LOCALIZATION OF COMPACT INVARIANT SETS OF A 4D SYSTEM AND ITS APPLICATION IN CHAOS

Transcription

1 IJRRAS () April wwwrppresscom/volumes/volissue/ijrras pdf LOCALIZATION OF COMPACT INVARIANT SETS OF A 4D SYSTEM AND ITS APPLICATION IN CHAOS Zhinn Wu * Fuchen Zhng & Xiowu Li 3 School of Mthemtics nd Computer ScienceYichun UniversitYichun 336PR Chin College of Mthemtics nd Sttistics Chongqing Universit Chongqing 444 P R Chin 3 Computer nd Informtion Engineerring CollegeGuihou Universit for NtionlitiesGuing 555PR Chin *E-mil: hi_nn_7@63com ABSTRACT We comine the gloll exponentill ttrctive set with the itertive theor to discuss the oundedness of Loren-Stenflo chotic sstem Firstl We get exponentill ttrctive set for this sstem Then we use itertive theor to get refined oundedness for this sstem Finll the oundedness for is pplied to chos snchronition Numericl simultions re presented to show the effectiveness of the proposed scheme Ke words: Chotic sstem; itertive theorem; invrint sets INTRODUCTION Since the discover of the Loren chotic sstem in 963 [] Chos hs een studied extensivel After tht mn chotic sstems hve een discovered such s Rössler sstem [] Chen sstem [3] nd Lü sstem [4] And these chotic sstems hve een widel studied [5-9] Efforts of mn reserchers hve een imed towrds investigtions of ifurctions chos oundr nd relted control prolems for these new chotic sstems In prticulr the oundness pls n importnt role in chotic sstems If we cn show tht sstem under considertion hs gloll ttrctive set then we know tht the sstem cnnot hve equilirium points periodic solutions qusi periodic solutions or other chotic ttrctors outside the gloll ttrctive set This gret -l simplifies the nlsis of the dnmicl properties of the sstem A noticele prog -ress hs een chieved in spite of sustntil complexit of mthemticl models of such sstems The pper [] nd [] hs een devoted to nlsis the pitchfork nd Hopf ifurctions sed on using ifurctions theor nd the centrl mnifold theorem otining n pproximte stilit oundr nd some other topics Recentl theoreticl efforts hve een performed to find loclition domins contining ll comp -ct invrint sets of nonliner continuous-time sstem possessing complex ehvior [] [3] [4] nd [5] Here we recll studies of Loren sstem [6] nd the perm nentmgnet motor sstem [3] Bounds for domin contining ll compct invri nt sets otined in these ppers in mn cses cn e used not onl for theoreticl stu -dies of chotic ttrctors eg ut lso for estimting for the Husdorff Dimension [7] or for the numericl serch of ttrctors The ultimte ound lso pls n importnt role in designing scheme for chos control nd chos snchronition [8] Recentl new Loren-Stenflo chotic sstem hs the following eqution [9]: x x dw cx x x () w x w Where > > c> d> re prmeters; c d re the Rleigh nd the rottion numers respectivel nd > is geometric prmeter When prmeters = =7 c=6 d=5 the Lpunov exponent for sstem is The lrgest Lpunov exponent is 3665> So the sstem () is chotic s = =7 c=6 d=5[9] Fig shows the phse portrits of sstem () in three dimensionl spces see Fig Fig shows Hopf ifurction digrm for sstem () with sstem prmeters ==7c=6 nd prmeter d rnges from to 3 5 see Fig

