Phase portraits of separable Hamiltonian systems

XX Cong eso de Ecuaciones Di e enciales y Aplicaciones X Cong eso de Ma em´ a ica Aplicada Se illa, 24-28 sep iemb e 2007 (pp. 1–8) Phase po ai s o sepa able Hamil onian sys ems A. Guillamon1, Ch. Pan azi2, 1Dep . de Ma em`a ica Aplicada I, Uni e si a Poli `ecnica de Ca alunya, D . Ma a´on n.44-50, 08028, Ba celona, Ca alonia, Spain,E-mail: [email p o ec ed]. 2Dep . de Ma em`a ica Aplicada I, Uni e si a Poli `ecnica de Ca alunya, A da. Diagonal 647 08028 Ba celona E-mails: [email p o ec ed]. Resumen We s udy some gene aliza ions o po en ial Hamil onian sys ems (H(x, y) = y2+ F(x)) wi h one deg ee o eedom. In pa icula , we a e in e es ed in Hamil onian sys ems wi h Hamil onian unc ions o ype H(x, y) = F(x) + G(y) a ising in applied mechanical p oblems. We p esen an algo i hm o plo he phase po ai (include he beha io a in ini y) o any Hamil onian sys em o ype H(x, y) = F(x)+G(y), whe e Fand Ga e a bi a y polynomials. We a e able o gi e he ull desc ip ion in he Poinca ´e disk acco ding o he g aphs o Fand G, ex ending he well-known me hod o he “ ini e”phase po ai o po en ial sys ems. 1. In oduc ion The algo i hm o plo he ini e phase po ai o po en ial sys ems is a well-known and classical example o quali a i e heo y o o dina y di e en ial equa ions. The clue o be able o opologically classi y his ype o Hamil onian sys ems is, ob iously, he simplici y o i s ene gy unc ion, H(x, y) = y2 2+F(x), wi h F∈C1(IR) (see Figu e 1). The ac ha H“depends”basically on F(x) allows o ela e he phase po ai o {x0=−Hy(x, y), y0=Hx(x, y)}wi h he g aph o he one- a iable unc ion F(x). This kind o educ ion was also explo ed in [2] o Hamil onian sys ems o ype H( , θ) = 2/2 + n+1 g(θ). The aim o his wo k is o ex end his ype o esul s o Hamil onian sys ems de- pending o mo e han one unc ion. We ha e chosen a na u al gene aliza ion o po en ial sys ems, hose wi h ene gy unc ion H(x, y) = F(x) + G(y), which co e s a wide ange 1 Ch. Pan azi, A. Guillamon F .... . . Figu a 1: Fini e ep esen a ion o he phase po ai o a po en ial sys ems; minima o F coincide wi h cen e poin s while maxima coincide wi h saddles. o conse a i e physical models. We p o ide an algo i hm o ob ain he global (along he pape , “global”means ha includes he beha iou a in ini y) phase po ai o polynomial Hamil onian sys ems o he o m x0=−G0(y), y0=F0(x),(1) whe e Fand Ga e a bi a y polynomials. We will e e o his ec o ield as XH. Ou desc ip ion includes he classical classi ica ion o ini e c i ical poin s o smoo h po en ial sys ems. To ix no a ion, we w i e: F0(x) = anxn+· · · +a1x, an6= 0; G0(y) = bmym+· · · +b1y, bm6= 0.(2) In he nex wo sec ions we p o ide he esul s ha suppo he algo i hm. Fo he sake o space, his documen only p esen s he main s eam o apply he algo i hm, bu does no co e all he p oo s and de ails. 2. Bounded dynamics Nex wo esul s a e use ul o de e mine he beha iou o bounded o bi s o he sys em (1). P oposi ion 1 (Fini e c i ical poin s) Le P= (x0, y0)be a ini e singula poin o sys em (1). Then, (a) Pis a saddle i and only i Fhas a maximum ( esp., a minimum) a x0and Ghas a minimum ( esp., a maximum) a y0. 2 Phase po ai s o sepa able Hamil onian sys ems (b) Pis a cen e poin i ei he Fhas a maximum a x0and Ghas a maximum a y0 o Fhas a minimum a x0and Ghas a minimum a y0. (c1) Pis a cusp poin i ei he Fhas an in lec ion poin a x0o Ghas an in lec ion poin a y0. (c2) I Fhas an in lec ion poin a x0and Ghas an in lec ion poin a y0, hen Pis a c i ical poin o med by he union o wo hype bolic sec o s. Saddle poin s will be deno ed by S; cen e s a ising om wo maxima will be deno ed by C−, while hose coming om wo minima will be deno ed by C+. All o he ypes o ini e c i ical poin s will be deno ed by D. The bounded sepa a ices o a saddle o ganize hemsel es acco