Abelian integrals of quadratic hamiltonian vector fields with an invariant straight line

Publicacions Ma em`a iques, Vol 39 (1995), 355–366. ABELIAN INTEGRALS OF QUADRATIC HAMILTONIAN VECTOR FIELDS WITH AN INVARIANT STRAIGHT LINE* Chengzhi Li, Jaume Llib e and Zhi en Zhang Abs ac We p o e ha he lowes uppe bound o he numbe o isola ed ze os o he Abelian in eg als associa ed o quad a ic Hamil onian ec o fields ha ing a cen e and an in a ian s aigh line a e quad a ic pe u ba ions is one. 1. In oduc ion Le H(x, y) be a eal polynomial o deg ee n+ 1, and le P(x, y) and Q(x, y) be eal polynomials o deg ee a mos m. The p oblem o finding an uppe bound N(n, m) o he numbe o isola ed ze os o he Abelian in eg als (1.1) I(h)=Γh Q(x, y)dx −P(x, y)dy, whe e Γh a ies in he compac componen s o H−1(h) is called he weakened 16 h Hilbe p oblem. I was posed by A nold in [1]. The weakened 16 h Hilbe p oblem is closely ela ed o he p oblem o de e mina ing an uppe bound o he numbe o limi cycles o he pe u bed Hamil onian sys em (1.2)ε dx d =∂H ∂y +εP(x, y), dy d =−∂H ∂x +εQ(x, y), *The au ho s a e pa ially suppo ed by DGICYT g an s, he fi s and hi d au- ho s a e also pa ically suppo ed by NSF o China, he second one is also pa ially suppo ed by a CIRIT g an . 356 C. Li, J. Llib e, Z. Zhang whe e 0 <|ε|<< 1.The ela ionship be ween bo h p oblems comes om he ollowing wo ac s: (1) I I(h∗)=0andI(h∗)= 0, hen he e exis s a hype bolic limi cycle Lh∗o sys em (1.2)εsuch ha Lh∗→Γh∗as ε→0; and con e sely, i he e exis s a hype bolic limi cycle Lh∗o sys em (1.2)εsuch ha Lh∗→Γh∗as ε→0, hen I(h∗)=0. (2) The o al numbe o isola ed ze os o (1.1) ( aking in o accoun hei mul iplici y) is an uppe bound o he numbe o limi cycles o sys em (1.2)ε ending o some pe iodic o bi Γho sys em (1.2)ε=0 when ε→0. Kho ansky [16] and Va chenko [24] p o ed independen ly ha N(n, m) is fini e, bu an explici exp ession o N(n, m) is unknown. Many au ho s ha e con ibu ed o es ima e he numbe N(n, m) o some alues o nand m, and o some classes o polynomial unc ions H(x, y), see o ins ance Bogdano [3] and [4], Pe o [20] and [21], Cushman and Sande s [10], Dumo ie , Roussa ie and So omayo [13], D achman, an Gils and Zhang [12], Li and Rousseau [18], Ga ilo and Ho ozo [14], Li, Llib e and Zhang [17], ... Aquad a ic Hamil onian ec o field is a ec o field o he o m (∂H/∂y,−∂H/∂x) whe e H=H(x, y) is a eal polynomial o deg ee 3. Ou main esul is o show ha N(2,2) = 1 o he quad a ic pe u - ba ions o he quad a ic Hamil onian ec o fields ha ing an in a ian s aigh line unde he flow defined by he ec o field. This esul is p o ed in Sec ion 2. In Sec ion 3 we cha ac e ize he phase po ai s o he quad a ic Hamil onian ec o fields ha ing an in a ian s aigh line, we will see ha he e a e six o such phase po ai s. We wan o hank o A mengol Gasull his commen s on a p elimina y e sion o his pape . 2. S a emen o he main esul s Le (∂H/∂y,−∂H/∂x) be a quad a ic Hamil onian ec o field, as usual we say ha dx dy =∂H ∂y ,dy d =−∂H ∂x , Abelian in eg als o quad a ic hamil onian . . 