EUSTAJEIA DUNAMIKWN SUSTHMATWN 1 Eisagwg O skop c tou par ntoc kefala ou e nai na parousi sei th basik jewr a gia th mel th thc eust jeiac en c mh gra

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1 EUSTAJEIA DUNAMIKWN SUSTHMATWN Eisagwg O skop c tou par ntoc kefala ou e nai na parousi sei th basik jewr a gia th mel th thc eust jeiac en c mh grammiko sust matoc. 'Opwc e nai gnwst, h genik l sh en c mh grammiko sust matoc e nai exairetik pol plokh kai den e nai dunat na ekfrasje sunart sei twn gnwst n sunart sewn. Gia to l go aut, ja periorisjo me sth mel th thc sumperifor c en c mh grammiko sust matoc sthn perioq eidik n l sewn, pwc e nai ta shme a isorrop ac kai oi periodik c troqi c. Oi l seic aut c kajor zoun kat tr po kr simo thn topolog a tou q rou twn f sewn kai sunep c, par to ti oi l seic aut c e nai memonwm nec, h mel th touc d nei poll c qr simec plhrofor ec gia th makroqr nia sumperifor tou sust matoc. Shmei noume ak ma ti, se ant jesh me th genik l sh tou sust matoc, h e resh twn anwt rw eidik n l sewn e nai sqetik e kolh. O sun jhc tr poc thc mel thc en c mh grammiko sust matoc sthn perioq en c shme ou isorrop ac mi c periodik c troqi c, e nai na grammikopoi soume to s sthma sthn perioq aut n twn l sewn kai na melet soume to ant stoiqo grammik s sthma. Dedom nou ti h majhmatik jewr a twn grammik n susthm twn e nai gnwst, aut ja mac d sei thn pl rh l sh tou grammikopoihm nou sust matoc. Sthn paro sa mel th ja doje idia terh mfash sth met bash ap to grammikopoihm no sto arqik s sthma kai ja melet soume poi basik qarakthristik param noun anallo wta. H ex lixh en c dunamiko sust matoc mpore na parastaje gewmetrik wc mia suneq c ro sto q ro twn f sewn. Me aut ton

2 tr po mporo me na perigr youme tic basik c idi thtec tou sust matoc, pwc p.q. an e nai sunthrhtik mh sunthrhtik, me apl c gewmetrik c nnoiec oi opo ec e nai e kola katanoht c. Ep shc, h suneq c ro en c dunamiko sust matoc mpore na perigrafe, isod nama, wc mia di krith apeik nish se nan upoq ro tou q rou twn f sewn, me th m jodo thc epif neiac tom c Poincare. Aut e nai idia tera qr simo se sust mata me d o bajmo c eleujer ac, pou o arqik c q roc twn f sewn e nai tess rwn diast sewn, en h epif neia tom c Poincare e nai d o diast sewn. Oi periodik c troqi c, oi opo ec e nai kleist c kamp lec ston tetradi stato q ro twn f sewn, antiproswpe ontai ap shme a isorrop ac, kai e nai faner ti aut aplouste ei th mel th. Sta ep mena ja parousi soume th basik jewr a thc sumperifor c en c mh grammiko sust matoc sthn perioq en c shme ou isorrop ac kai ja d soume apl parade gmata gia k je tupik per ptwsh, gia thn kal terh katan hsh thc jewr ac. 2 Basiko orismo Ja d soume t ra meriko c basiko c orismo c: Aut nomo s sthma 'Ena aut nomo dunamik s sthma perigr fetai ap na s sthma diaforik n exis sewn thc morf c _x = f(x); () pou x 2 R n f : E R n, E anoikt upos nolo tou R n kai f 2 C (E). To dianusmatik ped o f dhmiourge th ro t : E R n ; (2) h opo a ekfr zetai ap th l sh tou sust matoc twn diaforik n exis sewn (), thn opo a ja parist nome wc t (x) = (t; x). H l sh aut epalhje ei ek taut thtoc to s sthma (): d dt (t; x)= t= = f((; x)): (3) Epeid to s sthma e nai aut nomo, oi arqik c sunj kec mporo n na jewrhjo n ti antistoiqo n sto qr no t = : x() = x 2 E kai h l sh 2

3 pou antistoiqe se aut c tic arqik c sunj kec e nai h (t; x ), t toia ste (; x ) = x. Sta ep mena, th l sh ja th sumbol zome wc x(t; x ) apl c wc x(t). Dunamik s sthma 'Ena dunamik s sthma sto E or zetai wc m a C apeik nish : R E E; (4) pou E anoikt upos nolo tou R n kai an t (x) = (t; x), t te h ro t epalhje ei tic sq seic: (i) (x) = x gia k je x 2 E (ii) t s (x) = t+s (x) gia k je s; t 2 R kai x 2 E. Shme a isorrop ac kai periodik c troqi c Ac jewr soume to s sthma twn diaforik n exis sewn _x = f(x); x 2 R n : (5) 'Ena shme o tou q rou twn f sewn x or zetai wc shme o isorrop ac kr simo shme o an h x(t) = x e nai l sh tou sust matoc, dhlad e n to x plhro thn algebrik ex swsh f(x ) =. M a l sh (t) = t (x ) or zetai wc periodik troqi k kloc e n epalhje ei th sq sh pou T e nai h per odoc. t+t (x ) = t (x ); (6) 3 Grammikopo hsh 3. Sthn perioq shme ou isorrop ac 'Estw shme o isorrop ac x. Mporo me, qwr c ap leia thc genik thtac, na upoj soume ti h arq twn ax nwn qei metateje sth j sh tou shme ou isorrop ac, tsi ste na e nai x =. M a diataragm nh l sh e nai t ra h x(t), pou x e nai h metat pish ap th j sh isorrop ac kai jewre tai mikr. H l sh aut epalhje ei to s sthma (5), _x = f() + Df()x + :::; x 2 R n ; (7) 3