ding o he ules o he ollowing p oposi ion. P oposi ion 2 (Bounded sepa a ices) Any cen e o ype C+is emb aced by he p oximal sepa a ices o he neighbou ing saddle S∗which ene gy le el sa is ies H(S∗) = m´ın{S:H(S)> H(C+)}. Any cen e o ype C−is emb aced by he p oximal sepa a ices o he neighbou ing saddle S∗which ene gy le el sa is ies H(S∗) = m´ax{S:H(S)< H(C+)}. I he ou sepa a ices o a saddle Semb ace he pe iod annuli o a se o cen e s C:= {C1, . . . , Cn}, hen C ∪ S∪Wu,s(S)can be ea ed as a new cen e wi h he same ene gy le el han S. 3. Unbounded dynamics In o de o s udy he beha iou o sys em (1) nea in ini y we use Poinca ´e compac- i ica ion (see o ins ance [3]). I we call {X, Y, Z} he coo dina es in which he sphe e is exp essed, he equa o lies on Z= 0. Along he pape we a e going o use bo h cha s (U1, F1) ({X > 0}) and (U2, F2) ({Y > 0}) o he ec o ield (1) ex ended o he sphe e. Addi ionally, we deno e by (V1, G1) ({X < 0}) and (V2, G2) ({Y < 0}) he wo cha s on he opposi e side o (U1, F1) and (U2, F2) espec i ely. Depending on he sign and pa i y o n−mwe will use ei he cha , as i can be app ecia ed om he ollowing cases: Case 1: n > m. In he (U1, F1) cha he ec o ield akes he o m      z1 0=zn 2³z1G0(z1 z2) + F0(1 z2)´ =an+an−1z2+· · · +a1zn−1 2+bmzm+1 1zn−m 2+···+b1z2 1zn−1 2, z2 0=zn+1 2G0(z1 z2) = zn−m+1 2(bmzm 1+bm−1zm−1 1z2+· · · +b1z1zm−1 2). (3) 3 Ch. Pan azi, A. Guillamon We no e ha he e a e no c i ical poin s on {z2= 0}since an6= 0. In he (U2, F2) cha he ec o ield can be w i en as      z0 1=−zn 2³G0(1 z2) + z1F0(z1 z2)´ =−bmzn−m 2− · · · − b1zn−1 2−anzn+1 1− · · · − a1z2 1zn−1 2, z0 2=−zn+1 2F0(z1 z2) = −anzn 1z2− · · · − a1z1zn 2, (4) which has a unique c i ical poin , z1=z2= 0, on {z2= 0}. Case 2: n=m On he (U1, F1) cha , he ec o ield becomes ½z0 1= (an+bnzn+1 1)+(an−1+bn−1zn 1)z2+· · · + (a1+b1z2 1)zn−1 2, z0 2=z2(bnzn 1+· · · +b1z1).(5) We no e ha c i ical poin s a in ini y a e gi en by zn+1 1=−an bn. On he (U2, F2) cha he ec o ield is w i en ½z1 0=−bn−bn−1z2− · · · − b1zn−1 2−anzn+1 1−an−1zn 1z2− · · · − a1z2 1zn−1 2, z0 2=−z2(anzn 1+···+a1z1zn−1 2),(6) and he c i ical poin s a in ini y a e gi en by zn+1 1=−bn an. When nis odd and bn an<0, bo h (5) and (6) ha e 4 c i ical poin s a in ini y, which coincide; i bn an>0, he e a e no c i ical poin s a in ini y. When nis e en, bo h (5) and (6) ha e 2 c i ical poin s a in ini y, which coincide. Case 3: n < m. A e an app op ia e change o a iables, his case is he equi alen o Case 1. Wi hou loss o gene ali y, hen, we will omi i s s udy and concen a e on sys em (6), de ined on cha (U2, F2). One o he c ucial s eps o ob ain he global phase po ai is o de e mine which sepa a ices end (bo h o posi i e and nega i e imes) o c i ical poin s a in ini y. Fo his pu pose, we need o dis inguish special maxima and minima o unc ions Fand G, which apply o he case ha S= (x∗, y∗) is a saddle: De ini ion 1 Gi en a c i ical poin S= (x∗, y∗)o sys em (1), we say ha : x∗sa is ies he p ope y l( esp., L) i F(x∗)is a minimum ( esp., maximum) o F and F(x∗)< F(x)( esp., F(x∗)> F(x)) o each x < x∗. x∗sa is ies he p ope y ( esp., R) i F(x∗)is a minimum ( esp., maximum) o F and F(x∗)< F(x)( esp., F(x∗)> F(x)) o each x > x∗. y∗sa is ies he p ope y d( esp., D) i G(y∗)is a minimum ( esp., maximum) o G and G(y∗)< G(y)( esp., G(y∗)> G(y)) o each y < y∗. y∗sa is ies he p ope y u( esp., U) i G(y∗)is a minimum ( esp., maximum) o G and G(y∗)< G(y)( esp., G(y∗)> G(y)) o each y > y∗. Addi ionally, we de ine he ollowing p ope ies: 4 Phase po ai s o sepa able Hamil onian sys ems The saddle S∗= (x∗, y∗)sa is ies p ope y (h1) i x∗sa is ies bo h he p ope ies and l. Analogously, we will say ha i sa is ies p ope y (H1) i x∗sa is ies bo h he p ope ies Rand L. The saddle S∗= (x∗, y∗)sa is ies p ope y (h2u) ( esp., (H2U), (H2d) o (H2D)) i