Limit Cycle Bifurcations in a Class of Cubic System near a Nilpotent Center *

Transcription

1 Appled Mateatcs ttp://dxdoorg/6/a75 Publsed Onlne July (ttp://wwwscrporg/journal/a) Lt Cycle Bfurcatons n a Class of Cubc Syste near a Nlpotent Center * Jao Jang Departent of Mateatcs Sanga Marte Unersty Sanga Cna Eal: jaojang8@yaoocn Receed Aprl 5 ; resed June ; accepted June ABSTRACT In ts paper we deal wt a cubc near-haltonan syste wose unperturbed syste s a sple cubc Haltonan syste ang a nlpotent center We proe tat te syste can ae 5 lt cycles by usng bfurcaton teory Keywords: Near-Haltonan Syste; Nlpotent Center; Hopf Bfurcaton; Lt Cycle Introducton In te Internatonal Congress of Mateatcs eld n Pars n Hlbert ade a lst of probles Te second part of Hlbert s 6t proble s stll an open and dffcult queston: to fnd a upper bound of te nuber of lt cycles and ter relate locatons n polynoal ector felds of order n If te sngular pont of te syste s a non-saddle nor nlpotent te related Hopf bfurcatons are eleentary see [-] and ter references Hopf bfurcatons fro te eleentary focus type of sngulartes ae found broad and portant applcatons n bology cestry and pyscs and engneerng see [-7] for exaples Yet for te bfurcaton of lt cycles fro a non-eleentary center n a ore general planar ector feld ts ntrnsc dynacs s stll far away fro understandng due to te coplexty and tecncal dffcultes n dealng wt suc bfurcatons Ten t was natural to restrct te study of te nlpotent center by assung te syste s a perturbaton of a Haltonan syste Consder te followng syste x Hy P x y y H Q x y x were H xy Px y and () Q x y are C functons s sall and D wt D a copact set Wen syste () becoes x H y H () y wc s Haltonan syste Now suppose tat te * Te researc was partally supported by Natonal Natural Scence Foundaton of Cna (788) x Haltonan syste () as a nlpotent center at te orgn naely te functon H satsfes te followng condtons: H xy s a C functon satsfyng (H) H Hx Hy (H) te equaton H xy ; defnes a closed cure L surroundng te orgn and L approaces te orgn as goes to zero; Hy Hx Hy Hx (H) det xy xy It follows tat te expanson of H at te orgn as te for H xy y xy j j Assue tat te equaton H xy poste x-axs at A a Let ntersects te B denote te frst ntersecton pont of te poste orbt of () startng at A wt te poste x-axs Ten we ae were HB H A d H M O () AB M QdxPd y () L Te Abelan ntegral M aboe s called te frst order Melnko functon of syste () Fro Han [8] we ae a general teore as follows Teore Suppose tat te orgn s nlpotent sngular pont and tat L approaces te orgn as goes to zero If tere exst an nteger k and j Copyrgt ScRes

2 J JIANG 77 suc tat and Bj j k Bk B Bk rank k k ten we ae ) M as at ost k zeros near for and all near and k zeros can appear for soe near ) Syste () as at least k lt cycles near te orgn for soe near Man Results and Proof Consder te followng near-haltonan syste: x yaxybx y cy p x y y x ay bxy q x y () were and p and q are cubc polynoals We can wrte j p q c x y () x y j j Ten unperturbed syste () s a Haltonan syste wt Haltonan H xy x axbxy cy () syste () as a nlpotent center at te orgn Let L be te closed cure defned by H xy Ten t can be presented as ax bx y cy x () Assue tat te poste soluton of te aboe equaton n y s 5 6 y x u u u u u u O u 5 x C were u x (5) and Ten by () and (5) we obtan a 5a x ax bx ab x 5a 6a bb x a a b6ab x a 6a ba b b x a 86a b6a b 8ab x a a b77a b 8a b 56b x a 58a b77a b 66a b 5ab x a 87a b66a b a b 55a b 6 b x Ox 56 x cacx ba cx a ab cx 8a 6a bb cx a a b ab cx a a b a b b cx Ox x c acx a bcx a bacx a ab b cx a a b b ac x a a b a b b c x Ox x c acx a bcx a a bcx a ab b cx O x x c acx a bcx a bacx a a b b c x Ox 6 Copyrgt ScRes

