03 Nonlinear

8/7/2019 03 Nonlinear

1/30

N-l

--2. Non \ ~ v ' \ ( ~ o . r s:qc4em. f " q ~ c . . : ( { o V \ $ Q. b , . f ' . . r ~ ' ~ \ f \ - h c . J E ' 9 1 A t . d ? o ~ s . (v. p3 )i : ' + l ~ J t )

bEt. . ( l ; ~ O I l \ : h O ' M . 0 1 . A b 61 5 kw-- LA se:u.&l fobe.. ~ ~ C M e - - ~ " ..p . .fJ.pllCi ~ V \ ~ -I:::b O ~ V J ) " ~ . ) V\OV\ ~ n o W l ~ . ulJ . . p ( ~ j ~ t ) ? , r+ ,il+) -spD-,./hPd,iliiA CJ r o ~ ~ b l.st. +0 c l f ~

L-t) + C > b ) ~ ) Y)
http:///reader/full/hO'M.01.Abhttp:///reader/full/hO'M.01.Ab


2/30

N-2

Df2+/h t ~ O Y l : SI..oL 'oS pet e:- M LA C\. l.o


3/30

N-3


4/30

N-4Pc!\- ~ V ' \ c.A';Qt\s. a( f.'W\iL

. OrO ,Yp1 1 ~ ( t ) l I p ( f ( ~ L ~ J I f > J 1 ; ' ] : lphOr"" -66- [\)00) . ~ . ~ . ~ E i t . p w L t ~ 1/?:llp h { ~ ~ ~ r t o r V V \ . . = . H . 1 C ' ( { o O ~ : - ! ~ t J . ~ L - t ) l .. ~ d . Gi ..,(,clA II ' t 1 l ~ e . K ' ; ! f . > 1 - s


5/30

N-5 '3. ,,"2. ?t-op4.f"M% Ff.A)1\ cJ..\0 C o V \ + ; ~ (3,"2, ~ ) ' 4 ~ ~ ( J ; ~ ' ) ~ . ~ : L o ) ~ ) -+ 1< U . o ~ l , t \ M , ~ .. [fJ,oD t . { & o . t ~ ~ ' j . 1 J >9 ..... '3- cS ((QJ +1))'" "fO.i -+:: ,1'( (Q).J.0 ::)/f (+') -,c .-t-o) J .( 0Ji(i) ,. .' . '

+'(,Q' . S ~ ~ , : , ~ ,--,"',',, '1--...

L ; p S ~ ~ 1 - + 2 (3 ..2 ,6). A ~ V \ c < h o ~ C(J bJ -b 1R. ~ l ' p S : : ~ L 4 1 . [ ~ I b}Zi: ..
http:///reader/full/t.%7B&o.t~~'j%E0%AE%B1Jhttp:///reader/full/t.%7B&o.t~~'j%E0%AE%B1J


6/30

~ Q . . . h I ~ O ~ ' : ~ ["?t[1..P:, ",) "?'t.) '1C3J .. - . )1'lE:,X$,.Qq...u...tJol'\ CA..G L::' i ; )i.. 0. no f .......J V4-


7/30

N-7

~ , , ~ f.Lf ()( J /1- Jix) f (y) 11 /l y) kV 1 ( ) 1 ~ c f 1 1 ~ w . . . - 5 f C ( ~ j ~ t : : i . SVPfo$JI- ,.f.: X-i>.Y.s c v . ~ J +0 6 i c o V \ ~ ~ ~ a..* '}C.;:, e ) 0 /3- (A ((,) sA-,II .()11d - ~ ( ~ ~ J 1/ '( (. f. .... ""'- II "){- /1 t


8/30

f'J-8

C o ~ h ( A ~ ~ ' o H ~ f 2 i l ' l ( j ~ O { , . e V o " ' ? V\S v PfO T LA a l'\ op . f J - 1 . . ~ J ,:e) T'l-z:)) 'tK"'......-.b tf


