Variational Equation or Continuous Dependence on Initial Condition or Trajectory Sensitivity & Floquet Theory & Poincaré Map

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Jesse Hicks
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1 Vaiaioal Equaio o Coiuous Dpc o Iiial Coiio o Tajco Ssiivi & Floqu Tho & Poicaé Map. Gal ia o ajco ssiivi.... Homogous Lia Tim Ivaia Ssm No - Homogous Lia Tim Ivaia Ssm...3 Eampl (LTI:.... Homogous Lia Tim Vaig Ssm...5 Eampl (LTV: No Homogous Lia Tim Vaig Ssm Nolia ssm...6 Eampl 3 (NL:...6 Summa Sabili Aalsis...9 Summa: Floqu Tho... Summa Poicaé Map... Ovall Summa:...3 Eampl: Poicaé Map vs Floqu Tho.... Iiial umical aalsis.... Pola Cooias...6. Equilibia a soluio o ssm i pola cooias...7. Poica map a Jacobia om pola cooias Casia cooias Soluio i Casia cooias Jacobia i Casia cooias Pubaio calculaio usig h Jacobia... Floqu Tho...3. Jacobia aou h pioic obi Soluio o Mai Diial Equaio Calculaio o Mooom mai Pubaio calculaio usig h Mooom mai...6 /7

2 . Gal ia o ajco ssiivi Assum a soluio ( o a IVP How will h obi bhav i w sa v clos o i? δ ( δ ( δ ( δ ( δ W i: ( Figu a. δ ( δ ( Δ( b. Δ δ Now w hav o pss δ δ 3 Δ TS Δ as a ucio o h oigial pubaio: ( Δ δ δ Δ ( This is h vaiaioal quaio. Fo w hav ha Δ ( ( ( ( Δ I /7

3 So i w wa o s h ssiivi o h ICs w o i. Homogous Lia Tim Ivaia Ssm I w hav a lia ssm ( A & h h soluio is: A( (. Th o cous: A( ( Noic ha hough ha las quaio w ca giv aoh iiio o h STM (i his poial mai: h paial ivaiv o h soluio w h iiial coiio: A( Thus: Δ Δ( 3. No - Homogous Lia Tim Ivaia Ssm I w hav a lia ssm ( A BU & h h soluio is: A( A ( τ BU ( τ τ Th o cous: ( A( (3 A( Thus: Δ Δ( (s ampl i h ollowig pag 3/7

4 Eampl (LTI: % iiial clc cla clos sms au ; ; A[ -; 3]; B[;]; Usi(;Uausi(au; % omial obi [.;.]; solpm(a*(-*i(pm(a*(au*b*uauau; c; :.:.5; o ; sol(c:subs(sol; cc; % pub obi [.;.].5; solpm(a*(-*i(pm(a*(au*b*uauau; c; :.:.5; o ; sol(c:subs(sol; cc; % plo h ic subplo( plo(sol(:-sol(:sol(:-sol(: lim([mi(sol(:-sol(: ma(sol(:- sol(:] % Calcula h i D[.5;.5]; Jpm(A*(-; c; :.:.5; o ; D(c:subs(J*D; cc; subplo( plo(d(:d(:'' lim([mi(d(: ma(d(:] /7

STM Φ STM I Φ STM Noic ha hough ha las quaio w ca giv aoh iiio o h STM

5 . Homogous Lia Tim Vaig Ssm I w hav a lia ssm &( A a hc o cous Φ Th h soluio is ( STM ( Φ STM ( To i ow his STM w hav o umicall solv: A Φ STM Φ STM I Φ STM Noic ha hough ha las quaio w ca giv aoh iiio o h STM (i his cas: h paial ivaiv o h soluio w h iiial coiio: Eampl (LTV: Figu 5/7

6 5. No Homogous Lia Tim Vaig Ssm Th aalsis h is acl h sam as bo so o ampls a giv. 6. Nolia ssm I w hav a olia ssm: ( I I Jacobia Y τ τ τ τ τ τ τ τ τ τ τ τ 3 3 Now h RHS mus b qual o h LHS o which mas ha I. Also i a lia ssm ( A which mas agai ha is ohig mo ha h STM. Eampl 3 (NL: Cosi h ssm. Th umical soluio o ( a ( is: 6/7

