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Page 1: 127 5XÚ - PKUaerocontrol/wangjinzhi/08ode/Lecture_27.pdf · •••bbb333¢¢¢SSSóó󧧧¥¥¥LLLyyy‘‘‘XXXÚÚÚ˙˙˙flflfl›››§§§XXX•••bbb$$$˜˜˜‹‹‹ƒƒƒ¯¯¯>>>XXX

127ù ��5XÚ

December 28, 2010

() December 28, 2010 1 / 25

Page 2: 127 5XÚ - PKUaerocontrol/wangjinzhi/08ode/Lecture_27.pdf · •••bbb333¢¢¢SSSóó󧧧¥¥¥LLLyyy‘‘‘XXXÚÚÚ˙˙˙flflfl›››§§§XXX•••bbb$$$˜˜˜‹‹‹ƒƒƒ¯¯¯>>>XXX

��5X�A5

kkk������<<<ººº���mmm;

õõõ������ááá���²²²ïïï:::;

444������¶¶¶

©©©���yyy���¶¶¶

···bbbyyy���"""

() December 28, 2010 2 / 25

Page 3: 127 5XÚ - PKUaerocontrol/wangjinzhi/08ode/Lecture_27.pdf · •••bbb333¢¢¢SSSóó󧧧¥¥¥LLLyyy‘‘‘XXXÚÚÚ˙˙˙flflfl›››§§§XXX•••bbb$$$˜˜˜‹‹‹ƒƒƒ¯¯¯>>>XXX

��5X�A:

(I) kkk������<<<ººº���mmm(Finite escape time)"""

���555���ØØØCCCXXXÚÚÚµµµx = Ax"""ØØØ­­­½½½������555XXXÚÚÚ������mmmtªªªuuuÃÃá¡¡

���,,,)))ªªªuuuÃÃá¡¡"""

������555XXXÚÚÚ���)))���333kkk������mmmSSSªªªuuuÃÃá¡¡§§§XXXx = x2.

(II) õõõ������ááá���²²²ïïï:::"""

~~~µµµüüü{{{XXXÚÚÚml� = −mg sin � − kl�.

x1 = x2

x2 = −gl sinx1 −

kmx2

²²²ïïï:::§§§²²²ïïï:::���­­­½½½555"""

() December 28, 2010 3 / 25

Page 4: 127 5XÚ - PKUaerocontrol/wangjinzhi/08ode/Lecture_27.pdf · •••bbb333¢¢¢SSSóó󧧧¥¥¥LLLyyy‘‘‘XXXÚÚÚ˙˙˙flflfl›››§§§XXX•••bbb$$$˜˜˜‹‹‹ƒƒƒ¯¯¯>>>XXX

��5X�A:

(III) 444������

���555���ØØØCCCXXXÚÚÚ§§§���ÄÄĵµµXXXJJJ���§§§333666ÄÄÄeeeØØØ���UUU���±±±§§§���ÌÌÌ���

666uuuÐÐЩ©©���"""XXXµµµx = y, y = −x"""£££¥¥¥%%%¤¤¤

­­­½½½������ÄÄÄ777LLLddd������555XXXÚÚÚ���)))"""

dx

dt= y + x(1− x2 − y2)

dy

dt= −x+ y(1− x2 − y2)

---x = r cos �, y = sin �§§§���§§§zzz���µµµdr

dt= r(1− r2), d�

dt= −1. ���

ííí���r = 1===x2 + y2 = 1´­­­½½½���444������"""

() December 28, 2010 4 / 25

Page 5: 127 5XÚ - PKUaerocontrol/wangjinzhi/08ode/Lecture_27.pdf · •••bbb333¢¢¢SSSóó󧧧¥¥¥LLLyyy‘‘‘XXXÚÚÚ˙˙˙flflfl›››§§§XXX•••bbb$$$˜˜˜‹‹‹ƒƒƒ¯¯¯>>>XXX

©�

���ÄÄÄXXXÚÚÚµµµdx

dt= f(x, �)

ÙÙÙ¥¥¥x���GGG���CCCþþþ���þþþ§§§����������ëëëêêê"""

©©©���¯KKKµµµ���XXX������ëëëêêê����UUUCCC§§§f���äää���'''ÝÝÝ������NNNAAA���

����i(�) òòò333EEE²²²¡¡¡(Re�, Im�)þþþ$$$ÄÄħ§§���������ëëëêêê������,,,

����0���§§§XXXÚÚÚ���­­­½½½555òòòuuu)))CCCzzz§§§ùùùÒÒÒ´©©©���¯KKK"""

() December 28, 2010 5 / 25

Page 6: 127 5XÚ - PKUaerocontrol/wangjinzhi/08ode/Lecture_27.pdf · •••bbb333¢¢¢SSSóó󧧧¥¥¥LLLyyy‘‘‘XXXÚÚÚ˙˙˙flflfl›››§§§XXX•••bbb$$$˜˜˜‹‹‹ƒƒƒ¯¯¯>>>XXX

�.©�

���ëëëêêê�CCCzzz���§§§ÏÏÏLLLRe� = 0������¹¹¹���©©©���nnn«««aaa...µµµ

(I) ���...©©©���µµµ

AAA������÷÷÷¢¢¢¶¶¶BBBLLLJJJ¶¶¶"""ÙÙÙ¥¥¥����¢¢¢êêꧧ§���dddKKKCCC���"""