9/30

' - ----rLvv--: (X) \loll) k 0-- ~ t { t - ; ~ S ~ U 1 - ) i.e.*- B:: '?C : / / ~ - ~ I I ! : : orJ 4l< C l o ~ bJJ) r ~ X J r - ~ o $OHo' l lLL.d- X ~ ) 4 '1-: ro r V V \ f ~ 8 I'da ,/:e.... T(1(.) 6- -:B \J?cc 13 2) 3: r


10/30

" - 1 ' f ' ~ 50r w ~ k . : s ok :--P('1):. ':k +


11/30

~ ' ? 9 : J 4

3I ./,/

>/--------- ~ ~ / , / 21

"i


12/30

tJ - r'2..

(.) =-P ( t) -?C)) f 20 j ~ ' o ) : : 1t':o ) / ) C 6 - ~ V l J . . f ! : a < ? I R ~ R ~ c l ~ a .s CJUOII'\ - 0 } ( / ~ - :11'\?.J IA J0/24 d. h a. lA. 'M- :5 D/ tuf,b j,1 ]

J3 'Cr 0 . DUO,.., 0 vJJ-- Q"," I u-r:J [0 J TJ tv4... IM.Ldt"'\'Xl-) ~ i Q . b ~ . e ~ w ~ [0 , rJ c.-.", ; =-P(f I6 b t . f ) ~ ~ . + : '\.A c . . o V \ . J ; ~ . ~ ) -t

(2) i+-) = ' ; to +!.+ ( ~ J 1


13/30

-rL.OUIM.: Globo-Q & ' x , ~ , j . " ' C A . > + ( ) . . . u ~ ~ )5 " - f f o ~ J'cr.- ~ c o : l , T ro )o'b) :t ~ . ~ C o ~ ~ ~ ~ kT ~ c J hT s,-f.! ~ ( f : / - ( t , jll L kT /I?:- ~ I I ' : I ~ ) j e:tit'J \ I ~ [0) TJ1/ ( t. J 'X-o) II hr 'It e: [Q J T1

ttll\ (I) . J q c C 4 . - c + ~ "NL. -SoUOII\ [OJ 00)


14/30

A I'i-1-A- S 1- t;, +

1i ::: A(if) (-toJ ~ W 2 V ' , t,..;-E- k V \ c > ~ ~ ~ O I A -~ o v ~ w , 2 . I(0 Hl ( ~ o w . { , . ; : . =Nt)?C clr:cI {I1/ (t 1 -tOJ/! LIZ:) tI-t>t1:: /( p O ~ ~ ~ o V \ s - l o v . . : J - ; , .):;. $O'm( l .(iO All ~ ( ) ~ J : ; cd : i ~ A { t ) ? C !oJ -to -t-b oD(I /IItt.) to)I/-t:. 0 -t - t> oG'


15/30

fJ-IS

L I V ' \ , t ~ I l ~ - T n ~ C ! ~ ' 5 ~ 6 ~ ~ (LTI)~ O \ ( ' +-A.4- + e ~

....; ::. A ) -xJ.A-o) ~ , / ~ I I ' } k V \ ~ ~ O ~ LA:

2 .. 2.

(L) ALL ~ e ? ~ 1 :i=A?C b f i H . A ~ ~ iI adCWL +L.. h ~ V t V J l C ! v V s . cI A h().v.4... hol;'\ p o ~ : ~ / \ 1 L ~ Q..v\cA ~ f l ze.J\.C / l I 2 ~ p rhdJi QAL 1,1>'\'J.(Li) /4l{ ~ ' o V V J 01 - i ~ A ~ -k",d 10 -t11 aJ..{ ~ ~ V \ ~ A ( J J ) . crt A Y \ J 2 ~ ~ ~ / V ! ~ .