357 is i s associa ed quad a ic Hamil onian sys em, and ice e sa. Fo ab- b e ia ion we deno e by QH he class o quad a ic Hamil onian sys ems ha ing a cen e and an in a ian s aigh line. Lemma 1. E e y sys em in QH can be educed o ano he sys em in QH o he o m (2.1) dx d =∂H ∂y =2xy, dy d =−∂H ∂x =−y2+ (x), whe e (x)is a polynomial o deg ee 2. P oo : Le (2.2) dx d =∂H1 ∂y ,dy d =−∂H1 ∂x , be a sys em in QH. By a ansla ion we can pu he o igin o coo dina es on he in a ian s aigh line o sys em (2.2). Then by a o a ion we can ans o m he in a ian s aigh line o sys em (2.2) in he y-axis. Since bo h ans o ma ions a e canonical, hey p ese e he Hamil onian s uc u e o he sys em, and clea ly hey do no inc ease he deg ee o he Hamil onian unc ion. Thus sys em (2.2) can be w i en in he o m (2.3) dx d =∂H2 ∂y =x(ax +by +c), dy d =−∂H2 ∂x . No ice ha b= 0; o he wise ˙x=x(ax+c), and consequen ly sys em (2.3) would no ha e pe iodic o bi s. Now conside he change o a iables (2.4) x=x, by =ax +by +c. Clea ly de ∂(x, y) ∂(x, y)=1, 358 C. Li, J. Llib e, Z. Zhang and he ans o ma ion (2.4) is canonical. Fo simplici y ew i ing (x, y) ins ead o (x, y) in sys em (2.3) a e doing he change o a iables (2.4), we ge (2.5) dx d =∂H3 ∂y =bxy, dy d =−∂H3 ∂x . Since sys em (2.5) belongs o QH, i ollows ha dx d =∂H3 ∂y =bxy, dy d =−∂H3 ∂x =−b 2y2+g(x), whe e g(x) is a polynomial o deg ee 2. Now escaling he a iable y om y o 2y/b, he lemma ollows. No ice ha sys em (2.1) is in a ian unde he symme y (x, y, )→ (x, −y,− ) and ha i has x= 0 as an in a ian s aigh line. Now we p esen wo p elimina y esul s on he Abelian in eg als I(h) (see Sec ion 1). These esul s will allow us o simpli y he quad a ic pe u ba ion o a sys em in QH. They a e well-known bu since we canno find any e e ence o hem, we p o e hem he e. P oposi ion 2. The ollowing equali y holds I(h)=ΓhQ(x, y)+∂P(x, y) ∂x dydx. P oo : F om S okes heo em we ge ΓhP(x, y)dy +∂P(x, y) ∂x dydx =ln (Γh)∂P(x, y) ∂x dx ∧dy +∂ ∂y ∂P(x, y) ∂x dydy ∧dx =ln (Γh)∂P(x, y) ∂x −∂P(x, y) ∂x dx ∧dy =0. Then om (1.1) he p opesi ion ollows. F om P oposi ion 2 i ollows immedia ely. Abelian in eg als o quad a ic hamil onian . . 359 Co olla y 3. The Abelian in eg al I(h)associa ed o he sys em dx d =∂H ∂y , dy d =−∂H ∂x +εQ(x, y)+∂P(x, y) ∂x dy, is iden ical o he Abelian in eg al associa ed o sys em (1.2). F om P oposi ion 2 and Co olla y 3 we ge easily he ollowing esul . Co olla y 4. Assume ha he pe u ba ion (P,Q)o sys em (1.2)ε=0 is o deg ee 2. Then he Abelian in eg al I(h)associa ed o sys em (1.2)ε is iden ical o he Abelian in eg al associa ed o he ollowing sys em dx d =∂H ∂y , dy d =−∂H ∂x +ε(µ1y+µ2xy +µ3y2). Now we will apply Co olla y 4 o a sys em in QH. Lemma 5. E e y sys em in QH a e a quad a ic pe u ba ion has he same Abelian in eg al ha he ollowing pe u bed sys em o QH: (2.6) dx d =∂H ∂y =2xy, dy d =−∂H ∂x =−y2+ (x)+µ1y+µ2xy, whe e (x)is a polynomial o deg ee 2. P oo : By Lemma 1 e e y sys em in QH can be ans o med in he o m (2.1) wi h Hamil onian unc ion (2.7) H(x, y)=x(y2+F(x)), whe e F(x) is a polynomial o deg ee 2 such ha minus he de i a i e o xF(x) is equal o (x). F om Co olla y 4 he Abelian in eg al o a Hamil onian sys em (2.1) wi h an a bi a y quad a ic pe u ba ion is he same ha he Abelian in eg al o he sys em: (2.8) dx d =2xy, dy d =−y2+ (x)+µ1y+µ2xy +µ3y2. 