reader036/viewer/2022081513/5681315e550346895d97d513/html5/thumbnails/8.jpg)
行列形式
2W AW
0 1
1A
2
0
ww
wW
w
![Page 9: Periodic solution of van der Pol differential equation](https://reader036.vdocuments.site/reader036/viewer/2022081513/5681315e550346895d97d513/html5/thumbnails/9.jpg)
解の積分方程式
0
0 2t
A t sAtW t e W e s ds
2
0s
w s w s
![Page 10: Periodic solution of van der Pol differential equation](https://reader036.vdocuments.site/reader036/viewer/2022081513/5681315e550346895d97d513/html5/thumbnails/10.jpg)
のスペクトル表現
1 2A rP rP
1 2I P P 1 2 2 1 0PP P P
0 1
1 2A
21r i
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1 2
1At rt rt rt rt rt rte e P e P e e A re rer r
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基本行列のスペクトル表現
sin sincosAt t t t
e e A t
21 0 1
1 2A
![Page 13: Periodic solution of van der Pol differential equation](https://reader036.vdocuments.site/reader036/viewer/2022081513/5681315e550346895d97d513/html5/thumbnails/13.jpg)
Theorem 1.2
0 2 Stt
C
I te W t U W
I t
2
0
sints
S
sI t e w s w s ds
2
0
cost
sCI t e s w s w s ds
w t
W tw t
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定理1.2からの帰結(1)
• 2つの時点 , の関係
2 1
2 12 1t t
t te U W t e U W t
2
12
02
t sst
e U dsw s w s
1t 2t
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定理1.2からの帰結(2)
• 初期時点を
0
0 0t t
t te U W t W t
0t
0 20 00
sin2
cos
t t s
se w s t w s t ds
s
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Theorem1.2 の証明 (1)
sin0t
t
te W t U W
2
0
0sin2
ts
t s
t se U ds
w s w s
0
0 2t
A t sAtW t e W e s ds
![Page 17: Periodic solution of van der Pol differential equation](https://reader036.vdocuments.site/reader036/viewer/2022081513/5681315e550346895d97d513/html5/thumbnails/17.jpg)
Theorem1.2 の証明 (2)
sin
cos
sin cost t
tt
U Ut t
t s t sUU U 0U I
sint t
tU U
1 0
1
![Page 18: Periodic solution of van der Pol differential equation](https://reader036.vdocuments.site/reader036/viewer/2022081513/5681315e550346895d97d513/html5/thumbnails/18.jpg)
Theorem1.3
0 2 0S
C
IW
I
1 0
1
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Theorem1.3の証明
0 2 Stt
C
I te U W t W
I t
t
0 2 0S
C
IW
I
![Page 20: Periodic solution of van der Pol differential equation](https://reader036.vdocuments.site/reader036/viewer/2022081513/5681315e550346895d97d513/html5/thumbnails/20.jpg)
Theorem1.4 周期条件
W t W t
2 0S S
C C
I t I te U e U I W
I t I t
0
![Page 21: Periodic solution of van der Pol differential equation](https://reader036.vdocuments.site/reader036/viewer/2022081513/5681315e550346895d97d513/html5/thumbnails/21.jpg)
Theorem1.5 周期条件
1 0 2 S
C
Ie U W
I
W t W t
0
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Theorem1.6周期条件
S S S
C C C
I t I t Ie U
I t I t I
W t W t 0
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に同等な積分方程式
sin sin, cost t t
w t e t
2
0
sin2 , ,
tt s t s
e w s w s ds
21 0,w 0,w
22 1 0w w w w
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同等な積分方程式
sin sin, cost t t
w t e t
2
0
sin2 cos , ,
tt s t s
e t s w s w s ds
0,w 0,w 21
![Page 25: Periodic solution of van der Pol differential equation](https://reader036.vdocuments.site/reader036/viewer/2022081513/5681315e550346895d97d513/html5/thumbnails/25.jpg)
周期条件 , ,w t w t
sin sincose
2
0
sin2 , ,s s
e w s w s ds
0,w 0,w
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周期条件
sin sincose
2
0
sin2 cos , ,s s
e s w s w s ds
0,w 0,w
![Page 27: Periodic solution of van der Pol differential equation](https://reader036.vdocuments.site/reader036/viewer/2022081513/5681315e550346895d97d513/html5/thumbnails/27.jpg)
0
0 0lim 0,w
0 0lim 0,w
0 0 0 0 0cos sin
0 0 0 0 0sin cos
00lim
0 2 (mod 2 )n
0 0 0 0 0cos sin
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cos 1 sin
2 2
e e
22
0
sin, ,
s se w s w s ds
![Page 29: Periodic solution of van der Pol differential equation](https://reader036.vdocuments.site/reader036/viewer/2022081513/5681315e550346895d97d513/html5/thumbnails/29.jpg)
2
20 0 1
0
sin 0s w s w s ds
1
0lim
0
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2
20 0 1
0
cos 0s w s w s ds
0
1
0lim
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Limit cycle から決まる円
• の解は任意の半径を持つ円
0 0cos sinw t t t
0 0sin cosw t t t
0
lim ,w t w t
0w w
2 20 0 4
![Page 32: Periodic solution of van der Pol differential equation](https://reader036.vdocuments.site/reader036/viewer/2022081513/5681315e550346895d97d513/html5/thumbnails/32.jpg)
Limit cycle の様子
2 2
0lim , , 4w t w t
2 o
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Another Aproach
211 0
2x x x x
0 dxx
dt
2
2
d xx
dt
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2 22 1 0xx x x xx
2 2 2 24 1x x x x dt
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is a period of the solution
2 2 22 2 2
0
0 0 4 1t
x t x x t x x s x s ds
2 2
0
0 1 x s x s ds
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0 2
0 0 0cos sinx t x t t t
2
2 20 0
0
0 1 x s x s ds
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2
2 2
0 0 0 0
0
0 1 cos sin sin cost t t t dt
2 20 0 4
2 2
0lim 4x t x t