~~~1 x = �x− x3§§§äää���'''ÝÝݵµµ�− 3x2"""

���� < 0���§§§���������²²²ïïï:::x = 0­­­½½½¶¶¶

���� > 0���§§§nnn���²²²ïïï:::x = 0£££ØØØ­­­½½½¤¤¤§§§x =√�£££­­­

½½½¤¤¤§§§x = −√�£££­­­½½½¤¤¤"""

©©©���:::µµµ� = 0"""����dddKKKCCC������§§§AAA�������÷÷÷¢¢¢¶¶¶BBBLLLJJJ¶¶¶§§§���

²²²ïïï:::x = 0ddd­­­½½½CCC���ØØØ­­­½½½§§§lll ©©©���ÑÑÑ###���­­­½½½���²²²ïïï

:::x = ±√�"""ùùùÒÒÒ´���...©©©���"""() December 28, 2010 6 / 25

Page 7: 127 5XÚ - PKUaerocontrol/wangjinzhi/08ode/Lecture_27.pdf · •••bbb333¢¢¢SSSóó󧧧¥¥¥LLLyyy‘‘‘XXXÚÚÚ˙˙˙flflfl›››§§§XXX•••bbb$$$˜˜˜‹‹‹ƒƒƒ¯¯¯>>>XXX

¿Å©�

(II) ¿¿¿ÅÅÅ(Hopf)©©©���µµµAAA������÷÷÷EEE²²²¡¡¡(Re�, Im�)���þþþ���½½½eee���BBBLLL

JJJ¶¶¶"""ÙÙÙ¥¥¥����EEEêêꧧ§���ÙÙÙ¢¢¢ÜÜÜdddKKKCCC���"""