16/30

U i ~ s ( ; t ~ ~ - r ~

A b ~ Se."o f\J O t r ~ S= ~ ( 1 ; ) 'j) S)~ t - { A 6 ~ ~ I o ~ lCl := '::.l;) "Y 1..' 'I j+-t.... ......

'1C / ::: '1C 2 i" ::: i::) 'Jt,,) ?(,.)

1C,

b


17/30

N-I.

Table 2.1 summarizes the various kinds of equilibrium points forsecond-order linear systems.TABLE 2.1

E i g ~ n v Q l u ~ , 01A T y ~ o l Equilibrium PointA., Al real, A. < 0, Al < 0Ait Al real. A, > O. Al > 0A.. Al real. A,A2 < 0A.. A2 complex conjugates, Re A, > 0A" Al complex conjuaates, Re A. < 0A.. Al imaainary

Stable nodeUnstable nodeSaddle pointUnstable rocusStable rocusCenter


18/30

~ - / 8

TABLE 2.2 !f.qllilibrium Point I = 0, J = 0O/tM Lwariud System (7)-(8)

Stable nodeUnslable nodoSaddle pointStable focusUnslable focusCenter

EqllJllbrilun Point XI == 0, XJ = 0o/tll4 NonllMlITiud System ( /H2)Stable nodeUnstable nodoSaddle pointStable focusUnstable focus

7

e - K ~ ~ C r z : Vay\ Po I 's ~ ~ " . . l t D I I \


19/30

2... N l . ( ~ c c J r V \ . . . t e . < t ' ~ o V ' :3 . c..rthe.$At'\ J5 0 C t.i lb tkf'v-..R J loc:::..v..$. ot pO I kt ; (A

UJ'lt{ a. ~ ~ V \ fall l(fd- S/Op-tL, 4 ~ I ' f > O c . . I , ; " " , ~ + L 610f"L 0{ u L J.,f';.JJc! S ( ~ / l ~ l . ) = J . ~ ? = 17. : ~ ' l - , : : : ~ ) ='0< . ' ~ ~ ( 1 C I J ~ t , ) .

! ..

11 :: 1C'l.. i c ~ =- - '1(, ( /- ) -;r2

5 = :. ft:. I +.P (. / - ~ , 't) a. .:: c(~ ~ 2 . .
http:///reader/full/rV/...tehttp:///reader/full/rV/...te


20/30

---- ---

--------

----

--- ---

---

tJ -11 /8 .. 2.

___- - - \""' -----. -- .......

'\ I, I\ I l\ I l... _ \ I l_. . . . ... \ I I

I . . . . . . . " , I I _ _ _ .... '\ I I- .... \ \ , ~ . o -....... " \ \ I I'''' ........ ' , \ \1 I.... \\\1 I... I/

,

--- ...

..., ....... ........... YJ ........ _ I ___ N

--!- -- ____ t

___ 0

o....

.... _0 _ - -.... ,

"'- ..." '\ ----.... ..._- -._..,. --


21/30

0.3

0.2

0.1

o

-0.1

Van der Pol Oscillator

" " '\ \

\ \ " , , - - I ,I I / I / / I II , I , jI / I / 1, , , - - - / / I I I I f

, , , - - - - - / / / / I I I I I I I I I


22/30

tJ-20

Van der Pol Oscillator

3 I, \ \ \ \\ \\ \ i\i\ \2Ii \

\ \i I

1 \ Ii i

o

I \-1\, \ \ \ \ \ \" i \ \ \


23/30

f'J -'2J

...

. ~ " G ) f.u:. a6 Li,..;j c+k . h ~ M O n A e o,sci 110)0,..