360 C. Li, J. Llib e, Z. Zhang Since o a pe iodic o bi Γho he Hamil onian sys em we ha e Γh∩ {x=0}=∅,i ollows om (2.7) ha Γh y2dx =Γhh x−F(x)dx =0. So he Abelian in eg al o sys em (2.8) is he same ha he Abelian in eg al o sys em (2.6). No ice ha sys em (2.6) only depends on wo pa ame e s. Now we s a e ou main esul . Theo em 6. We ha e ha N(2,2) = 1 o he quad a ic pe u ba ion o quad a ic Hamil onian ec o fields ha ing an in a ian s aigh line. P oo : We deno e by Ii(h)=Γh xiydx o i=0,1. Then, om Lemma 5, he Abelian in eg al o he quad a ic pe u ba ion o a sys em in QH is I(h)=µ1I0(h)+µ2I1(h), whe e h∈(h0,h 1),and h0and h1co espond o some equilib ium poin and some homoclinic o he e oclinic loop o he phase pa ai o he Hamil onian sys em dx d =∂H ∂y =2xy, dx d =−∂H ∂x =−y2+ (x), in he Poinca ´e disc. Since (2.6) is a quad a ic sys em ha ing an in a ian s aigh line, by he p ope ies o quad a ic sys ems (see [5], [6], [15], [25], [21], [8] and [9]), sys em (2.6) has a mos one limi cycle, and i i exis s hen i is hype bolic. So I(h) can ha e a mos one ze o. Hence, N(2,2) = 1. Abelian in eg als o quad a ic hamil onian . . 361 The numbe o isola ed ze os o he Abelian in eg al (1.1) associa ed o quad a ic pe u ba ions o a sys em in QH, and he numbe o limi cycles o he pe u bed sys em in gene al is no he same. I may ex- is limi cycles o he pe u bed sys em such ha when he quad a ic pe u ba ion goes o ze o hey end o he equilib ium poin ( he cen e o he sys em in QH) o o he homoclinic o he e oclinic loop o ming he bounda y o he cen e in he Poinca ´e disc. Bo h possibili ies has been confi med by Zoladek in [26] and [27]. Hence, in o de o ob ain all he limi cycles which can be bi u ca ed om a cen e i is necessa y o s udy simul aneously h ee cases. The fi s one is o con ol he num- be o limi cycles which can be bi u ca ed om he equilib ium poin , secondly hose which can be bi u ca ed om pe iodic o bi s o he cen e ( o ins ance, h ough Abelian in eg als), and finally hose which can be bi u ca ed om he homoclinic o he e oclinic loop a he bounda y o he cen e . Fo mo e de ails see [4], [19], [18] and [22]. 3. Classifica ion o he Sys em in QH We shall use he classifica ion o all he phase po ai s o quad a ic Hamil onian ec o fields gi en by A es and Llib e in [2] o cha ac e ize he phase po ai s o he sys ems in QH, i.e. he phase po ai s o he quad a ic Hamil onian ec o fields ha ing an in a ian s aigh line and a leas one cen e . By Lemma 1 e e y sys em in QH can be ans o med o a sys em (3.1) dx d =∂H ∂y ,dy d =−∂H ∂x , wi h H(x, y)=x(y2+ax2+bx +c). In o de o classi y he phase po a i s o sys em (3.1) in QH we dis inguish h ee main cases. Case 1. a=0.I b= 0 hen sys em (3.1) has no cen e . So we assume ha b= 0. A e escaling he a iable x he pa ame e bcan be educed o 1. The e o e H(x, y)=x(y2+x+c) = 0 has wo b anches, he s aigh line x= 0 and he pa abola y2+x+c=0. I c≥0 hen sys em (3.1) has no cen e s. So we assume c<0. Then he wo b