©©©���:::µµµ� = �0÷÷÷vvvµµµ���� = �0���§§§äää���'''ÝÝÝ333²²²ïïï:::äääkkk���

XXXJJJ���"""

~~~2 ⎧⎨⎩ x = −y + x(�− x2 − y2)

y = x+ y(�− x2 − y2)

() December 28, 2010 7 / 25

Page 8: 127 5XÚ - PKUaerocontrol/wangjinzhi/08ode/Lecture_27.pdf · •••bbb333¢¢¢SSSóó󧧧¥¥¥LLLyyy‘‘‘XXXÚÚÚ˙˙˙flflfl›››§§§XXX•••bbb$$$˜˜˜‹‹‹ƒƒƒ¯¯¯>>>XXX

¿Å©�

(0, 0)������²²²ïïï:::§§§ÙÙÙäää���'''ÝÝÝ���µµµ

J =

⎡⎣ � −1

1 �

⎤⎦ÙÙÙAAA���������� = �± i"""���ëëëêêê�dddKKKCCC������§§§�©©©OOO÷÷÷¢¢¢¶¶¶���þþþ���

ÚÚÚeee���BBBLLLJJJ¶¶¶§§§:::(0, 0)ddd­­­½½½������:::CCC¤¤¤ØØØ­­­½½½������:::"""

() December 28, 2010 8 / 25

Page 9: 127 5XÚ - PKUaerocontrol/wangjinzhi/08ode/Lecture_27.pdf · •••bbb333¢¢¢SSSóó󧧧¥¥¥LLLyyy‘‘‘XXXÚÚÚ˙˙˙flflfl›››§§§XXX•••bbb$$$˜˜˜‹‹‹ƒƒƒ¯¯¯>>>XXX

¿Å©�

444���III���§§§���µµµ ⎧⎨⎩ r = r(�− r2)

� = 1

444���IIIXXX¥¥¥���� > 0������kkk,,,���²²²ïïï:::r =√�§§§éééAAA������IIIXXX¥¥¥

���444������x2 + y2 = �"""

ooo(((µµµ���� < 0���§§§���������²²²ïïï:::���­­­½½½������:::§§§���� > 0���§§§

:::(0, 0)CCC¤¤¤ØØØ­­­½½½������:::§§§lll ©©©���ÑÑÑ444������x2 + y2 = �§§§ùùù«««

ddd���­­­��� ÑÑÑyyy���444���������©©©���§§§ÒÒÒ´¿¿¿ÅÅÅ©©©���"""

() December 28, 2010 9 / 25

Page 10: 127 5XÚ - PKUaerocontrol/wangjinzhi/08ode/Lecture_27.pdf · •••bbb333¢¢¢SSSóó󧧧¥¥¥LLLyyy‘‘‘XXXÚÚÚ˙˙˙flflfl›››§§§XXX•••bbb$$$˜˜˜‹‹‹ƒƒƒ¯¯¯>>>XXX

Q(:©�

(III) QQQ(((:::©©©���µµµAAA������÷÷÷¢¢¢¶¶¶üüü>>>ªªªuuuJJJ¶¶¶"""ÙÙÙ¥¥¥����¢¢¢êêꧧ§����ddd

���KKKüüü>>>ªªªuuu""""""

~~~3

x = �+ x2

y = y

²²²ïïï:::P (√−�, 0)ÚÚÚQ(−

√−�, 0)§§§P:::éééAAA���äää���'''ÝÝÝ���AAA���

�������1 = 2√−�, �2 = 1¶¶¶Q:::éééAAA���äää���'''ÝÝÝ���AAA������

����1 = −2√−�, �2 = 1¶¶¶

���� < 0���§§§P:::���(((:::¶¶¶Q:::���QQQ:::¶¶¶���� = 0���§§§P:::ÚÚÚQ:::ÜÜÜ

���������§§§===(((:::ÚÚÚQQQ:::ÜÜÜ������:::(0, 0)¶¶¶���� > 0���§§§ÃÃò²²ïïï:::"""

() December 28, 2010 10 / 25

Page 11: 127 5XÚ - PKUaerocontrol/wangjinzhi/08ode/Lecture_27.pdf · •••bbb333¢¢¢SSSóó󧧧¥¥¥LLLyyy‘‘‘XXXÚÚÚ˙˙˙flflfl›››§§§XXX•••bbb$$$˜˜˜‹‹‹ƒƒƒ¯¯¯>>>XXX

áÚf

���ÄÄÄXXXÚÚÚµµµdx

dt= F (x), x ∈ RN .