2 IJRRAS () April Fig Phse portrits of sstem () with sstem prmeters ==7c=6 d=5 Fig Hopf ifurction digrm for sstem () with sstem prmeters ==7c=6 nd prmeter d rnges from to 3 5 Moreover in the sense defined Vne c ek nd C elikovský [] It is immeditel cler tht the sstem is topologicll nonequivlent to the originl Loren nd the other Loren-like sstems Therefore it is interesting is to further find out wht kind of new dnmics this sstem hs In the following we will discuss the ultimte ound nd the loclition set of chotic sstem () Some sic dnmicl properties were studied in [9] But mn properties of this new sstems remins to e uncovered In this pper we investigte the ultimte ound nd the compct invrint sets for the new chotic sstem vi Lpunov functions theor extremum theor nd itertion theorem [3-4] The rest of this pper is orgnied s follows: Section is some nottions In section we give n ellipsoidl loclition nd we denote the rdius for the ellipsoid In section 3 it is out pplictions of the itertion theorem In section 4 nd 5 we use nother two locliing functions In section 4 we Loclie using the circulr clinder In section 5 we Loclie with help of clindricl surfce In section 6 we give one exmple of the loclition of chotic ttrctor for the Loren-Stenflo sstem At the sme time our work minl focus in these sections It is evident tht we cn get smller ound of the chotic ttrctor for this chotic sstem In section 7 we stud snchronition using the ound for Section 8 is simultion stud The conclusion is drwn in Section 9 SOME NOTAIONS AND PRELIMINARIES Let us introduce rel polnomil right-side sstem () x f x n Here x R is the stte vector Let us tke rel polnomil h of n rel vriles wh -ich is not first integrl of () The function h tht used in the solution of the locli- tion prolem of the compct invrint sets is clled n locliing B h we denote the restrictions of function h on set B R B Lhwe f denote the Lie derivtive of the function h And here f B L h x f x Let us define f n h x i n i L h x is the Lie derivtive []: x i f x f x f x f x

3 IJRRAS () April inf inf sup h h x x S h sup h h x x S A loclition of ll compct invrint sets of the sstem () is descried the follo- wing results [3-4] Proposition Let W S h locted in the set If K x h W hx hsupw W h Then ech compct invrint set of the sstem () contined in W is inf (3) W S h then sstem hs no compct invrint sets tht contined in W Lemm Ech compct invrint set of () tht contined in W hs common points with the set The function h used in the formultion of these results is clled locliing Theorem Let Set K K h x h m ( K m Km Km m n m ) e sequence of functions from C R m with Km m x hm inf hm x hm sup Sh K m m hmsup sup hm x Sh K m m hminf inf hm x Then we hve K K K nd m m contin ll compct invrint sets of the sstem () K K K (4) 3 ELLIPSOIDAL LOCALIZATION WITH PRECISE BOUNDS Sstem () hs n ellipsoidl gloll exponentill ttrctive set s follows: x w x c dw where is rel numer nd where c 4 c c 4 Proof: Define the following positive definite nd rdicll unounded Lpunov function: h x w V x w x c dw Then its derivtive long the orits of sstem () is V xx c dww x dw c (5) c c = x dw 4 Let V then we cn get the following ellipsoidl surfce : x dw c c 4 W S h 3

4 IJRRAS () April Outside V while inside V Therefore the ultimte ound for sstem () cn e reched on In the following we will discuss the sstem chotic ttrctor o -und out sstem different prmeter (i)when let f on From (5) we cn get I c c f c leds to I f c In the following we estimte the mximum vlue of Then let Furthermore we notice tht f sup f R c 4 so we cn otin tht From (5) we cn derive V x c dw c c x c dw f c x c dw f c x c dw V When x c dw According to the comprison theorem we hve t tt t V X t V X e e d t V X e e t t t t WhenV X t t t there exists exponentil estimtion given s follows: t t V X t V X e (6) (ii) When from (5) we cn get V x dw c x dw c x c dw c V (when x c dw ) Similrl when V X t t t we cn get n exponentil estimtion given s follows: t t V X t V X e (7) (iii)when Similrl to (i) (ii) we cn lso get: 4