7 Figu 3 Th Jacobia is: 3 3 alog h obi ( i o o i h STM: which s o b valua ( ( I Th Jacobia is: Hc a s h STM has h valu: Figu >> [DPA' DPB'] as /7

8 So l s a a small pubaio Δ a s o b:.. ( a I pc h pubaio >> [DPA' DPB']*[.;.] as oigial To Wokspac To Wokspac oigial >> [; ] as Summa 8/7

9 So o summais: To i how h pubaios volv w us: Δ Δ Thus w which is h STM. Hc o i h STM: ( o Fo LTI ssms A o Fo LTV ssms Φ STM A Φ STM ( Φ STM I o Fo NL ssms agai solv h pvious quaio bu ow ( ( A. ( 7. Sabili Aalsis Now l s assum ha w hav a pioic obi h w ca sill appl h pvious mho a w ca o i h pubaios usig: Δ Δ o Δ ( Φ Δ Noic ha w us Φ isa o Φ as i h gal olia cas ma sill b a ucio o. This mai ow ca b ou b solvig Φ STM ( Φ STM ( Φ STM ( I L s valua h abov pubaios a T: Δ ( T Φ( T Δ( Th w call h mai ccl. ( T Φ h Mooom Mai (MM o h limi 9/7

10 ( ( ( ( ( T Φ ( T Δ( Δ T Φ T Δ T Now Δ T Φ T Φ T Δ Δ A also Δ ( T Φ( T Δ( Thus w immial s ha Φ( T Φ ( T ha: Δ Φ ( kt Φ T k ( k ( kt Φ ( T Δ ( o ha which implis k Usig igvalu composiio: Δ( kt U Λ UΔ( Which implis ha i h all igvalus o h mooom mai hav absolu valu lss ha h obi is sabl. Summa: So o summais: Δ ( Thus w ( Δ ( Hc o i h STM: o Fo LTI ssms which is h STM. ( A( o Fo LTV ssms Φ STM A Φ STM ( Φ STM I o Fo NL ssms agai solv h pvious quaio bu ow ( ( A. ( Evalua h STM a T 8. Floqu Tho Aoh wa o s h abov is o liais aou h limi ccl: /7

11 & & & ( ( p ( ( ( ( Δ( ( & ( Δ p ( p ( ( ( ( ( ( ( ( ( ( ( p p p Δ Δ ( ( p p Δ ( Which i h pvious oaio his ca b wi as: Δ ( ( Δ ( ( Φ STM Δ( a hus h soluio is: Δ which agai implis ha Φ STM ( ( ( Φ ( ( STM ( a ha a ha ( I Φ STM. Hc w hav a simila quaio o h o bo a hc w g h sam sul. Thus o i h sabili popis w hav o i h mooom mai a ach igvalus. Summa: Liais h ssm aou h pioic obi a hc pss h pubaios hough a LTV iial quaio To solv ha w h STM which ca b ou b solvig a mai iial quaio: ( ( Φ STM Φ STM Φ ( Th sabili is ou b valuaig Φ STM a T. I STM /7

12 9. Poicaé Map Goig back o h gal soluio o a olia (i ca also b lia ssm: ( ( ( τ τ τ Now h soboscopic Poica map is ohig mo ha: T T T T FP ( ( τ FP τ τ FP ( τ FP τ τ ( T ( ( τ τ τ P( ( ( τ τ Th i poi is τ Th sabili aalsis is simila o h pvious cas: P T ( FP ( ( τ τ I ( τ FP ( τ τ Hc h mooom mai is ohig mo ha h Jacobia o h Poicaé map! Uoual agai i h gal cas w hav o umicall solv h mai iial quaio: ( ( o [ T ] FP FP FP FP Summa:. Evalua h soluio a T im: ( T Th sabili ca b ou om To calcula his agai w hav o solv a mai iial quaio.. /7