4 pou anapt xame to dexi m loc se seir Taylor kai apale yame touc rouc deut rac kai anwt rac t xewc. Lamb nontac up yin ti f() =, qoume telik to grammik s sthma pou o p nakac A = Df() d detai ap th @x B : : n C Df() : : : : : : : : : : : : _x = A x; n x= : (9) H l sh auto tou sust matoc ja kajor sei th sumperifor tou sthn perioq tou shme ou isorrop ac, se grammik pros ggish. Kr simo r lo sthn poiotik sumperifor thc l sewc pa zoun oi idiotim c tou p naka A, ap t c opo ec exart tai h eust jeia tou sust matoc. Ap th jewr a twn grammik n diaforik n exis sewn e nai gnwst ti h l sh e nai peratwm nh m non tan lec oi idiotim c qoun mh jetik pragmatik m roc. 'Ena shme o isorrop ac or zetai wc uperbolik shme o tan den up rqei idiotim tou A me mhdenik pragmatik m roc. 3.2 Sthn perioq periodik c troqi c 'Estw h periodik troqi : x = (t), peri dou T. Mia geitonik troqi mpore na grafe up th morf x(t) = (t) + (t); () pou (t) e nai h metat pish thc diataragm nhc troqi c ap thn periodik troqi, gia thn dia qronik stigm. To grammikopoihm no s sthma sthn perioq thc periodik c troqi c d detai ap to s sthma pou o p nakac A(t) @x B : : n Df((t)) : : : : : : : : : : : _ = A(t) ; n C A x=(t) : (2) O p nakac A e nai nac periodik c p nakac me per odo T. H l sh tou sust matoc () eur sketai ap th jewr a Floquet kai kajor zei thn eust jeia thc periodik c troqi c, se grammik pross ggish. 4

5 4 Genik l sh grammiko sust matoc Jewro me to grammik s sthma _x = A(t) x; x 2 R n : (3) H genik l sh tou e nai grammik c sunduasm c n grammik anex rthtwn l sewn kai mpore na ekfrasje up th morf x(t) = (t) C; (4) x(t) = h (t)() i x ; (5) pou (t) e nai nac n n p nakac pou oi st lec tou e nai grammik anex rthtec l seic kai C e nai na n di nusma. E nai faner ti kai oi st lec tou p naka (t)() e nai ep shc grammik anex rthtec l seic. K je p nakac (t) pou oi st lec tou qoun aut thn idi thta onom zetai jemeli dhc p nakac l sewn. An e nai kai () = I, t te h l sh ekfr zetai sunart sei twn arqik n sunjhk n x() wc x(t) = (t) x(): (6) Sta ep mena ja melet soume ton jemeli dh p naka l sewn sto grammik s sthma me stajero c suntelest c _x = A x, A stajer c. 'Ena basik er thma pou ja exet soume e nai to p c sqet zetai h ro tou grammikopoihm nou sust matoc sthn perioq tou shme ou isorrop ac me th ro tou mh grammiko sust matoc. P te sump ptoun poiotik kai p te qi; 5 Eust jeia shme wn isorrop ac grammiko sust matoc Jewro me to mh grammik s sthma _x = f(x); x 2 R n kai upoj toume ti h arq x = e nai shme o isorrop ac, f() =. To grammikopoihm no s sthma sthn perioq tou shme ou isorrop ac e nai to _x = A x; (7) 5

6 Sq ma : To portr to f sewc tou sust matoc (9) pou A = Df() e nai nac stajer c n n p nakac. auto tou sust matoc e nai h H genik l sh x(t) = e ta x ; (8) pou e ta e nai o jemeli dhc p nakac l sewn. Wc portr to f sewc (phase portrait) or zoume to s nolo lwn twn l sewn sto R n. Par deigma: Jewro me to s sthma _x _x 2 = 2 x x 2 ; (9) h l sh tou opo ou e nai h x (t) = x e t, x 2 (t) = x 2 e 2t. To portr to f sewc d detai sto sq ma. 5. Genik l sh tou sust matoc _x = A x O p nakac A e nai diag nioc 'Estw ti A = : : n C A : (2) Orism c: e B = I + B + B2 2 + ::: + Bn n + ::: 6

7 Sthn per ptwsh aut up rqoun n grammik anex rthtec l seic thc morf c x i = e it x i ; i = ; 2; :::n: (2) O p nakac A den e nai diag nioc Oi idiotim c tou A e nai pragmatik c kai di kritec: Onom zoume U ; U 2 ; ::U n ta idiodian smata tou p naka A, ta opo a sqhmat zoun b sh ston R n. O p nakac P = U ; U 2 ; ::U n ] e nai antistr yimoc kai o metasqhmatism c P A P diagwniopoie ton A, P A P = diag ; 2 ; :: n ]: O metasqhmatism c y = P x metasqhmat zei to s sthma _x = A x sto s sthma _y = B y, pou o p nakac B e nai diag nioc, B = P A P me diag nia stoiqe a tic idiotim c tou p naka A. H l sh wc proc y e nai h y(t) = e t : : e nt C A y(); kai telik, h l sh wc proc thn arqik metablht x, e nai h x(t) = h i P (t) P x() pou (t) = diag h e t ; :::e nti. O p nakac P (t) P e nai o jemeli dhc p nakac l sewn e ta. Oi idiotim c tou A e nai migadik c kai di kritec: 'Estw A nac 2n 2n p nakac me 2n di kritec idiotim c j = j i j kai W j = U j iv j, j = ; 2; ::n, ta ant stoiqa idiodian smata. O 2n2n p nakac P = V U :::V n U n ] e nai antistr yimoc kai " # P j A P = diag j : j j H l sh tou sust matoc e nai h " x(t) = P diag e j t cos j t sin j t sin j t cos j t # P x(): O 2 2 p nakac sthn anwt rw sq sh parist nei strof kat j t rad. 7