5 IJRRAS () April WhenV X t t t we hve t t V X t V X e (8) From formul (6) (7) (8) we otin lim V X t t Hence x w x c dw is the gloll expone- ntill ttrctive set of sstem () At the sme time ccording (6) (7) (8) it implies tht is lso the ultimte ound for sstem () Especill let us tke we cn get the following theorem Theorem All compct invrint sets of the sstem () re contined in the ellipsoid defined With K h x w x c dw hsup c c 4 c 4 4 APPLICATIONS OF THE ITERATION THEOREM K h is contined in the poltope defined We note tht x ; ; c ; w d Now we cn ppl dditionl locliing functions let us tke h x w w x x Then Sh is given w h Sh x S h K h x w w x c dw x contined in the set S h x w w x Therefore hsup hinf And K x w w It is clerl tht the ltter cn e refined the formul K x w w min d Therefore the set which is 5

6 IJRRAS () April Kij Here nd elow we use the sme nottions for sets nd sets with pproximte corresponding ounds Hence K x w x c dw w min (9) d Also concerning the lower ound on w it follows (9) tht we cn get w d for w d d 5 LOCALIZING BY USING THE CIRCULAR CYLINDER let us tke h3 x w c then Sh3 is given c nd c h3 R c S h 3 with R c c 4 So we get h R c if () h 3sup 3inf R c if () Hence ll compct invrint sets of sstem () re locted in prolic clinder Kh3 defined K x w h h ; K x w h h 3 3 3sup () is fulfilled 3 3 3inf in the dependence on which pir of inequlities () 6 LOCALIZING WITH HELP OF CYLINDRICAL SURFACE Here we tke locliing function h4 x w x dw x dw therefore using this formul we cn get tht x dw h4 x dw Sh4 d x w So h if () 4sup h if (3) 4inf then the set Sh 4 is given the eqution Hence ll compct invrint sets of sstem () re locted in the prolic clinder Kh4 defined K x w h h ; K x w h h in the dep -endence on which pir of inequlities 4 4 4sup () (3) is fulfilled 4 4 4inf Theorem 3 All compct invrint sets of the sstem () re contined in the set defined 6

7 IJRRAS () April x w x c dw w min K3 K4 d Provided restrictions tht imposed on prmeters hold Tht is to s in the dependence on which pir of inequlities () () () (3) is fulfilled With hsup c c 4 c 4 7 ONE EXAMPLE OF THE LOCALIZATION OF A CHAOTIC ATTRACTOR In this section we descrie one exmple of the loclition of chotic ttrctor which ws found [9] Prmeters of the sstem () re chosen = =7c=6 d = 5 For convenience we depict some ellipsoid ccording to Theorem According to Theorem we cn ttin c 7 We show figure of the chotic ttrctor estimted in Theorem nd the phse portrit for sstem () tht projected into the (x ) (x w) (x w) ( w ) spce see Fig3 Fig3 The projections of the chotic ttrctor into the (x ) (x w) (x w) ( w ) spce ccording to in Theorem 8 SYNCHRONIZATION OF THE NEW CHAOTIC SYSTEM Let the following the sstem () e the driver sstem x x cw dx x x () w x w nd the response sstem is: 7

8 IJRRAS () April c4 d k x Sstem () cn snchronie the sstem (4) djusting prmeter k From theorem then we cn get the oundness of tht is denote M d M (4) then we hve the following theorem Theorem 4 Sstem () nd sstem (4) re gloll completel snchronie When d M k 4 M ( here M ) 4 Proof: Let the stte errors e sstem () nd (4) is e e e ce4 e de e 3 e k e e 3 e e e3 e 4 e e4 Let V e e e3 ce4 d For convenience let us e x e e e w then the error dnmics of (5) M where is rel prmeter nd 4 sstem (5) is V e e e e e e ce e e e e ce e de e e e e e k e 4 3 e e e e e ce e ce e k e e ce d e e e e e k e e ce d M e e M e e EPE E e e e e Where 3 4 T d M M d M k P M c B some elementr clcultion we know tht the mtrix P is positivel definite when M d M k 4 M 4 Then its time derivtive long the This implies tht the origin of the error sstem (5) is smptotic stle which is equivlent to s tht the sstem () cn snchronie the sstem (4) completel 8