y∗sa is ies p ope y u( esp. U,do D) and i s maximal ( esp., maximal, minimal o minimal) wi h espec o his p ope y. Finally, we say ha (x( ), y( )) is an uppe ( esp., lowe ) sepa a ix o he saddle Si he e exis s ² > 0such ha o |y( )−y∗|< ², we ha e y( )> y∗( esp., y( )< y∗). Nex esul (and he co olla ies no shown he e) gi es he beha io o he main sepa- a ices ha each o come om a c i ical poin a in ini y. P oposi ion 3 Le S= (x∗, y∗)be a ini e saddle poin . Then: I Ssa is ies he p ope ies (h1) and (H2U), o (H1) and (H2u), he uppe sepa a- ices o S o m an ellip ic sec o Ea +∞( ha is, he c i ical poin o he ec o ield on cha (U2, F2)on {z2= 0}). I Ssa is ies he p ope ies (h1) and (H2D), o (H1) and (H2d), he lowe sepa a- ices o S o m an ellip ic sec o Ea −∞ ( ha is, he c i ical poin o he ec o ield on cha (V2, G2)on {z2= 0}), see Figu e 4. Finally, o he in ini e c i ical poin s we can use Ha mann Theo em in case hey a e hype bolic. Howe e , since he deg ees n,mand he coe icien s anand bmgi e all he in o ma ion abou he index o he ec o ield on R2, we can deduce he index o he in ini e c i ical poin s om Poinca ´e-Hop Theo em. Then, wi h he help o P oposi ion 3 and addi ional easonings, we can gi e he opological classi ica ion o all in ini e c i ical poin s. Mo e p ecisely he in o ma ion abou he ini e index allows o es ablish he ollowing esul : P oposi ion 4 Suppose i s ha n−mis odd and le q∞a c i ical poin on he equa o o he Poinca ´e sphe e. Then, I q∞does no ha e ellip ic sec o s, q∞is a node. I q∞has one ellip ic sec o s, q∞is he union o his ellip ic sec o and a hype bolic one, each o hem lying on a di e en hemisphe e o he Poinca ´e sphe e (see Figu e 2)(a). Suppose now ha n−mis e en. Then: I iR2=−1,q∞is he union o wo ellip ic sec o s (see Figu e 2)(b). I iR2= 1,q∞is he union o wo hype bolic sec o s (see Figu e 2)(c). 5 Ch. Pan azi, A. Guillamon . . Ellip ic and Two ellip ic hype bolic hype bolic sec o s sec o s sec o s Figu a 2: Di e en ypes o c i ical poin s on he equa o o he Poinca ´e sphe e. 4. Algo i hm Combining p oposi ions 1, 2, 3 and 4 one can desc ibe he comple e phase po ai on he Poinca ´e sphe e. In Figu es 3 and 4 we gi e a couple o examples, explaining he whole p ocedu e o build up he global phase po ai . Ag adecimien os Pa ially suppo ed by he DGES g an numbe MTM2005-06098-C02-1, he CONA- CIT g an numbe 2005SGR-986 and he MCyT/FEDER g an MTM2006-00478. Re e encias [1] S. Dilibe o,On sys ems o o dina y di e en ial equa ions, Con ibu ions o he Theo y o Nonlinea Oscilla o s 1(1950), 1–38. [2] A. Gasull, A. Guillamon, V. Ma˜ nosa Phase po ai o Hamil onian sys ems wi h homogeneous nonlinea i ies, Nonlinea Analysis: Theo y, Me hods & Applica ions 42 (2000), 679–707. [3] J. So omayo ,Li¸coes de equa¸co´es di e enciais o din´a ias, IMPA, Rio de Jane io, 1979. 6 Phase po ai s o sepa able Hamil onian sys ems F’=(x-6)(x-1)x(x-3)(x+1) 2 G’=y(y-9)(y-4) .. .. . . . . . . . . C C S S S S C C SC CS + + + + - - 461 10 7 12 9 11 8 3 5 2 .. . . . . . . . . . . .. . . .. . . S8 S9 S5 S6 S3 + S10 - . . . . S8S9 S5S6 . . Figu a 3: The saddles S10 and S3become S− 10 and S+ 3 espec i ely, see P oposi ion 2. The e is a connec ion be ween each o he saddles S9,S8,S6,S5and a c i ical poin a in ini y. C i ical poin s a in ini y a e nodes, see P oposi ion 4. 7 Ch. Pan azi, A. Guillamon F‘=(x-4)(x-3)(x+5) G’=(y-1)(y+2) ... ... CSC SCS ++ - 143 265 . . . .. CS C ++ ... SC S - Figu a 4: The saddle S2sa is ies he p ope ies (h1) and (H2D). F om P oposi ion 3, he lowe sepa a ices o he saddle S2 o m an ellip ic sec o a −∞ and so S2becomes S+ 2. Then, he saddle S4emb aces he “closed he e oclinic loop”S+ 2and so S4becomes S+ 4. A he end, he saddle S5emb aces S+ 4. Addi ionally, he singula poin a in ini y is a union o ellip ic and hype bolic sec o s, see also Figu e 2(a). 8