8 Oi idiotim c tou A e nai pragmatik c kai migadik c, di kritec: Ac upoj soume ti o p nakac A qei k pragmatik c idiotim c ; 2 ; ::: k me ant stoiqa idiodian smata ta U ; U 2 ; :::U k kai migadik c idiotim c j = j i j ; j = k + ; k + 2; :::n me ant stoiqa idiodian smata ta U j iv j. O p nakac P = U ; U 2 ; :::U k ; V k+ ; U k+ ; :::V n ; U n ] diagwniopoie ton p naka A, P A P = diag ; ::: k ; B k+ ; B n ] ; pou B j = " j j j j # : H l sh ekfr zetai up th morf x(t) = h P (t) P i x(); (22) pou (t) = diag " e t ; :::e kt ; e j t cos j t sin j t sin j t cos j t # : par deigma 'Estw ti A = :4 : 6 : C A : Oi idiotim c tou e nai oi = :4; 2;3 = : 4i kai ta ant stoiqa idiodian smata e nai ta U = ; W 2;3 = U 2 iv 2 = i : O p nakac P e nai o P = 4 C A 8

9 ; ; ; Sq ma 2: To portr to f sewc tou sust matoc (23) kai h l sh e nai telik h x = e :4t x ; x 2 = e :t x 2 cos 4t 4 x 3 sin 4t; ] (23) x 3 = e :t 4 x 2 sin 4t + x 3 cos 4t : To portr to f sewc d detai sto Sq ma 2. Pollapl c idiotim c: 'Estw = 2 = ::: = m ; m < n mia pollapl pragmatik idiotim. E n sthn idiotim aut antistoiqo n m grammik c anex rthta idiodian smata, t te gia th genik l sh isq ei h prohgo menh jewr a, pou lec oi idiotim c e nai apl c. E n mwc sthn pollapl idiotim, pollapl thtac m, antistoiqo n lig tera tou m idiodian smata, t te isq oun ta ex c: Or zoume wc genikeum no idiodi nusma tou p naka A pou antistoiqe sthn pollapl idiotim, th mh mhdenik l sh U tou sust matoc (A I) k U = ; k m: Ep shc or zoume nan p naka N wc mhdenod namo (nilpotent) t xewc k e n N k 6= ; N k = Je rhma 'Estw n n p nakac A, me pragmatik c idiotim c ; 2 ; ::: n, merik c 9

10 twn opo wn e nai pollapl c. - Up rqei b sh U ; U 2 ; :::U n ] genikeum nwn idiodianusm twn sto R n. - O p nakac P = U ; U 2 ; :::U n ] e nai antistr yimoc. - Up rqei p nakac S t toioc ste A = S + N, pou P SP = diag ; 2 ; ::: n ] kai o p nakac N = A S e nai mhdenod namoc t xewc k n kai SN = NS. T te h l sh e nai h x(t) = P diag h i " e j t P I + Nt + ::: + N # k t k x : (24) k Eidik tera gia thn per ptwsh pou h idiotim e nai pollapl thtac n, " x(t) = e t I + Nt + ::: + N # k t k x : (25) k An oi idiotim c en c 2n 2n p naka A e nai migadik c kai pollapl c, oi j = j i j ; j = ; 2; ::n, t te isq oun ta ex c: - Up rqoun genikeum na migadik idiodian smata, ta W j = U j iv j kai o p nakac P = V ; U ; :::V n ; U n ] e nai antistr yimoc. - Up rqei p nakac S t toioc ste A = S + N, pou " # P j SP = diag j ; - o p nakac N +A S e nai mhdenod namoc t xewc k 2n kai SN = NS. T te h l sh e nai h " # " x(t) = P diag e j t cos j t sin j t P I + ::: + N # k t k : (26) sin j t cos j t k j j 6 Grammik sust mata sto R 2 -Portr ta f sewc Jewro me to grammik s sthma _x = B x; (27)

11 Sq ma 3: s gma (saddle) pou B e nai nac stajer c 2 2 p nakac kai x = e nai shme o isorrop ac. Je melet soume thn topolog a tou q rou twn f sewn, sthn perioq tou shme ou isorrop ac x =, gia di forec peript seic tou p naka B. SAGMA (saddle) B = ; < < ; (28) H l sh auto tou sust matoc e nai h B = et e t x ; (29) kai to portr to f sewc d detai sto sq ma 3: KOMBOS (node) per ptwsh A B = ; < : (3) H l sh d detai ap tic sq seic (29), kai ta portr ta f sewc sthn per ptwsh aut d dontai sta sq mata 4a kai 4b.

12 . Sq ma 4: K mboc (node). Gia > h for e nai ant strofh per ptwsh B B = H l sh d detai ap th sq sh B = x(t) = e t (I + N t) x = e t t ; < < : (3) ; < : (32) x : (33) To portr to f sewc d detai sto sq ma 4g. Sthn per ptwsh < o k mboc e nai eustaj c, en an e nai >, h for sta sq mata 4 e nai ant strofh kai o k mboc e nai astaj c. ESTIA (focus) H l sh e nai h B = x(t) = e t cos t sin t sin t cos t ; < : (34) x : (35) To portr to f sewc d detai sto sq ma 5a, gia > kai 5b gia <. Sthn per ptwsh aut ( < ), h est a e nai eustaj c. E n e nai >, h for sta sq mata 5 e nai ant strofh kai h est a e nai astaj c. 2

13 . Sq ma 5: Est a: (a) >, (b) <.. Sq ma 6: K ntro: (a) >, (b) <. KENTRO (center) kai h l sh e nai h x(t) = B = cos t sin t sin t cos t ; (36) x : (37) To portr to f sewc d detai sta sq mata 6a,b gia > kai <, ant stoiqa. Mh grammik s sthma 'Ena shme o isorrop ac mh grammiko sust matoc or zetai wc s gma, k mboc, est a k ntro an to portr to f se c tou sthn perioq tou shme ou isorrop ac x e nai isod namo, se grammik pros ggish, me m a ap tic parap nw peript seic, ant stoiqa. 3

