DYNAMIC BEHAVIOUR OF THE SOLUTIONS ON A CLASS OF COUPLED VAN DER POL EQUATIONS WITH DELAYS

Transcription

1 Jol of Mheml Sees: Adves d Applos Volme 6 Pges 67-8 Avlle hp://sefdveso DOI: hp://ddoog/86/jms_778 DYNAMIC BEHAVIOUR OF THE SOLUTIONS ON A CLASS OF COUPLED VAN DER POL EQUATIONS WITH DELAYS CHUNHUA FENG Depme of Mhems d Compe See Alm Se Uves Mogome AL 6 USA e-ml: feg@lsed As I hs ppe lss of opled v de Pol eqos wh me dels s vesged B mes of mheml lss ppoh some sffe odos o gee he esee of osllo solos fo he model e oed Compe smlos e povded o demose he poposed esls Iodo Fo lss of opled v de Pol eqos of he fom: () ( () ) () () () () () () ( () ) () () () () () () ( () ) () () () () () Mhems Sje Clssfo: K Kewods d phses: opled v de Pol eqo del osllo Reeved Noveme 6 6 () 6 Sef Adves Plshes

2 CHUNHUA FENG 68 whee j R j > M good esls hve ppeed (see ()-(9)) Fo he se Ho d R [] hve vesged he esee of lm les sg S degee heo d he gme ems vld fo he se h > Rompl e l [8] osdeed ssem of hee v de Pol osllos whh opled s follows: () ( ()) [ ] () ( ()) [ ] () ( ()) [ ] [ ] µ µ µ µ w w w p w w w w w () Fo g of fo mll opled ologl ssems desed opled v de Pol osllos: () ( ()) ( ) ( ) () ( ()) ( ) ( ) () ( ()) ( ) ( ) () ( ()) ( ) ( ) () The sl odes d he m dml ses hve ee osdeed o he sl mps he ( ) ple [9] I s well ow h he me del s evle m phsl d ologl pheome sh s mfg poess le eos oe moos mehl oollg ssems poplo dms d so o Nll he me del opled v de Pol eqos lso hve ee eesvel sded m esehes [-] Fo emple Ws e l [] hve osdeed he dms of wo wel opled v de Pol osllos whh he oplg ems hve me dels:

3 DYNAMIC BEHAVIOUR OF THE SOLUTIONS ON 69 () ( () ) ( ) () ( () ) ( ) α α () L e l [] hve sded he opled v de Pol osllos wh wo ds of dels: ( ()) () [ ( ) ( )] ( ()) () [ ( ) ( )] α α w w (5) Zhg d G [] sed he heo of oml fom d el mfold heoem o dsss he followg me del ssem: () ( () ) ( ) () [ ] () ( () ) ( ) () [ ] α α (6) Moved he ove models hs ppe we osde he followg me dels model: () ( () ) () ( ) ( ) ( ) () ( () ) () ( ) ( ) ( ) () ( () ) () ( ) ( ) ( ) (7) whee < d < fo eh ( ) j R j ( ) e me dels O m s o vesge he dm ehvo of opled osllos Pelmes Fo oveee leg ( ) ( ) ( ) ( ) ( ) j j j he he opled ssem (7) e we s he followg:

4 CHUNHUA FENG 7 () ( ) ( ) ( ) () () ( ) ( ) ( ) () () ( ) ( ) ( ) () () ( ) ( ) ( ) () (8) Ovosl he og ( ) s eqlm of ssem (8) The lezo of he opled ssem (8) og s

5 DYNAMIC BEHAVIOUR OF THE SOLUTIONS ON 7 () ( ) () () ( ) () () ( ) () () ( ) () (9) The ssem (9) e epessed he followg m fom: () ( ) ( ) BX AX X () whee () [ ()] ( ) [ ( ) ( ) X X T ( ) ( ) ] 5 5 T Boh ( ) j A d ( ) j B e mes s follows:

6 CHUNHUA FENG 7 A B Lemm Assme h he m ( ) B A C s osgl m he ssem (9) hs qe eqlm mplg h ssem (8) hs qe eqlm

7 DYNAMIC BEHAVIOUR OF THE SOLUTIONS ON 7 Poof A eqlm po [ ] T os solo of he followg lge eqo: of ssem (9) s A B ( A B) C () Se C ( A B) s m odg o he le lge owledge f C s sgl m Eqo () m hve fel m solos Howeve f C s osgl m Eqo () hs ol oe solo mel he vl solo Implg h ssem (9) hs qe eqlm Nog h ssem (8) he ole ems () ()( ) e hghe ode fesml whe d ( ) ed o zeo Theefoe ssem (9) hs qe eqlm sggess h ssem (8) hs qe vl solo Lemm All solos of ssem (8) e foml oded Poof To pove he odedess of he solos ssem (8) we os Lpov fo V () () Cllg he devve of V () hogh ssem (8) oe ge V () ( ) () 8 () { ( )} { ( )} { ( )}

8 7 CHUNHUA FENG j j j () Nog h s d j ( j ) ed o f ( ) e hghe ode f h d Theefoe hee ess sl lge L > sh h j V () ( ) s L( ) se < ( ) 8 < Ths mes h ll solos of ssem (8) d hee ssem (7) e oded M Resls Theoem Assme h lezed ssem (9) hs qe eqlm po fo gve pmees vles of j ( j ) d ( j ) j Le µ ( A) m { } d B m If he followg odo holds: ( B ) e ep ( µ ( A) ) > () whee m{ } The he qe eqlm of ssem (9) s sle Ssem (7) geees lm le mel peod solo Poof We shll show h he qe eqlm of ssem (9) s sle Cosde Le z() () The we hve ssem () s follows: X () AX ( ) BX ( ) () dz() d µ ( A) z( ) B z( ) (5)

9 DYNAMIC BEHAVIOUR OF THE SOLUTIONS ON 75 Spell fo eqo dw() d µ ( A) w( ) B w( ) (6) If he qe eqlm of ssem (6) s sle he he hes eqo ssoed wh (6) gve λ λ µ ( A ) B e (7) wll hve el egve oo s λ d we hve fom (7) λ λ A B e µ ( ) (8) Usg he foml e e fo > oe ge λ µ ( A) ( µ ( A) λ ) B µ ( A) ( B e ) e e B e e µ ( A) λ ( µ ( A) λ ) (9) The ls eql ods Eqo () Hee o lm egdg he sl of eqlm of ssem (6) s vld Aodg o he ompso heoem of dffeel eqo we hve ( ) w( ) Se he vl solo s sle ssem (6) hs mples h he vl solo of (5) (hs ssem ()) s sle Nog h me del ssem s he vle of del eses he sl of he qe eqlm sll ms [ ] So fo vles of me dels he qe eqlm of ssem () (o ssem (9)) s sle Rell h ssem (8) he ole ems () ()( ) e hghe ode fesml whe d ( ) ed o zeo So he qeess d sl of he eqlm of ssem (9) mples h ssem (8) hs qe sle eqlm Ths sggess h he qe eqlm of ssem (8) (o eqvle ssem (7)) s sle Se ll solos of ssem (7) e oded hs ssem (7) geees lm le sed o []

10 76 CHUNHUA FENG Theoem Assme h lezed ssem (9) hs qe eqlm po fo gve pmees vles of d j ( j ) If hee ess sl lge posve me L sh h he followg eql holds: L L µ ( A) B e > () The he qe eqlm of ssem (5) s sle mplg h he qe eqlm of ssem (9) s lso sle Ssem (7) geees lm le mel peod solo Poof We sll osde ssem (6) d wll pove h he vl solo of ssem (6) s sle I s sffe o show h he hes eqo ssoed wh (6) gve (7) hs el posve oo Nog h Eqo (7) s sedel eqo he hes vles m e omple mes Howeve we lm h Eqo (7) hs el posve oo Le λ f ( λ) λ µ ( A) B e () Ovosl f ( λ) s oos fo of λ Nog h > µ ( A ) > B s oded d f ( ) µ ( A) B < O he ohe L hd f ( L) L µ ( A) B e > Aodg o he emede vle heoem of oos fo hee ess posve me M sh h f ( M ) ( M ( L) ) I he ohe wods hee ess posve hes oo of hes Eqo (7) Theefoe he vl solo of ssem (6) s sle Sml o Theoem oe pove h ssem (8) geees lm le mel peod solo Smlo Resls Fs we osde hee opled v de Pol eqo d sed he fom of ssem (8) fo smlo We fed Theefoe