13 Ovall Summa:.. I all cass w sa om assumig ha w hav h soluio. Epss h pubaios as: Δ Δ( 3. To i Φ STM solv h mai DE: Φ STM A Φ STM ΦSTM ( I. Th soluio o h abov quaio will giv Φ a hc w ca g h MM: Φ ( T mi h sabili as Δ ( kt STM k Δ 5. Fo h Floqu Tho w mus is iv pss ( ( Δ ( ( Δ Δ ( Δ. ( 6. Fo h Poica map i is acl h sam as bo bu ow w wi icl: ( T STM. Th igvalus o his will which is LTV a hc w h STM o ( T Δ. ( Δ Thus Th Floqu Tho s a a sp ha h Tajco Ssiivi Th Poicaé map is jus a spcial cas o h Tajco Ssiivi 3/7

14 Eampl: Poicaé Map vs Floqu Tho. Iiial umical aalsis A ssm is giv b Th EPs a: ( Fom h s q: ( ( A b placig ha a h q: ( ( ( ( which is icosis Hc h is ol o quilibium poi a h oigi. Th ssm has a sabl limi ccl as i ca b s b is umical simulaio To su h ssm w ca us pola cooias: cos( θ which will giv: a θ a si( θ /7

15 /7

16 . Pola Cooias Ls calcula h mol i pola cooias: ( ( ( ( ( ( cos( θ Also so Hc si( θ ( ( ( ( ( ( ( ( ( ( ( ( ( ( A θ a θ ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( 6/7

17 So h ssm i pola cooias is: ( Simulik valia his as wll. θ -u(u(*(-u(^-u(^ MATLAB Fucio s sq(u(^u(^ MATLAB Fc u(u(*(-u(^-u(^ Igao MATLAB Fucio MATLAB Fc MATLAB Fucio MATLAB Fc s Igao aa(u(/u( MATLAB Fucio MATLAB Fc3 ha Zo-O Hol. Equilibia a soluio o ssm i pola cooias Th quaio has o EP bu h is has a: ( 3 : Th Jacobia is ( 3. So h oigi is usabl a is : sabl. Th ODE i pola cooias ca b solv a w hav: θ θ θ So w ca s ha h agl θ will coiuousl icas whil h ampliu will covg o.. Poica map a Jacobia om pola cooias Bu gaig h oigial ssm h cosi/si o h agl will b pioic o pio a as h ampliu will covg o a sa sa valu h his implis ha h ssm will hav a limi ccl which will b a pc cicl o aius c a h oigi a a pio o. 7/7

18 So i w sa a h h poi a o pio (Poica map is: P which ca b wi as: P To i h i poi: As h FP is h h Jacobia a ha poi is: P P A hc: < P a hc i is sabl. O ha simpl h pubaios aou h i poi a giv b: Δ Δ which has b cosschck usig Simulik. To o ha I simula h oigial ssm a a h sam ssm wih. a I sampl hi ic wih T. Th I simula h Δ Δ wih iiial coiio -.9. a I compa h pvious ic a ha oupu. Th small h IC h small hi ic: 8/7

19 3. Casia cooias 3. Soluio i Casia cooias Now l s i h soluio i h - pla: ( ( ( ( si cos si cos si cos si cos θ θ θ 9/7

20 This has b valia b Simulik. I.. I simula h pvious s o quaios ha giv aalical soluio a compa ha wih h umical ha I g om Simulik. Th o was -. I I o o assum ha θ : a si a cos a si cos si cos θ θ θ θ θ θ 3. Jacobia i Casia cooias Now l s i h Poica map i h - omai: /7

21 cos a P si a Th Jacobia o ha is ah cumbsom so I will us Malab clc cla sms T ^^; a(/-; ba*p(*t; ba*p(-*t; c/sq(b; a/; haa(at; c*cos(h; c*si(h; Di(; Di(; Di(; Di(; Dsubs(D{T}{*pi}; Dsubs(D{T}{*pi}; Dsubs(D{T}{*pi}; Dsubs(D{T}{*pi}; DF[D D; D D] ig(df DF W g igvalus: Th s o is h sam as h o ha w go om h pola cooias a is om h ac ha h ssm is auoomous. Noic ha w coul hav calcula h Jacobia i w ha solv ( ( o [ T ] FP FP FP FP 3.3 Pubaio calculaio usig h Jacobia /7