14 'Ena shme o isorrop ac x mh grammiko sust matoc _x = f(x) onom zetai katab jra, (sink), an lec oi idiotim c tou p naka D f(x ) qoun arnhtik pragmatik m roc kai phg, (source), an qoun lec jetik pragmatik m roc. 7 Anallo wtoi upoq roi Jewro me to grammik s sthma _x = A x; x 2 R n ; (38) pou A n n stajer c p nakac. Ja ekfr soume th l sh auto tou sust matoc gia tic parak tw peript seic: Idiotim c tou A pragmatik c kai apl c 'Estw ti oi idiotim c tou A e nai oi ; 2 ; ::: n, oi opo ec e nai apl c. H l sh tou sust matoc (38) e nai h x(t) = P e t : e nt C A P x ; (39) pou P = U ; U 2 ; :::U n ] kai U i ta idiodian smata pou antistoiqo n stic idiotim c i. Anal oume t ra tic arqik c sunj kec x wc grammik sunduasm twn idiodianusm twn U i, x = P C = c U + ::: + c n U n ; (4) pou C e nai na n di nusma me stoiqe a c ; c 2 ; :::c n. H l sh, me th bo jeia twn exis sewn (39) kai (4), ekfr zetai up th morf x(t) = P e t : e nt C A C; x(t) = c U e t + ::: + c n U n e nt : (4) 4

15 Idiotim c tou A migadik c kai apl c 'Estw ti oi idiotim c tou p naka A e nai migadik c kai apl c, oi z j = j i j kai W j = U j iv j ta ant stoiqa idiodian smata, j = ; 2; :::n. H l sh e nai h " # x(t) = P diag e j t cos j t sin j t P x sin j t cos j t ; pou P = V U ; :::V n U n ] 2n 2n p nakac. Oi arqik c sunj kec x mporo n na ekfrasjo n wc grammik c sunduasm c twn dianusm twn V j, U j, x = P C = c V + c 2 U + ::: kai h l sh e nai h x(t) = P " diag e j t cos j t sin j t sin j t cos j t # C; kai telik x(t) = e t c (V cos t + U sin t) + c 2 ( V sin t + U cos t)]+:::: (42) Ap tic parap nw d o peript seic (pragmatik n kai migadik n idiotim n) prok ptei ti an oi arqik c sunj kec e nai kat m koc en c idiodian smatoc, t te diege retai m non o ant stoiqoc ekjetik c roc. Eidik tera: Pragmatik c idiotim c 'Estw ti e nai x = c U : H l sh e nai, s mfwna me ton t po (4), h x(t; cu ) = c U e t : (43) To s nolo aut n twn shme wn apotelo n mia euje a par llhlh proc to idiodi nusma U, h opo a apeikon zetai ston eaut thc me th l sh tou sust matoc, dhlad param nei anallo wth. O monodi statoc aut c q roc onom zetai anallo wtoc upoq roc. An <, onom zetai eustaj c anallo wtoc upoq roc kai sumbol zetai wc E s kai an > onom zetai astaj c anallo wtoc upoq roc kai sumbol zetai wc E u. 5

16 Migadik c idiotim c 'Estw ti e nai x = c V + c 2 U : H l sh tou sust matoc e nai, sumfwna me ton t po (42), h x(t) = e t (c V cos t + U sin t] + c 2 V sin t = U cos t]) : (44) Ap th l sh (44) prok ptei ti to s nolo lwn twn shme wn pou apotelo n th l sh aut eur skontai se na ep pedo to opo o or zetai ap ta dian smata V kai U. O disdi statoc aut c q roc e nai anallo wtoc wc proc th l sh kai onom zetai anallo wtoc upoq roc. E n e nai < onom zetai eustaj c kai sumbol zetai wc E s kai e n e nai > nom zetai astaj c kai sumbol zetai wc E u. E n stic anwt rw d o peript seic, pragmatik c migadik c idiotim c, e nai = =, o anallo wtoc upoq roc onom zetai kentrik c upoq roc kai sumbol zetai wc E c. An loga isq oun kai gia pollapl c idiotim c. H morf thc l shc e nai an logh me tic parap nw. H diafor e nai ti p ran tou ekjetiko pou pou emfan zetai se k je per ptwsh, kai o opo oc kajor zei to e doc thc eust jeiac, emfan zontai kai roi me dun meic tou t. Parade gmata anallo wtwn upoq rwn Ja parousi some di fora parade gmata anallo wtwn upoq rwn se dunamik sust mata pou or zontai ap to s sthma twn diaforik n exis sewn _x = Ax, pou A stajer c p nakac. Par deigma Jewro me ton p naka A, A = 4 o opo oc qei tic idiotim c = ; 2 = 4 kai ta ant stoiqa idiodian smata U = ; ]; U 2 = ; 4]. H l sh d detai ap tic sq seic x = x + y 4 4 e 4t ; y = y e 4t ; ; 6

17 kai parathro me ti stic arqik c sunj kec x = k; y =, pou k auja reto, antistoiqe h l sh x(t) = k; y(t) =. Sunep c, h gramm h par llhlh proc to di nusma ; ] param nei anallo wth sth ro e ta. To di nusma aut e nai to idiodi nusma U pou antistoiqe sthn idiotim =. O monodi statoc aut c q roc e nai o anallo wtoc kentrik c upoq roc E c. Ep shc, parathro me ti stic arqik c sunj kec x = k; y = 4k, antistoiqe h l sh x(t) = ke 4t ; y(t) = 4ke 4t kai sunep c h gramm h par llhlh proc to di nusma ; 4] param nei anallo wth sth ro e ta. To di nusma aut e nai to idiod nusma U 2 kai o anwt rw monodi statoc q roc e nai o anallo wtoc upoq roc E s. Par deigma 2 Jewro me ton p naka A, A = 2 Oi idiotim c tou e nai oi ;2 = i; 3 = 2 kai o jemeli dhc p nakac l sewn e nai o " # cos t sin t e ta B e t C sin t cos t A : e 2t Ap ta parap nw eur sketai ti qoume na monodi stato astaj upoq ro, ton E u = ; ; ] pou antistoiqe sthn pragmatik idiotim 3 = 2 kai na disdi stato eustaj anallo wto upoq ro, ton E s = span (; ; ); (; ; )], pou antistoiqe sth migadik idiotim ;2. Par deigma 3 Jewro me ton p naka A, A = Oi idiotim c tou e nai oi = ; 2 = ; 3 = kai ta ant stoiqa idiodian smata e nai ta U = ; ; ]; U 2 = ; ; ]; U 3 = ; ; ], op te C A : C A : 7