ÅÅÅðððXXXÚÚÚ���������mmmNNNÈÈÈ333$$$ÄÄÄLLL§§§¥¥¥´ØØØCCC���§§§ÑÑÑÑÑÑXXXÚÚÚÙÙÙ������

mmmNNNÈÈÈ333$$$ÄÄÄLLL§§§¥¥¥´ØØØäää   ���§§§������mmm���NNNÈÈÈ

���dV = dx1dx2 ⋅ ⋅ ⋅ , dxN���CCCzzzÇÇÇAAA÷÷÷vvv

1

dV

d

dt(dV ) =

N∑i=1

∂xi

dxidt

=

N∑i=1

∂Fi(x)

∂xi< 0

þþþªªª���������ÑÑÑÑÑÑXXXÚÚÚ���½½½ÂÂÂ"""

ddduuuÑÑÑÑÑÑXXXÚÚÚÙÙÙ������mmmNNNÈÈÈ333$$$ÄÄÄLLL§§§¥¥¥´ØØØäää   ���§§§$$$ÄÄÄ;;;

������������áááÚÚÚ���������kkk������������mmm¥¥¥���§§§ùùùÒÒÒ´áááÚÚÚfff"""

() December 28, 2010 11 / 25

Page 12: 127 5XÚ - PKUaerocontrol/wangjinzhi/08ode/Lecture_27.pdf · •••bbb333¢¢¢SSSóó󧧧¥¥¥LLLyyy‘‘‘XXXÚÚÚ˙˙˙flflfl›››§§§XXX•••bbb$$$˜˜˜‹‹‹ƒƒƒ¯¯¯>>>XXX

áÚf

������mmm¥¥¥���������fff888A¡¡¡���áááÚÚÚfff£££XXX­­­½½½���²²²ïïï:::§§§­­­½½½���444

������¤¤¤§§§XXXJJJ§§§äääkkkXXXeee555���µµµ

(i) Aéééuuu���§§§���666´ØØØCCC���;

(ii) ���333������A���������§§§§§§333���§§§¤¤¤(((½½½���666ÄÄÄeee   ���A¶¶¶

(iii) 333Aþþþ���666´ÌÌÌ������§§§===éééx £££x ∈ A¤¤¤:::���???ÛÛÛ������§§§333vvv���

������mmmSSS§§§ÏÏÏLLLA¥¥¥x:::���;;;���ooo¬¬¬222ggg²²²LLLùùù���������¶¶¶

(iv) AØØØUUU©©©)))¤¤¤üüü���ØØØ­­­UUU���ÜÜÜ©©©"""

() December 28, 2010 12 / 25

Page 13: 127 5XÚ - PKUaerocontrol/wangjinzhi/08ode/Lecture_27.pdf · •••bbb333¢¢¢SSSóó󧧧¥¥¥LLLyyy‘‘‘XXXÚÚÚ˙˙˙flflfl›››§§§XXX•••bbb$$$˜˜˜‹‹‹ƒƒƒ¯¯¯>>>XXX

áÚf

ddduuuÑÑÑÑÑÑXXXÚÚÚ������mmmNNNÈÈÈ���ªªª���   ���"""§§§¤¤¤±±±áááÚÚÚfff���N���NNN

ÈÈÈ���"""§§§===áááÚÚÚfff������êêê���$$$uuu������mmm������êêê"""

­­­½½½���²²²ïïï:::´���{{{üüü���áááÚÚÚfff§§§ÙÙÙ���êêê���"""¶¶¶���­­­½½½���²²²ïïï:::

ÏÏÏLLLHopf©©©���¤¤¤���444���������§§§§§§BBB´������áááÚÚÚfff¶¶¶

äääkkk������êêê���êêê���áááÚÚÚfff¤¤¤���ÛÛÛ%%%áááÚÚÚfff£££···bbbáááÚÚÚfff¤¤¤"""

333ÑÑÑÑÑÑXXXÚÚÚXXXÚÚÚ¥¥¥~~~kkkooo«««áááÚÚÚfffµµµ½½½~~~áááÚÚÚfff§§§±±±ÏÏÏáááÚÚÚfff§§§

[[[±±±ÏÏÏáááÚÚÚfffÚÚÚ···bbbáááÚÚÚfff"""