22 >> DF*[.;.] as.-3 * u(u(*(-u(^-u(^ MATLAB Fucio MATLAB Fc s Igao u(u(*(-u(^-u(^ MATLAB Fucio s MATLAB Fc Igao Subac Scop -u(u(*(-u(^-u(^ MATLAB Fucio MATLAB Fc u(u(*(-u(^-u(^. s Igao Displa MATLAB Fucio MATLAB Fc3 s Igao3. No: I I i o kow h IC o h Poica map i.. is i poi I hav o us a Nwo Rapsho mho. /7

23 Floqu Tho. Jacobia aou h pioic obi So ow I hav a olia ssm wih a sabl pioic obi wih T6.8 a also I kow is soluio. Now I will o us h Floqu Tho. Th olia vco il is: ( a hc So h Jacobia aou h obi is: ( which will giv: 3 3 Now his mus b valua aou h obi: ( ( si cos saig o h poi as his is a poi o h ccl. ( ( ( ( p p si cos so: ( ( ( ( ( ( 3si cos si cos si cos si 3cos P O: 3/7

24 cos( cos si cos si cos( si( cos( si( si( cos( cos( si( cos( si( si( P P cos( si( si si P Fo o T6.8 Th lms o h Jacobia mai a: 5. Soluio o Mai Diial Equaio 5. Calculaio o Mooom mai I o o i h mooom mai I o o solv h MDE: Φ ( Φ Φ A Wih h IC I P ( Φ Th ollowig mol will calcula h mai A om o T. /7

25 si Clock Gai Tigoomic Fucio -K- Gai5 -K- Gai6 Pouc Mai Mulipl si Tigoomic Fucio u Mah Fucio - Gai -K- Gai s Igao Pouc Mai Mulipl DP To Wokspac cos Tigoomic Fucio u Mah Fucio - Gai -K- Gai3 s Igao DP To Wokspac >> DP[DP' DP'] DP Noic ha his sul is v clos o h o ha w go om h Poica Map. Now ha h mooom mai is ou w ca i h Floqu muliplis: >> ig(dp as Which a v clos o h os om h PM. Aoh wa o o ha is o umicall solv h oigial ssm a h o us h 3 umical soluio o wih 3 ( cos (. si I is impoa hough o mmb ha h ssm mus sa om. 5/7

26 s oigial Pola I MATLAB Fucio -K- [] Jacobia o Poica I Numical o aaliic MATLAB Fc MATLAB Fucio -K- Gai7 [] [] Pouc3 [] Mai M ulipl MATLAB Fc3 Gai8 [] MATLAB [] -K- [] Fucio MATLAB Fc Gai9 [] [] s Igao Pouc [] Mai M ulipl DPA To Wokspac MATLAB Fucio MATLAB Fc [] -K- Gai s DPB To Wokspac3 Igao3 Clock Gai si Tigoomic Fucio -K- [] Gai5 [] -K- Gai6 [] si Tigoomic Fucio u Mah Fucio - Gai [] -K- Gai [] [] [] [] s Igao Pouc [] Mai Mulipl Pouc [] Mai Mulipl DP To Wokspac cos Tigoomic Fucio u Mah Fucio - Gai [] -K- Gai3 s Igao DP To Wokspac >> ig(dpa as Hc w g h sam suls. 5. Pubaio calculaio usig h Mooom mai So I hav ou Φ ow I will us his mai o pov ha Δ( Φ Δ b kowig ha Δ [.. So o.t I hav go om h NL ssm: A om h pouc >> E[DP' DP']*[.;] E ] Ths a i a v goo agm. 6/7

27 7/7

Phase plane method is an important graphical methods to deal with problems related to a second-order autonomous system.

Phase plane method is an important graphical methods to deal with problems related to a second-order autonomous system. NCTU Dpam of Elcical ad Compu Egiig Sio Cous