18 h l sh mpore na ekfrasje wc x(t) = c C A + c 2 C A + c 3 e t Parathro me ti oi anallo wtoi upoq roi e nai oi C A : E c = span (; ; ); (; ; )] ; E u = ; ; ]; E s = : Par deigma 4 Jewro me ton p naka A, A = 2 Oi idiotim c tou e nai oi ;2 = i; 3 = 2 kai ta ant stoiqa idiodian smata e nai ta W ;2 = ; ; ] i; ; ]; U 3 = ; ; ], kai h l sh mpore na ekfrasje wc x(t) = cos t sin t sin t cos t Oi anallo wtoi upoq roi e nai oi C B x x 2 C A : C A + e 2t x 3 C A : E c = span (; ; ); (; ; )] ; E s = ; E u = ; ; ]: 8 Ap to grammikopoihm no sto mh grammik s sthma Ja jewr soume to mh grammik s sthma _x = f(x); x 2 R n ; (45) kai upoj toume ti up rqei shme o isorrop ac x, op te kai f(x ) =. 'Opwc anaf rame kai prohgoum nwc, mporo me p ntote na upoj soume, 8

19 E u U S E s Sq ma 7: Oi anallo wtoi upoq roi kai oi ant stoiqec eustaje c kai astaje c topik c pollapl thtec qwr c ap leia thc genik thtac, ti x =. Ja xekin soume ap th mel th tou grammikopoihm nou sust matoc sto x, to pou A e nai nac n n stajer c p nakac. _x = Ax; A = Df(x ); (46) Ja exet soume pr ta thn per ptwsh pou oi idiotim c tou p naka A qoun lec mh mhdenik pragmatik m roc. Aut shma nei ti qoume m no eustaje c astaje c anallo wtouc upoq rouc E s ; E u sto grammik s sthma. Ja melet soume p c metasqhmat zontai oi q roi auto tan metaba noume ap to grammik sto mh grammik s sthma. Ja d soume pr ta meriko c basiko c orismo c: Topik eustaj c pollapl thta S S = fx 2 S= t (x) x kaj c t kai t (x) 2 S gia t g: (47) Topik astaj c pollapl thta U U = fx 2 U= t (x) x kaj c t kai t (x) 2 U gia t g: (48) Apodeikn etai ti sto mh grammik s sthma, sthn perioq tou shme ou isorrop ac x, up rqei mia eustaj c pollapl thta S kai mia astaj c pollapl thta U oi opo ec ef ptontai, sto x, stouc q rouc E s kai E u, ant stoiqa, kai qoun tic diec diast seic pwc ka ta E s, E u. Aut apeikon zetai sqhmatik sto sq ma 7. 9

20 Ta prohgo mena anaf rontai sthn topolog a tou q rou twn f sewn sthn perioq tou shme ou isorrop ac kai or same thn topik eustaj kai astaj pollapl thta. Ja exet some t ra p c epekte nontai oi nnoiec aut c se ol klhro to q ro. D nome touc parak tw orismo c: Eustaj c pollapl thta (global stable manifold) W s = t t (S): (49) Astaj c pollapl thta (global unstable manifold) W u = t t (U): (5) H W s eur sketai an af some ta shme a sthn S na kinhjo n p sw sto qr no kai h W u an af some ta shme a thc U na kinhjo n empr c sto qr no. Ja parousi some t ra ta basik shme a thc jewr ac me orism na parade gmata. par deigma Jewro me to s sthma twn diaforik n exis sewn _x = x _x 2 = x 2 + x 3 _x 3 = x 3 + x 2 ; sto opo o up rqei to shme o isorrop ac x =. To grammikopoihm no s sthma sthn perioq tou shme ou isorrop ac e nai to kai h l sh tou e nai h _x = x _x 2 = x 2 _x 3 = x 3 ; x = x e t ; x 2 = x 2 e t ; x 3 = x 3 e t : Ap th l sh aut prok ptei ti h eustaj c pollapl thta E s e nai to ep pedo x ; x 2 kai h astaj c pollapl thta E u e nai o xonac x 3. 2

21 E u U S E s Sq ma 8: H sq sh metax twn eustaj n kai astaj n upoq rwn E s ; E u kai twn eustaj n kai astaj n pollaplot twn S; U. H l sh tou mh grammiko sust matoc e nai h x = x e t ; x 2 = x 2 e t + x 2 e t e 2t ; x 3 = x 3 e t + x2 e t e 2t : 3 Ap th l sh aut prok ptei ti h eustaj c pollapl thta E s e nai h S = fx 2 R 3 = x 3 = 3 x2 g kai h astaj c pollapl thta e nai h U = fx 2 R 3 = x = x 2 = g: H l sh h opo a antistoiqe stic arqik c sunj kec thc eustajo c pollapl thtac e nai h x = x e t ; x 2 = x 2 e t + x 2 e t e 2t ; x 3 = x2 3 e 2t ; kai h l sh h opo a antistoiqe stic arqik c sunj kec thc astajo c pollapl thtac e nai h x = ; x 2 = ; x 3 = x 3 e t : 2

22 U E u E s S Sq ma 9: Oi q roi S, U, E s kai E u. Oi eustaje c kai astaje c pollapl thtec S; U prok ptoun ap tic l seic aut c kai parousi zontai sto sq ma 8, maz me touc eustaje c kai astaje c upoq rouc E s ; E u. par deigma 2 Jewro me to mh grammik s sthma _x = x x 2 2 ; _x 2 = x 2 + x 2 ; to opo o qei to shme o isorrop ac x = (; ). To grammik s sthma sthn perioq tou shme ou isorrop ac e nai to kai h l sh tou e nai h _x = x ; _x 2 = x 2 ; x = x e t ; x 2 = x 2 e t : E nai faner ti ti oi anallo wtoi upoq roi E s kai E u e nai oi xonec x kai x 2, ant stoiqa. Gia na bro me thn eustaj topik pollapl thta, eur skoume th l sh tou mh grammiko sust matoc proseggistik gia mikr x: Xekino me ap na shme o x = ; x 2 = ;, tou eustajo c 22