() December 28, 2010 13 / 25

Page 14: 127 5XÚ - PKUaerocontrol/wangjinzhi/08ode/Lecture_27.pdf · •••bbb333¢¢¢SSSóó󧧧¥¥¥LLLyyy‘‘‘XXXÚÚÚ˙˙˙flflfl›››§§§XXX•••bbb$$$˜˜˜‹‹‹ƒƒƒ¯¯¯>>>XXX

·bXÚ

(IV) ···bbbXXXÚÚÚ

···bbbyyy���´ØØØNNN\\\???ÛÛÛ���ÅÅÅÏÏÏ������(((½½½555������555XXXÚÚÚ£££½½½���½½½555

KKK¤¤¤���)))������«««   uuukkk���«««������;;;���[[[ØØØ­­­EEE���$$$ÄÄħ§§···bbb$$$ÄÄÄ

´ØØØ555KKK���§§§aaaqqq���ÅÅÅ������«««111���§§§���äääkkk���±±±ÏÏÏ���kkkSSS555"""

1963cccLorenz§§§���ííí���ÆÆƧ§§ïïïÄÄÄUUUíííýýý������(((JJJÚÚÚ���íííëëë666yyy���

���§§§nnn���CCCþþþggg£££���ÛÛÛÔÔÔ[[[���§§§§§§ÙÙÙ)))´���±±±ÏÏÏ���ééé···ÏÏϧ§§´111

������333ÑÑÑÑÑÑXXXÚÚÚ¥¥¥������(((½½½������§§§���ÑÑÑ···bbb)))���~~~fff"""���ÑÑÑíííÿÿÿ

ØØØUUU°°°(((­­­üüü���UUUíííÃÃÃ{{{���ÏÏÏýýý������mmm���333XXX���«««éééXXX§§§£££ããã

ÐÐÐ������¯aaa555"""

ÙÙÙ§§§+++���¥¥¥kkkéééõõõaaaqqq���yyy���"""

() December 28, 2010 14 / 25

Page 15: 127 5XÚ - PKUaerocontrol/wangjinzhi/08ode/Lecture_27.pdf · •••bbb333¢¢¢SSSóó󧧧¥¥¥LLLyyy‘‘‘XXXÚÚÚ˙˙˙flflfl›››§§§XXX•••bbb$$$˜˜˜‹‹‹ƒƒƒ¯¯¯>>>XXX

