Existence of Higher Dimensional Invariant Tori for Hamiltonian Systems

Transcription

1 JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS, ARTICLE NO. AY Exstence of Hgher Densonal Invarant Tor for Haltonan Systes Cong Fuzhong and L Yong Departent of Matheatcs, Jln Unersty, Changchun 33, Chna Subtted by George Hagedorn Receved January 3, 997 Ths paper s on the persstence of hgher-densonal nvarant tor for Haltonan systes. A KAM type theore s establshed. 998 Acadec Press Key Words: KAM tor; Haltonan syste; degeneracy.. INTRODUCTION The celebrated KAM theory says that under a certan nondegeneracy condton, a nearly ntegrable Haltonan syste preserves a faly of nvarant tor. Recently ths theory has undergone soe consderable developents. However, the study of the exstence of hgher densonal nvarant tor Ž KAM tor. for Haltonan systes sees qute rare. Parasyuk 7 and Herann 3 ade nterestng observatons on such probles. In the present paper, otvated by ther work, we shall consder ore general stuatons. Now let us consder a Haltonan functon, HŽ x, y. NŽ y. PŽ x, y., Ž.. where N s a real analytc functon defned on soe closed bounded and connected regon G R l ; P s a real analytc functon defned on T G. Here l s even and l; T R Z denotes the torus of denson. Ž Let T G,. be a syplectc anfold, and I an analytc Haltonan hoeoorphs fro the -for space to vector felds, that s, let I be an antsyetrc atrx such that the relaton Ž, I. Ž., for Ž. all -for defned on T G see. Then the -for can be 55-7X98 $5. Copyrght 998 by Acadec Press All rghts of reproducton n any for reserved.

2 56 FUZHONG AND YONG deterned n the followng way: f, f df Ž Idf. Ž Idf, Idf., for all sooth functons f and f defned on T G, where, denotes the usual Posson bracket. Let be nvarant relatve to T. Thus the coeffcents of and the atrx I are ndependent of the coordnate x. Hence the Haltonan syste correspondng to Ž.. has the followng for : z IŽ y. grad T HŽ z., Ž.. where z Ž x, y., and the superscrpt T denotes the transpose of a vector. By Lea Ž see Appendx., we have that T T I y grad N y y,,...,. ž / l Throughout ths paper, we ake the followng hypotheses: ž / a l ½ y 5 rank r, on G, Ž.3. y rank, : Z, r, on G, Ž.. where Z stands for the set of nonnegatve ntegers, and y Ž y,...,. y T. Set G y:reyg,iy, x:rext,ix. In what follows, for a vector, denotes ts axu nor n coponents; for a functon, denotes the usual supreu nor; ², : denotes the usual nner product n the correspondng Eucldean spaces. Now we are n the poston to state our an result: THEOREM A. Assue that HŽ x, y. s a real analytc functon on G, and NŽ y. satsfes the condtons Ž.3. and Ž.. on G. Then there exst and a nonepty Cantor set G G such that, wheneer P, Ž.. preseres a faly of narant tor IT y, y G, whose frequency Ž y. satsfes 3 c, and the followng easure estate holds: es G G c Ž r., l

3 INVARIANT TORI FOR HAMILTONIAN SYSTEMS 57 where c s a constant ndependent of and, as. depends only on, and Reark. In 7, a nondegeneracy condton on Ž y. s needed, and that condton does not see obvous. But n our result, the condtons Ž.3. and Ž.. are obvous, and as r, the unperturbed syste possesses a certan degeneracy. Reark. If l r, and ž / E IŽ y. Ž E s the unt atrx of order., E then Theore A s the classcal KAM theore. Reark 3. The dscusson of the case l s slar. The only dfference s the easure estate of nvarant tor, whch can be copleted n a certan way Žsee... A SMALL DIVISOR PROBLEM In ths secton we shall gve an auxlary result that deals wth a sall dvsor proble. Throughout the paper we also use the followng notatons. Set y G. Wrte Thus f DŽ, s. Ž x, y.:rext,ix, yy s, PŽ y. H PŽ x, y. dx. T PŽ x, y. Ý ' ² k, x: Pe k, kz then P P. We shall use c s to denote soe postve constants dependng only on the constants,,, M,, l, and, where,,, and M wll be gven n the sequel. Let F and N be real analytc functons defned on DŽ, s., and whch satsfes IŽ y. grad T NŽ y. Ž,., ² : r k, k, kz, where r s constant,, and k k k.