23 anallo wtou upoq rou E s, pol kont sthn arq, kai h l sh se grammik pros ggish e nai h H ep menh pros ggish e nai h x = e t ; x 2 = : x = e t + O( 4 ); x 2 = 3 2 e 2t + O( 4 ); op te h ex swsh thc kamp lhc pou parist nei thn eustaj pollapl thta e nai h S : x 2 = x2 3 + O(x5 ): Omo wc br sketai kai h astaj c topik pollapl thta U : x = x O(x5 2): Oi q roi S, U, E s kai E u gia to s sthma aut d dontai sto sq ma 9. Gia na bro me thn prosseggistik l sh ergaz maste wc ex c: Gr foume to mh grammik s sthma up th morf pou x = x x 2 _x = B x + G(x); ; B = ; G = x2 2 x 2 kai G(x) e nai o m grammik c roc. O jemeli dhc p nakac l sewn (t) tou grammiko m rouc _x = Bx e nai o (t) = e t e t kai gr fetai up th morf (t) = U(t) + V (t); pou U(t) = e t + V (t) = e t : 23

24 Parathro me ti isq oun oi sq seic _ = B; _U = BU; _ V = BV: H l sh u tou mh grammiko sust matoc plhro thn oloklhrwtik ex swsh u = U(t) + Z t Z U(t s)g(u(s))ds V (t s)g(u(s))ds; kai antistoiqe sth l sh me arqik c sunj kec pol kont stic x =, x 2 =, (apode xte to). H l sh thc oloklhrwtik c ex swshc mpore na breje proseggistik, me diadoqik c prosegg seic wc ex c: Upoj toume ti h l sh gr fetai up morf seir c, wc U = u + u + u 2 + ::: kat tic dun meic tou kai qrhsimopoio me tic anadromik c sq seic u n+ = U(t) + Z t gia n = ; ; 2; ::, xekin ntac ap u =. t Z U(t s)g(u n (s))ds V (t s)g(u v (s))ds; H eustaj c kai h astaj c pollapl thta W s kai W u, ant stoiqa, eur skontai ap thn arijmhtik olokl rwsh tou mh grammiko sust matoc kai d dontai sto sq ma. Wc arqik c sunj kec gia thn eustaj pollapl thta W s ja p roume na shme o ep nw ston eustaj anallo wto upoq ro E s, pol kont sto shme o isorrop ac x = (; ) kai ja oloklhr soume gia t. Omo wc, gia thn astaj pollapl thta W u ja p roume na shme o ep nw ston astaj anallo wto upoq ro E u kai ja oloklhr soume gia t. Ap th l sh tou grammiko sust matoc kai thn proseggistik l sh tou mh grammiko sust matoc br kame touc anallo wtouc upoq rouc sthn perioq tou shme ou isorrop ac, oi opo oi d dontai sto sq ma 9. Parathro me ti kat th met bash ap to mh grammik sto grammik s sthma, qoume diat rhsh tou eustajo c kai tou astajo c upoq rou. t 24

25 W u W s. Sq ma : (a) H eustaj c pollapl thta W s, (b) h astaj c pollapl thta W u, (g) O q roc twn f sewn, pou h eustaj c kai h astaj c pollapl thta t mnontai sto omoklinik shme o. Aut h idi thta, pou e dame se aut to sugkekrim no par deigma, isq ei genik, gia uperbolik shme a isorrop ac, pwc prok ptei ap to ep meno je rhma (to opo o parousi zoume qwr c ap deixh): Je rhma eustajo c pollapl thtac Jewro me to dunamik s sthma _x = f(x); x 2 R n, to opo o qei to shme o isorrop ac x = ; (f() = ). D dontai ta ex c: - E anoikt upos nolo tou R n pou peri qei to. - f 2 C (E). - t h ro tou dunamiko sust matoc. - Upoj toume ti o p nakac Df() tou grammiko sust matoc qei k idiotim c me arnhtik pragmatik m roc kai n k idiotim c me jetik pragmatik m roc. T te up rqei mia k-dim diafor simh pollapl thta S, efapt menh ston eustaj upoq ro E s sto shme o, tsi ste gia k je t, e nai t (S) 2 S, kai gia k je x 2 S e nai lim t t(x ) = : Ep shc up rqei mia (n k)-dim diafor simh pollapl thta U, efapt menh ston eustaj upoq ro E u sto shme o, tsi ste gia k je t, e nai t (U) 2 U, kai gia k je x 2 U e nai lim t t(x ) = : S mfwna me to je rhma aut, diathro ntai o eustaj c kai o astaj c upoq roc sthn perioq tou shme ou isorrop ac, an o 25

26 p nakac Df(x ) tou grammiko sust matoc den qei idiotim c me mhdenik pragmatik m roc. Ta prohgo mena anaf rontan sto p c metasqhmat zetai eustaj c kai o astaj c upoq roc, tan metaba nome ap to grammik sto mh grammik s sthma. Ja exet soume t ra pwc metasqhmat zetai to s nolo twn troqi n, sthn perioq tou shme ou isorrop ac, kat th met bash ap to grammik sto m gramik s sthma. Ja d soume pr ta nan orism : Orism c: D o aut noma sust mata diaforik n exis sewn, p.q. ta (): _x = f(x) kai (2): _x = Ax, x 2 R n e nai topologik isod nama sthn perioq thc arq c (dhlad qoun thn dia poiotik dom ) an up rqei omoiomorfism c H, pou apeikon zei na anoikt s nolo U pou peri qei thn arq, se na anoikt s nolo V pou peri qei thn arq, o opo oc apeikon zei tic troqi c tou () sto U se troqi c tou (2) sto V kai diathre ton prosanatolism. An ep pl on o H diathre thn parametropo hsh me to qr no, t te ta d o sust mata l gontai topologik suzhg. Wc par deigma anaf roume ta sust mata () : _x = Ax; A = 3 3 ; kai (2) : _y = By; B = 2 4 Parathro me ti oi p nakec A kai B sund ontai metax touc me th sq sh B = RAR, R = p 2 kai epom nwc o metasqhmatism c y = Rx metasqhmat zei to s sthma () sto s sthma (2). 'Ara, ta d o aut sust mata e nai topologik isod nama, kai o metasqhmatism c y = Rx matasqhmat zei to portr to f sewc tou en c sust matoc sto portr to f sewc tou llou, pwc fa netai sto sq ma. Gia to p c metab lletai h topolog a tou q rou twn f sewn kat th met bash ap to grammik sto mh grammik s sthma, isq ei to ak loujo je rhma: : 26