3 77 J JIANG By [8] te negate soluton of () n y satsfes x u x u Tus two solutons of () are x u uu u u u O u x u u u u u u O u (6) On te oter and te ntersecton ponts of L and x- axs ae te x-coordnates a Ten by () we can wrte were and x y L H ci j j M qdx pdy p q dxdy j a (7) Here j a j H a I xydd dx xydy j j j a j x a d x x x u u x Introduce j (8) Wj x x d x () Ten slar to te etod of Han [8] we ae j W ( ) j j j j x x dx Terefore n turn by (6)-() we ae () I dx dy u u u O u dx W W W W W W W W W 6W6 6 5W 5W O O d d 5 d I x x y x u u u O u x W W W 5 7W8 W 8 W W 5W6 6 5W O O 5 7 d d 5 d 6 I x x y x u u u O u x W W W 5 6W8 8W 8 W W W6 6 5W O O Copyrgt ScRes

4 Notng tat 5 7 u u O u ten slarly we ae 5 7 I dx ydy u u O u dx J JIANG 775 W W W W W W O O 5 7 I xdx ydy x u u O u dx W W W W 5 O O u 6 u 5 O u 7 we ae In te sae way usng 5 7 I dx y dy u u Ou dx W W 8 W W W W O O Copyrgt ScRes

5 776 J JIANG Hence we ae 5 M BB BB B B5 O were B c c c c c c c And c B c c B c 8 c c 8 c c B c c c c c c 8 c B c c c 8 c c 8 B c c c 8 c Now t s drect tat B B B B B B det 66abc 8abc c c c c c c abc 8abc abc abc abc c c 76abc 7878abc 55abc 86866abc abc abc 8abc 8556abc 58abc abc 88abc 686abc abc a bc 586a bc 8677abc abc abc abc a bc a bc 88abc abc Copyrgt ScRes

6 J JIANG abc abc abc 56abc 5655abc abc a bc 57a bc abc abc 75abc 858abc a bc a bc 7568abc abc abc abc 8 Here f let a b c ten for soe cubc syste () we can obtan te aboe deternant s not zero ten te functon M can ae 5 sple zeros n > near = for soe near For exaple let a b c we obtan fro te aboe forula B B B B B B5 det c c c c c c Here ten we can obtan B B B B B B5 det 8865 () c c c c c c By Teore we ae: Teore Te functon M as at ost 5 zeros n near and for sall te cubc syste () can ae 5 lt cycles near te orgn REFERENCES [] A Androno E Leontoc I Gordon and A Maer Teory of Bfurcatons of Dynacal Systes on a Plane Israel Progra for Scentfc Translatons Jerusale 7 [] M Han On Hopf Cyclcty of Planar Systes Journal of Mateatcal Analyss and Applcatons Vol 5 No pp - do:6/jaa6758 [] Y A Kuznetso Eleents of Appled Bfurcaton Teory Sprnger-Verlag New York 5 [] T Caron R Uzdn C Pger Z Musslan M Sege and A Neponyascy Rotatng Propeller Soltons Pyscal Reew Letters Vol 87 No p do:/pysrelett87 [5] J Guckeneer and P Holes Non-Lnear Oscllatons Dynacal Systes and Bfurcaton of Vector Felds Sprnger-Verlag New York 8 [6] B D Hassard N D Kazarnoff and Y H Wan Teory and Applcatons of Hopf Bfurcaton Cabrdge Unersty Press Cabrdge 8 [7] S Wggns Global Bfurcatons and Caos: Analytcal Metods Sprnger-Verlag New York 88 [8] M Han J Jang and H Zu Lt Cycle Bfurcatons n Near-Haltonan Systes by Perturbng a Nlpotent Center Internatonal Journal of Bfurcaton and Caos Vol 8 No 8 pp -7 do:/s8786 Copyrgt ScRes