anches o H(x, y)=0 in e sec a wo saddle poin s o sys em (3.1). The o he singula poin (−c/2,0) o sys em (3.1) is a cen e . Hence, om [2], sys em (3.1) has he phase po ai o ype Vulpe 5 (see Figu e 1). Case 2: a>0. A e escaling he a iable x he pa ame e acan be aken equal o 1. So H(x, y)=x[y2+(x+b/2)2+c−b2/4]. Subcase 1. c−b2/4<0 . Then H(x, y) = 0 has wo b anches, he s aigh line x= 0 and he ci cle cen e ed a he poin (−b/2,0) wi h 362 C. Li, J. Llib e, Z. Zhang adius equal o b2/4−c.I c<0 hen bo h b anches in e sec a wo saddles o sys em (3.1). The e o e, om [2], sys em (3.1) has he phase po ai o ype Valpe 3 (see Figu e 1). I c= 0 he s aigh line x=0 is angen o he ce cle y2+(x+b/2)2=b2/4, and om [2] sys em (3.1) has he pha e pa ai o ype Vulpe 2 (see Figu e 1). I c>0 he wo b anches o H(x, y) = 0 do no in e sec , hen om [2] sys em (3.1) has he phase po ai o ype Vulpe 2 (see Figu e 1). Subcase 2. c−b2/4≥0.Then H(x, y) = 0 has a unique b anch, he s aigh line x=0. I b2−3c≤0 hen sys em (3.1) has no cen e s. The e o e we assume b2−3c>0. Now sys em (3.1) has exac ly wo singula poin s o coo dina es ((−b±√b2−3c)/3,0), a cen e and a saddle. The e o e, om [2] he phase po ai o sys em (3.1) is o ype Vulpe 2 (see Figu e 1). Case 3. a<0. A e escaling he a iable x he pa ame e acan be aken equal o −1. So H(x, y)=x[y2−(x−b/2)2+c+b2/4]. Subcase 1. c+b2/4=0.Then H(x, y) = 0 has h ee b anches, he s aigh lines x=0,y =x−b/2 and y=x+b/2. Each wo o hem in e sec s a a saddle. The e o e, om [2], sys em (3.1) has he phase po ai o ype Vulpe 10 (see Figu e 1). Subcase 2. c+b2/4= 0. Then H(x, y) = 0 has h ee b anches, he s aigh line x= 0 and he wo b anches o he hype bola y2−(x−b/2)2= −(c+b2/4). I b2+3c≤0 hen sys em (3.1) has no cen e s. So we assume ha b2+3c>0. I c+b2/4<0 hen x= 0 in e sec s he wo b anches o he hype bola, a wo saddles. The e o e, om [2] i ollows ha sys em (3.1) has he phase po ai o ype Vulpe 9 (see Figu e 1). I c+b2/4>0 hen x= 0 in e sec s only one b anch o he hype bola, a wo saddles. Hence, om [2], sys em (3.1) has he phase po ai o ype Vulpe 8 (see Figu e 1). Thus we ha e ob ained six diffe en opological phase po ai s o he sys ems in QH. We ema k ha he unique o hese six phase po ai s which can be ealized o a quad a ic Hamil onian sys em wi hou ha ing an in a ian s aigh line is he phase po ai o Vulpe 2. This is due o he p ope y ha i a quad a ic sys em has an o bi wi h α-limi one saddle and ω-limi ano he saddle, hen his o bi is con ained in an in a ian s aigh line. This esul is due o Dong [11], see also Ye Yanqian and o he s [25]. An imp o emen o his esul is due o Chicone and Sha e , see Theo em 2.8 o [7]. We also ema k ha Vulpe 2 wi hou in a ian s aigh line is e- la ed wi h he Bogdano -Takens bi u ca ion es ic ed o quad a ic ec- o fields. Abelian in eg als o quad a ic hamil onian . . 363 a=0 Vulpe 5 c−b2/4<0, c<0 Vulpe 3 c−b2/4<0, c=0 Vulpe 2 c−b2/4<0, c>0 Vulpe 2 a>0 c+b2/4=0 Vulpe 10 c+b2/4<0 Vulpe 9 c+b2/4>0 Vulpe 8 a<0 Figu e 1: The six diffe en opological phase po ai s o he quad a ic Hamil onian ec o fields ha ing an in a ian s aigh line and a leas one cen e .