27 \ + 5 \ Sq ma : Ta portr ta f sewc twn susthm twn () kai (2) Je rhma Hartman-Grobman D dontai ta ex c: - E anoikt upos nolo tou R n pou peri qei thn arq O. - f 2 C (E). - t : h ro tou _x = f(x). - f() =, dhlad to shme o e nai shme o isorrop ac. - A = Df(): h or zousa tou grammiko sust matoc sthn perioq tou shme ou isorrop ac. - Den up rqei idiotim tou A me mhdenik pragmatik m roc. T te up rqei omoiomorfism c H en c anoikto sun lou U pou peri qei thn arq, se anoikt s nolo V pou peri qei thn arq, tsi ste gia k je x 2 U up rqei anoikt di sthma I 2 R pou peri qei thn arq, ste gi la ta x 2 U kai t 2 I, H t (x ) = e ta H(x ); dhlad apeikon zei tic troqi c tou _x = f(x) sthn perioq tou se troqi c tou grammiko sust matoc _x = Ax sthn perioq tou mhden c, kai diathre thn parametropo hsh. S mfwna me to je rhma aut, tan den up rqoun idiotim c tou grammiko sust matoc me mhdenik pragmatik m roc, h topolog a tou q rou twn f sewn diathre tai, kat th met bash ap to grammik sto mh grammik s sthma, kai sunep c diathre tai kai h eust jeia ast jeia tou shme ou isorrop ac. 27

28 : F ( F Sq ma 2: H kentrik pollapl thta den or zetai monos manta. 'Ola ta parap nw anaf rontai sthn per ptwsh pou oi idiotim c tou grammiko sust matoc qoun pragmatik m roc mh mhdenik. To ep meno je rhma kajor zei to ti g netai sthn per ptwsh idiotim n me mhdenik pragmatik m roc. Je rhma kentrik c pollapl thtac Upoj toume ti: - E anoikt upos nolo tou R n pou peri qei thn arq. -f 2 C r (E); r. - t : h ro tou _x = f(x). - f() =, dhlad to shme o e nai shme o isorrop ac. - A = Df(): h or zousa tou grammiko sust matoc sthn perioq tou shme ou isorrop ac. - O p nakac A qei k idiotim c me arnhtik pragmatik m roc, j idiotim c me jetik pragmatik m roc kai m = n j k idiotim c me mhdenik pragmatik m roc. T te up rqei mia m-dim kentrik pollal thta W c () t xewc C r, efapt menh ston kentrik upoq ro E c tou _x = Ax sto, h opo a param nei anallo wth sth ro t tou _x = f(x). Par deigma kentrik c pollapl thtac Jewro me to s sthma _x = x 2 ; _x 2 = x 2 ; 28

29 to opo o qei to shme o isorrop ac x = ; x 2 =. To grammik s sthma sthn perioq tou shme ou isorrop ac e nai to _ = ; _ 2 = 2 ; me idiotim c tic = ; 2 = kai idiodian smata ta U = ; ]; U 2 = ; ]. Epom nwc, o xonac x 2 e nai o eustaj c upoq roc E s kai o xonac x o kentrik c upoq roc E c. H l sh tou mh grammiko sust matoc e nai h x = x tx ; x 2 = x 2 e t ; kai me apaleif tou t pa rnome to portr to f sewc x 2 = x 2 e x e x : Gia x < oi kamp lec prosegg zoun thn arq me parag gouc pou te noun sto. H kentrik pollapl thta parousi zetai sto sq ma 2 kai parathro me ti den or zetai monos manta. 9 Eust jeia uperbolik n shme wn isorrop ac Me b sh la ta prohgo mena mporo me t ra na sunoy soume ta sumper smata gia thn eust jeia twn shme wn isorrop ac en c mh grammiko sust matoc. Orism c: 'Ena shme o x e nai eustaj c an gia k je > up rqei > t toio ste gia la ta x 2 N (x ) kai t na e nai t (x) 2 N (x ): To x e nai astaj c an den e nai eustaj c. 'Estw x shme o isorrop ac tou mh grammiko sust matoc _x = f(x); x 2 R; 29

30 op te e nai f(x ) =. Jewro me ton p naka A = Df(x ) tou grammikopoihm nou sust matoc sto shme o isorrop ac. An to shme o isorrop ac e nai uperbolik, dhlad kam a idiotim tou den e nai sh proc mhd n, t te: To shme o isorrop ac e nai asumptwtik eustaj c an kai m no an lec oi idiotim c tou p naka A qoun pragmatik m roc arnhtik, Re( j ) < ; j = ; 2; :::n: To shme o isorrop ac e nai astaj c an m a toul qiston idiotim tou A qei pragmatik m roc jetik, Re() > : Ta sumper smata aut e nai sun peia tou jewr matoc thc eustajo c pollapl thtac kai tou jewr matoc Hartman-Grobman. An to shme o isorrop ac x en c mh grammiko sust matoc e nai mh uperbolik, dhlad up rqei m a toul qiston idiotim tou p naka A me pragmatik m roc so proc mhd n, t te h eust jei tou mpore na melethje me th m jodo tou Liapunov. H m jodoc tou Liapunov Me th m jodo tou Liapunov mporo me na melet soume thn eust jeia en c shme ou isorrop ac en c mh grammiko sust matoc. H m jodoc sthr zetai sthn e resh mi c sunart sewc Liapunov V (x), pwc ja de xoume sta ep mena. 'Estw to mh grammik s sthma _x = f(x); x 2 C (E) kai shme o isorrop ac x, f(x ) =. 'Estw ak ma kai mia sun rthsh V (x) 2 C (E), (V (x ; x 2 ; :::x n )). H metabol thc sun rthshc V (x) kat m koc miac troqi c d netai ap th sq sh dv _x n _x n ; 3