11 DYNAMIC BEHAVIOUR OF THE SOLUTIONS ON 77 µ ( A ) 68 d B We seleed 5 hs d ( B ) e ep ( µ ( A) ) 5 > Bsed o Theoem hs ssem geees peod solo (see Fge ) Whe we ese he me dels s 6 8 he 5 5 dml ehvo sll ms (Fge ) The we dsss fo opled v de Pol eqo ssem (8) We fed d 87 Ths µ ( A ) 9 B 65 me dels e So 5 I s es o he hee ess L > sh h odo () s ssfed Bsed o Theoem hee ess peod solo of hs ssem (see Fge ) Fge Dml ehvo of he solos dels: ( )

12 78 CHUNHUA FENG Fge Dml ehvo of he solos dels: (5 6 8) Fge Dml ehvo of he solos dels: ( )

13 DYNAMIC BEHAVIOUR OF THE SOLUTIONS ON 79 5 Colso I hs ppe we hve dsssed he dml ehvo of opled v de Pol eqo wh me dels Bsed o he mheml lss heo smple eo o gee he esee of peme osllos whh s es o he s omped o he fg mehod hs ee poposed Some smlos e povded o de he oeess of he eo Refeees [] N Ho d S R Esee of lm les fo opled v de Pol eqos J Dff Eqo [] R H Rd d P J Holmes Bfos of peod moos wo wel opled v de Pol osllos Ie J Nole Meh 5 (98) [] D W Sol d R H Rd Dms of wo sogl opled elo osllos SIAM J Mh Al 6() (986) [] A M Respose ool fo he eell eed v de Pol osllo wh o-lol feed J Sod d Vo (5) () [5] Z M Ge S C L S Y L d C M Chg Pgml dpve hos ool fom ew dole v de Pol ssem o ew dole Dffg ssem Appl Mh Comp () (8) 5-5 [6] Y H Q d S M Che Ae ppome ll solos fo mldegee-of-feedom opled v de Pol-Dffg osllos homoop lss mehod Comm Nole S Nme Sml 5 - [7] S Ds d K Mh Fol dml model fo he geeo of ECG le sgls fom fleed opled v-de Pol osllos Compe Mehods Pogms Bomede 9-57 [8] K Rompl R Rd d H Howld Dms of hee opled v de Pol osllos wh pplo o d hhms Comm Nole S Nme Sml (5) (7) 79-8 [9] H G Kdj J B Oo d P Wofo Shozo dms g of fo mll opled ologl ssems Comm Nole S Nme Sml (7) (8) 6-7 [] S Ws d R Rd The dms of wo opled v de Pol osllos wh del oplg Nole D 5- [] X L J J d C H Hse Dms of wo del opled v de Pol osllos Meh Res Comm (5) (6) 6-67

14 8 CHUNHUA FENG [] J M Zhg d X S G Sl d fo lss he del-opled v de Pol osllos Appl Mh Model (9) () 9-99 [] W Y Wg d L J Che We d o-eso dole Hopf fos m opled v de Pol osllos wh del oplg Appl Mh Model 9(-) (5) 9- [] A M A V de Pol s osllo de deled feed J Sod V 8() (999) -9 [5] X L J C J C H Hse d C X T The espose of Dffg-v de Pol osllo de deled feed ool J Sod d Vo 9(-5) (6) [6] I V Emov G V Sde d J De Semodo g lse sje o deled opl feed: Bfos d sl Comm Nole S Nme Sml [7] H B Jg Q S B d S Zheg Implsve osess deed ewos of del ole osllos wh swhg opologes Comm Nole S Nme Sml [8] Y H Q d S M Che Ae ppome ll solos fo mldegee-of-feedom opled v de Pol-Dffg osllos homoop lss mehod Comm Nole S Nme Sml 5 - [9] Y Q L W H Jg d H B Wg Dole Hopf fo d qs-peod os del-opled lm le osllos J Mh Al Appl 87-6 [] I D Ld F Boz R R Bmed d A V Besço Alss of ool elev opled ole osllo ssems Eope J Cool () (8) 6-8 [] Z L She d C R Zhg Dole Hopf fo of opled dsspve S Ld osllos wh del Appl Mh Comp 7(5) () [] J Fode d P Nelso Applos of Sm seqees o fo lss of del dffeel eqo models Pep [] D Ghosh A R Chowdh d P Sh O he vos ds of shozo deled Dffg-v de Pol ssem Comm Nole S Nme Sml () (8) 79-8 [] N Chfee A fo polem fo fol dffeel eqo of fel eded pe J Mh Al Appl 5() (97) -8 g