4 58 FUZHONG AND YONG Set ² : N grad T NŽ y., yy. We have the followng. PROPOSITION. The equaton has a unque real analytc soluton, and, N FF, Ž.. c DŽ, s. r DŽ, s. F. Let Proof. Wrte D DŽ, s.. Clearly, ² T, N dž IdN., grad :. Puttng. nto. yelds Ý ' ² k, x: k kz F F y e.. Fk Ž y. ' ² k, x: Ý e. ² k, : kz By Cauchy s estate we have FkŽ y. DŽ, s. Ý e ² k, : kz Ý kz k k e F D kž. j F see Ý j e j D c Ý j j F D Ž s the nteger part of. c F DŽ, s..

5 INVARIANT TORI FOR HAMILTONIAN SYSTEMS 59 Fx y G, and wrte 3. PROOF OF THEOREM A ; N T N NŽ y. ² grad NŽ y., yy: Ž y.ž yy., yy! y t N N N, ˆ Ž 3.. where y y tž yy.,t. t Reark. In general, we ay take N N N y Nˆ Ž x, y., where NˆŽ x, y. ² QŽ x, y.ž yy., y y :. By Ž.. t s easy to see that x, y : xt, yy s an nvarant torus of denson of Haltonan N. T Snce grad N y and N y are bounded on G, we ay assue that on DŽ, s., N s, Nˆ Ž s., ss, Ž 3.. where and are constants dependng on G.. Outlne of the proof and estate of the easure Choose convergent sequences: 98, 8Ž 3., s 8, 6,,,,..., where constant satsfes condtons Ž A. Ž H. lsted below, and. Set 7 j Dj D j, s 8 ž / n 7j ½ ž 8 / 5 x, y :RexT,Ixj, y s, j6. Assue that the Haltonan functon N P on D s, by a syplectc transforaton, reduced nto N P, where N N N N ˆ, Ž 3.3. N Ž s., Nˆ Ž s., ss Ž 3.. DŽ, s. DŽ, s. P. Ž 3.5.

6 6 FUZHONG AND YONG Set r ² : O yg: k, y k,kz. Take y O. Then IŽ y. grad T N Ž y. Ž y.,. Introduce a syplectc transforaton such that Ž N P. N P. We are gong to prove that N P satsfes the correspondng expressons Ž 3.3. Ž If k k y O, then the teraton processes can contnue. Set N N R Ž N P Ž y.. NŽ y. P Ž y., yy ž y ;/ Ž Nˆ Ž x, y. Rˆ Ž x, y.. N N N ˆ, Ž 3.6. where R s deterned by the followng Ž 3... Wthout loss of generalty, we assue I G, Ž 3.7. where Iz IG ax sup, z Ž x, y.. z Fro 3.5 yg z and Cauchy s estate, we derve By , we see that as c T 3 3 D s grad P. Ž ½ / c 3 5 n,, Ž A. ž

7 INVARIANT TORI FOR HAMILTONIAN SYSTEMS 6 the followng nequalty holds on D : where, and Ý k k k D k T k k Ý G k I grad P D 3 c3 Ý k, k s gven n the Appendx. Hence T I grad N M, Ž 3.. G G T where M ax I grad N, c, c, and c and c wll be deterned below. Set G G 5 5 G O. By 3.9 and Lea 3 we obtan es G G es Ž G O. Ý Here we have used the nequalty. KAM teraton l l / Ý l es Ž G O. c Žr. Ý c Žr.. 6 r r3 ž. Ž B. We prove only one cycle of teraton processes, to say, fro the th step to the th step. For splcty, we ot notaton and denote by.