Lorenz XÚ

~~~1 Lorenz XXXÚÚÚ ⎧⎨⎩x = �(y − x)

y = −xz + rx− y

z = xy − bz

ÙÙÙ¥¥¥xLLL«««ééé666rrrÝÝݧ§§yLLL«««���þþþ666ÚÚÚ���eee666333üüü   ������mmm���§§§ÝÝÝ

���§§§zLLL«««RRR���������§§§ÝÝÝ©©©ÙÙÙ���������555rrrÝÝݧ§§−xzÚÚÚxy���������555

���§§§r´aaa|||êêꧧ§§§§LLL«««ÚÚÚåååééé666ÚÚÚëëë666���°°°ÄÄÄÏÏÏ���ÚÚÚ³³³���ééé666

ÏÏÏ������'''§§§´XXXÚÚÚ���ÌÌÌ���������ëëëêêê"""� =�

�´ÊÊÊKKKAAAêê꣣£�ÚÚÚ�©©©

OOO���©©©fffÊÊÊ555XXXêêêÚÚÚ999DDD���XXXêêꤤ¤§§§b���LLL���ééé666pppîîî'''kkk'''���

///'''"""

() December 28, 2010 15 / 25

Page 16: 127 5XÚ - PKUaerocontrol/wangjinzhi/08ode/Lecture_27.pdf · •••bbb333¢¢¢SSSóó󧧧¥¥¥LLLyyy‘‘‘XXXÚÚÚ˙˙˙flflfl›››§§§XXX•••bbb$$$˜˜˜‹‹‹ƒƒƒ¯¯¯>>>XXX

Lorenz XÚ

LorenzXXXÚÚÚ���ÑÑÑÑÑÑXXXÚÚÚµµµ������mmmNNNÈÈÈ������mmmCCCzzzÇÇÇ���

1

V

dV

dt= divV =

∂x

∂x+∂y

∂y+∂z

∂z= −(� + b+ 1) < 0, (� > 0, b > 1)

²²²ïïï:::µµµ

£££1¤¤¤���0 < r < 1���§§§===kkk������²²²ïïï:::

O : (x, y, z) = (0, 0, 0)

£££2¤¤¤���r > 1���§§§kkknnn���²²²ïïï:::⎧⎨⎩O : (x, y, z) = (0, 0, 0)

c1 : (x, y, z) = (√b(r − 1),

√b(r − 1), r − 1)

c2 : (x, y, z) = (−√b(r − 1),−

√b(r − 1), r − 1)

() December 28, 2010 16 / 25

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Lorenz XÚ

���gggCCCqqqXXXÚÚÚ���­­­½½½555©©©ÛÛÛ²²²ïïï:::���­­­½½½555"""

(1) ���0 < r < 1���§§§²²²ïïï:::O:::���­­­½½½���"""

(2) ���1 < r < r0 = 24.74���§§§O���ØØØ­­­½½½���²²²ïïï:::§§§c1ÚÚÚc2���­­­½½½���

:::"""

(3) ���r = r0���uuu)))¿¿¿ÅÅÅ©©©���§§§ÑÑÑyyyØØØ­­­½½½���444������"""

(4) ���r > r0 = 24.74���§§§333,,,r������SSS§§§ÙÙÙ)))���±±±ÏÏÏ)))§§§333,,,���

r������SSS§§§ÙÙÙ)))���ÛÛÛ%%%áááÚÚÚfff§§§���)))···bbb$$$ÄÄÄ"""

() December 28, 2010 17 / 25

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Lorenz XÚ

0 2 4 6 8 10 12 14 16 18 20−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

t

x,y,

z

a=10, r=0.5, b=8/3

0 2 4 6 8 10 12 14 16 18 20−5

0

5

10

15

20

t

x,y,

z

a=10, r=12, b=8/3

ããã1 ããã2

ããã1µµµ���������­­­½½½²²²ïïï:::(0, 0).

ããã2µµµØØØ­­­½½½²²²ïïï:::(0, 0)§§§­­­½½½²²²ïïï:::(±5.416,±5.416, 11)

() December 28, 2010 18 / 25

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Lorenz XÚ

−20 −15 −10 −5 0 5 10 15 20

−50

0

50−10

0

10

20

30

40

50

x

a=10, r=28, b=8/3

y

z

0 10 20 30 40 50 60 70 80 90 100−30

−20

−10

0

10

20

30

40

50

t

x,y,

z

a=10, r=28, b=8/3

ããã3 ···bbbáááÚÚÚfff""" ããã4 GGG���x(t), y(t), z(t)

() December 28, 2010 19 / 25

Page 20: 127 5XÚ - PKUaerocontrol/wangjinzhi/08ode/Lecture_27.pdf · •••bbb333¢¢¢SSSóó󧧧¥¥¥LLLyyy‘‘‘XXXÚÚÚ˙˙˙flflfl›››§§§XXX•••bbb$$$˜˜˜‹‹‹ƒƒƒ¯¯¯>>>XXX

·bXÚ�A5

SSS333���ÅÅÅ555"""···bbb«««���111���LLLyyyÑÑÑ���ÅÅÅØØØ(((½½½555§§§ùùù«««ØØØ(((½½½555

ØØØ´555 uuuÜÜÜ���¸������ÅÅÅÏÏÏ���éééXXXÚÚÚ���KKK���§§§ ´XXXÚÚÚggguuu���

)))���"""

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() December 28, 2010 20 / 25

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() December 28, 2010 21 / 25

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() December 28, 2010 22 / 25

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() December 28, 2010 23 / 25

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() December 28, 2010 24 / 25

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() December 28, 2010 25 / 25