YI.(A f: cn'cJa iA aM.. pll ~ J . a . . i ..J (01'1 d.i.j,OIr\-' ~ " ' ~ ......."":'!;.. ""-,,2 \ \'\. J- ~ " " " - ~ .......- .--'" 1.5 \ '\.---""" . ~ ? 1 ' / " . , ?'11/

t "'" " \

\I

\ Vt1 . \/ ' " ,/ / ! / ' .,.. j,/ ! I

'" , I0.5a \ /1 f I-0.5 1 , I ;/1 ,1 /-1 ,/1'/j j


27/30

L / ~ t 4 c ~ ~ v W h ( ) " : i t ~


28/30

Bendixson"s theorem example

3 '\ " " .-'1" //\ \ ' \ . - ' " / /' / '2 \ \ " - ".'" . / /\ \ \ '" ,., I

, !

o

-1 I

/ ' , / / \ \


29/30

""iC1

Y\ok-: tJ LA < : : . o V \ ~ c ~ e f ) bd r\(J1 G i ~ ~ COY} hIlt:..feat . V'f = ~ - B .,... 2)"! =- 20


30/30

A point z e R" is said to be a limitpoinl of this trajectory if thereexists a sequence (/Ji in R+ such that I. - 00 as n - 00 andx(tJ - z as II - 0 0 . The set of all limit points of a trajectory x(t)is called the limit sel of the trajectory and is denoted by L.

42 .R.EMARK.S: Basically, a limit point of the trajectory x(t) is a pointz which has the property that, as time progresses, the trajectorypasses arbitrarily close to J infinitely many times. We sballencounter limit points and limit sets again in Chap. S.

\ 43 T l u t o t w ~ ( p o i n c a r 6 - B e n d i . x s o ~ r r Let44 S == (X(/), t O}

denote a trajectory in RI of the system [2.3(l)H2.3(2)]. and let Ldenote its limit set. If L is contained in a closed bounded region Min RI and if M contains no equilibrium points of [2.3(1)H2.3(2)].then either

(i) S is a periodic solution of [2.3(I)H2.3(2), or(0) L is a periodic solution of [2.3(1)H2.3(2)].We shall omit the proof because it is beyond the scope of thisboot.

45 REMA.u:.s: Roughly speaking. what Theorem (43) states is the following: Suppose we can find a closed bounded region M in RI suchthat M does no t contain any equilibrium points of [2.3(I)H2.3(2)]and such that all limit points of some trajectory S are contained inM. Then M contains at least one periodic solution of [2.3(l)}[2.3(2)]. In practice, it is very difficult to verify that M contains allthe limit points of a trajectory. However, because M is closed, it canbe shown that i f some trajectory x(t) is eventually contained in M,i.e., there exists a time t. < 00 such that x(t) e M V I 10 thenL is contained in Thus the theorem comes down to this: I f we canfind a closed bounded region M containing no equilibrum pointssuch that some tra jectory is eventually confined to M, then M contains at least one periodic solution. Now, a sufficient condition fora trajectory to be eventually confined to M is that, at every pointalong the boundary of M, the velocity vector field always points intoM. If this is the case, then any trajec tory originating within M mustremain in M, and hence M contains at least one periodic solutiontrajectory. (This is depicted in Fig. 2.19.)

XI .( \Ol '\ II ~ G I . . . I ' - ' FIG. 2.19 : : : : ~ 0 :>ci II (

46 Example. Consider once again the system (9}-( I 0), and let Mbe the annular region defined by

I M = ((xu xJ: 0.9 If .xi +xi 1.1p"}Then M contains no equilibrium points of the system (9}-(10). Furthermore, a sketch of the velocity vector field for this system reveals that,all along the boundary of M, the vector field always points into M, asdepicted in Fig. 2.19. Hence we can apply Theorem (43) and concludethat M contains a periodic solution.

.. Example. In applying Theorem (43), the condi tion that M shouldnot contain any equilibrium points is vr:ry important. To see this. consider the system.- XI = -X l + XI

50 X" = -X l - X iThe velocity vector field for this system is sketched in Fig. 2.20. I f wechoose M to be the unit disk centered at the origin. then all along theboundary of M the velocity vector field points into M. Hence all trajec

X2

-\ XI

- \

Documents

03 Nonlinear