31 suntom tera, _V (x) = DV (x) f(x): Ja parousi soume qwr c ap deixh to ep meno je rhma: Je rhma tou Liapunov :'Estw E anoikt s nolo tou R n pou peri qei to shme o isorrop ac x. 'Estw ak ma ti up rqei sun rthsh V 2 C (E) h opo a ikanopoie tic sq seic: T te V (x ) = kai V (x) > gia x 6= x : an _ V (x) gia k je x 2 E, to x e nai eustaj c. an _ V (x) < gia k je x 2 E fx g, to x e nai asumptwtik eustaj c. an _ V (x) > gia k je x 2 E, to x e nai astaj c. H sun rthsh V (x) me tic parap nw idi thtec onom zetai sun rthsh Liapunov. H sun rthsh Liapunov den br sketai, en g nei, e kola. An up rqei pr to olokl rwma thc k nhshc, e nai dunat n na qei tic idi thtec thc sun rthshc Liapunov, pwc ja do me sta parade gmata pou akoloujo n. Par deigma Jewro me to s sthma _x = x 3 2 ; _x 2 = x 3 ; to opo o qei to shme o isorrop ac x = (; ). H or zousa tou grammikopoihm nou sust matoc sto shme o isorrop ac e nai h Df() = ; (5) h opo a qei d o mhdenik c idiotim c, tic = kai 2 =. Parathro me ti h sun rthsh V (x) = x 4 + x 4 2 3

32 e nai sun rthsh Liapunov, di ti e nai _V (x) = 4x 3 _x + 4x 3 2 _x 2 = 4x 3 ( x 3 2) + 4x 3 2(x 3 ) = gia k je troqi (parathr sate ti h V (x) e nai na olokl rwma tou sust matoc). S mfwna me to je rhma tou Liapunov, to shme o isorrop ac x = (; ; ) e nai eustaj c shme o isorrop ac, all qi asumptwtik eustaj c. Pr gmati, oi l seic t (x) e nai oi kleist c kamp lec x 4 + x 4 2. Par deigma 2 Jewro me to s sthma _x = 2x 2 + x 2 x 3 ; _x 2 = x x x 3 ; _x 3 = x x 2 ; pou h arq x = (; ; ) e nai shme o isorrop ac. H or zousa tou grammikopoihm nou sust matoc e nai h h opo a qei tic idiotim c Df() = 2 C A ; (52) = ; 2;3 = 2i: (53) Parathro me ti to shme o isorrop ac x = (; ; ) e nai na mh uperbolik shme o isorrop ac. Gia thn e resh thc sun rthshc Liapunov pa rnoume th sun rthsh V (x) = c x 2 + c 2 x c 3 x 2 3 ; c ; c 2 ; c 3 > kai zhto me na upolog soume touc suntelest c c ; c 2 ; c 3 _V (x) >, V _ (x) < sto E. 'Eqoume ste na e nai An p roume 2 _ V (x) = (c c 2 + c 3 )x x 2 x 3 + ( 2c + c 2 )x x 2 : c 2 = 2c ; c 3 = c 2 ; 32

33 qoume _V (x) = kai sunep c h sun rthsh V (x) = x 2 + 2x x 2 3 e nai sun rthsh Liapunov. Gia th sun rthsh aut isq oun oi sq seic V (x) = gia x = ; V (x) > gia x 6= ; _V (x) = gia x 2 R 3 : (54) S mfwna me to je rhma tou Liapunov h arq (; ; ) e nai eustaj c shme o isorrop ac. Parathr ste ti lec oi troqi c ke ntai sto elleiyoeid c x 2 + 2x x 2 3 : Par deigma 3 Jewro me t ra to s sthma _x = 2x 2 + x 2 x 3 x 3 ; _x 2 = x x x 3 x 3 2 ; _x 3 = x x 2 x 3 3 : To s sthma aut qei thn arq x = (; ; ) wc shme o isorrop ac kai o p nakac A tou grammikopoihm nou sust matoc e nai o p nakac (52) tou prohgo menou parade gmatoc kai sunep c kai oi idiotim c tou e nai oi (54). To shme o aut e nai na mh uperbolik shme o3 isorrop ac kai wc sun rthsh Liapunov dokim zoume thn ant stoiqh sun rthsh tou parade gmatoc 2, V (x) = x 2 + 2x x 2 : 3 Parathro me ti e nai pr gmati sun rthsh Liapunov, di ti isq ei V (x) > kai _ V (x) < gia x 6= : Parathro me ti sthn per ptwsh aut h arq e nai na asumptwtik eustaj c shme o isorrop ac. 33

34 Ap ta parade gmata pou parousi same fa netai kajar ti tan to shme o isorrop ac e nai mh uperbolik, dhlad qei idiotim c m pragmatik m roc mhd n, h met bash ap to grammik sto mh grammik s sthma den e nai p ntote h dia. Aut shma nei ti h topolog a tou q rou twn f sewn, sthn perioq tou shme ou isorrop ac, metab lletai en g nei, kat th met bash ap to grammik sto mh grammik s sthma. Anafor c ] Guckenheimer, J. and Holmes, Ph., 983: Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Springer-Verlag. 2] Jordan, D.W. and Smith, P., 987: Nonlinear Ordinary Dierential Equations, Clarendon Press, Oxford. 3] Lichtenberg, A.J. and Lieberman, M.A., 983: Regular and Stochastic Motion, Springer-Verlag. 4] Mpo nth,a. 997: Mh Grammik c Sun jeic Diaforik c Exis seic, Ekd seic Pneumatiko, Aj na. 5] Perko, L., 996: Dierential Equations and Dynamical Systems, Springer-Verlag. 34