8 6 FUZHONG AND YONG We need to construct functons S S S, R R R Rˆ such that S, N PR, S, R R R R. Ž 3.. Let t be the flow deterned by the vector feld IdS. Defne. Then s a syplectc transforaton. On the bass of the fact and Taylor s forula, we have Usng 3. yelds d t F F,S dt H N P N t tp t R,S dt, 3. H t N NR, P tp Ž t. R,S dt. Ž 3.3. S, N PŽ x, y. R, S, R R, Ž 3.. P N Ž. ; Ž.Ž. y y ; S, N x, y, yy S, y yy, yy Take t ½ 5 R, S, R R. Ž 3.5. ; ; P y ; P RPŽ y. R Ž y., yy, Ž 3.6. y ½ 5 N RP ˆ ˆ S, N ˆ y yy, yy S, N ˆ Ž.Ž., Ž 3.7. y ˆPPP Ž x, y. Ž x, y., yy. Ž 3.8. Utlzng 3.5, 3., 3.6, and the proposton, we have c D r S. Ž 3.9.

9 INVARIANT TORI FOR HAMILTONIAN SYSTEMS 63 By 3.6, 3.9, and Cauchy s estate, we obtan ½ 5 P N Ž x, y., y y; S, Ž y.ž yy., yy R ; y y D ; P Ž x, y., yy y D ½ N Ž.Ž. ; 5 S, y yy, yy y D c c I G r3 c r3 c M Ž c c., c M. Ž 3.. Here we have used the nequalty c 5! The proof wll be gven n the sequel. Applyng the proposton yelds c c5 c6 S D. Ž rr3 5 Fro Ž 3.8. and Ž 3.. we see ½ 5 ˆ N Nˆ Pˆ D c7, ax, c8 c8m. Ž 3.. y y D D By Ž 3.7., Ž 3.9. Ž 3.., Ž 3.7., and Cauchy s estate, we have c c6 Rˆc c MI c MI 7 r D G G c 9. Ž Ž 3.9. and Ž 3.. ply S D 3 n6. Ž 3.. Fro Ž 3.3., Ž 3.3., Ž 3.., and Cauchy s estate, t follows that c P D P,S D R,S D c c 6 s s c 3. s

10 6 FUZHONG AND YONG By the choce of s and, we have provded P, Ž Ž 3. c. C As Now we verfy 3. for N. By Ž 3.6. we obtan N DŽ 6, s. c Ž s. Ž s., s s, s ˆ N DŽ 6, s. ž c5 Ž s. Ž s., s s. s / 8 we also have These ply 3.. ž / 8 / 6 ž, Ž D. 8 s s, 8 c M, c M Estates of coordnates By Ž 3.. and Cauchy s estate we derve Choosng S c S c5,. 6 6 x y s D D we obtan 8 Ž 5. c c, E c c5 38 d D s. Ž s

11 INVARIANT TORI FOR HAMILTONIAN SYSTEMS 65 Hence s well defned on DŽ, s. and. Copleton of the proof : DŽ, s. D 5. Ž 3.7. Choose y G and defne. Then as we have Set 6Ž r3. 8Ž r3. ½ž / 5 n,, Ž F. ž / 6 Ý. D x:rext,ix yy. Usng 3.6 and D yelds that on D, d Ý kk Ý. k k Hence s unforly convergent on D. As n Ž 3.9., fro Ž A. we see 6 3 y y c, y G, where l y. By 3.6 we have as c7 38 E, Ž x, y. s where E s the unty atrx. Thus whenever c, 7 / k 3 Ý Ž k. Ýž D k, Ž x, y. 3 / 3 Therefore x, y converges on D. Ž G. ž. Ž H.

12 66 FUZHONG AND YONG By and 3.3 we ay assue that ˆR Ž x, y. OŽ Ž yy.. unforly holds on D. Hence the syste z IŽ y. grad T N Ž z. has an nvarant torus x, y : x T, y y. Usng the convergence of and x, y, we derve that H H N, l, has an nvarant torus x, y : x T, y y wth the fre- quency Ž y.. Ths copletes the proof.. APPENDIX Now we lst soe leas that have been used n prevous sectons. LEMMA 7. Consder the Haltonan syste z IŽ y. grad T HŽ z., z Ž x, y.. Ž.. If H z H y, then. has the for Ž. T where y y,..., y. x Ž y., y, n LEMMA 9. Let O R be a bounded and connected regon, and let g and g be real analytc n-densonal ector-alued functons, on O, such that g rank r, ž y/ g ½ y 5 rank g, :, nr n; then there s such that as s suffcently sall and g, the set ² : n O yo: k, gž y. gž y. k,kz s a nonepty Cantor set, and where c s ndependent of, g. es Ž O O. c Ž nr., n

13 INVARIANT TORI FOR HAMILTONIAN SYSTEMS 67 LEMMA 3. Assue that Ž y. satsfes condtons Ž.3. and Ž.., and Ž y. s a real analytc -densonal ector-alued functon. Then there s such that as s suffcently sall and, the set ² : G yg: k, Ž y. Ž y. k,kz s a nonepty Cantor set, and where c s ndependent of and. Proof. es Ž G G. c Ž r., l Apply Lea and Fubn s theore. ACKNOWLEDGMENT The authors thank the referee for valuable suggestons. REFERENCES. V. I. Arnold, Proof of a theore by A. N. Kologorov on the preservaton of quas-perodc otons under sall perturbatons of the Haltonan, Uspekh Mat. Nauk. 8 Ž 963., 3.. J. Poschel, On the ellptc lower densonal tor n Haltonan systes, Math. Z. Ž 989., L. H. Elasson, Perturbatons of stable nvarant tor for Haltonan systes, Ann. Scuola Nor. Sup. Psa Cl. Sc. Ser. () 5 Ž 988., 57.. M. B. Sevryuk, Soe probles of the KAM-theory: condtonally perodc otons n typcal systes, Uspekh Mat. Nauk. 5 Ž 995.,. 5. C.-Q. Cheng and Y.-S. Sun, Exstence of KAM tor n degenerate Haltonan systes, J. Dfferental Equatons Ž 99., C.-Q. Cheng and Y.-S. Sun, Exstence of nvarant tor n three densonal easure-preservng appngs, Celestal Mech. Dyna. Astrono. 7 Ž 99., I. O. Parasyuk, On preservaton of ultdensonal nvarant tor of Haltonan systes, Ukran. Mat. Zh. 36 Ž 98., Z.-H. Xa, Exstence of nvarant tor n volue-preservng dffeoorphss, Ergodc Theory Dyna. Systes Ž 99., J. X. Xu, J. G. You, and Q. J. Qu, Invarant tor for nearly ntegrable Haltonan systes wth degeneracy, Math. Z. Ž 997., 6 Ž 997., Cong Fuzhong, L Yong, and Huang Mngyou, Invarant tor for neary twst appngs wth ntersecton property, Northeast. Math. J. Ž 996., V. I. Arnold, Matheatcal Methods of Classcal Mechancs, Sprnger-Verlag, New YorkHedelbergBerln, B. A. Dubrovn, A. T. Foenko, and S. P. Novkov, Modern Geoetry: Methods and Applcatons, Sprnger-Verlag, New YorkBerlnHedelbergTokyo, M. Herann, Exeples de flots Haltonan, C. R. Acad. Sc. Pars Ser. I Math. 3 Ž 99..