9 IJRRAS () April 9 SIMULATION STUDIES The numericl simultions re crried out using the MATLAB 74 The initil conditions of the driver ( ) nd response sstems (4) re 59 nd 893 When = =7 c=5 d=6[9] it is es to M 7 M 54 otin 7 So then the ound of the coefficients of feedck control k could e otined ccording to the Theorem 4 Choose k=86 The response sstem snchronies with the drive sstem s shown in Fig4 Fig4 Snchronition error of the two sstems under liner feedck control CONCLUSIONS To estimte domin contining ll compct invrint sets of dnmicl sstem i- s n importnt ut quite chllenging tsk in generl In this pper we stud the loclition prolem of compct invrint sets of the new chotic sstem with the help of the itertion theorem nd the first order extremum theorem Conclusions out itertion theorem is not otined from results of this pper it ws otined in ppers of Luis N Cori nd Konstntin E Strkov We lso estlish tht ll compct invrint sets of this sstem re locted in the intersection of ellipsoid with frust nd we lso compute its prmeters In ddition loclition using the two-prmeter set of the circulr clinder is descried Then domin contining ll compct invrint sets hs een estlished Finll the ound for is pplied to chos snchronition Numericl simultions show the effectiveness nd the dvntge of our methods REFERENCES [] Dmei Li Jun-n Lu Xioqun Wu Gunrong Chen Estimting the ultimte ound nd positivel invrint set for the Loren sstem nd unified chotic sstem Journl of Mthemticl Anlsis nd Applictions 6 33 (): [] Rösser OE An eqution for continuous chos [J] Phs Lett A (5): [3] Wen-Xin Qin Gunrong Chen On the oundedness of solutions of the Chen sstem Journl of Mthemticl Anlsis nd Applictions 7 39 (): [4] Lü J Chen G A new chotic ttrctor coined [J] Int J Bifurct Chos (3) : [5] Chen G Lü J Dnmics of the Loren Sstem Fmil: Anlsis Control nd Snchronition [M] Science Press Beijing (in Chinese) [6] Yu C Chen G Complex dnmics in Chen s sstem [J] Chos Solitons & Frctls 7 () (6) pp75-86 [7] Li C Chen G A note on Hopf ifurction in Chen s sstem [J] Int J Bifurct Chos 33 (6):69-65 [8] Li T Chen G Tng Y On stilit nd ifurction of Chen s sstem [J] Chos Solitons & Frctls 49 (5):69-8 [9] Yu Y Zhng S Hopf ifurction in the Lü sstem [J] Chos Solitons & Frctls 3 7 (5): 9-96 [] Zhujun Jing Chng Yu Gunrong Chen Complex dnmics in permnent mgnet snchronous motor model Chos Solitons & Frctls 4; : [] Li Z Prk JB Joo YH Zhng B Chen G Bifurctions nd chos in permnent mgnet snchronous motor IEEE Trns Circ Sst : Fundment Theor Appl 49: [] Alexnder P Krishchenko Konstntin E Strkov Loclition nlsis of compct invrint sets of multi-dimensionl nonliner sstems nd smmetricl prolongtions Communictions in Nonliner Science nd Numeicl Simultions 5 (5):59-65 [3] Luis N Cori Konstntin E Strkov Bounding domin contining ll compct invrint sets of the permnent-mgnet motor sstem Communictions in Nonliner Science nd Numeicl Simultions 94 (): [4] Konstntin E Strkov Bounds for compct invrint sets of the sstem descri- ing dnmics of the nucler spin genertor Communictions in Nonliner Science nd Numeicl Simultions 9 4 (6): [5] Konstntin EStrkov Bounds for domin contining ll compct invrint sets of the sstem descriing the lser plsm interction Chos Solitons & Frctls 9 39 (4): [6] Alexnder P Krishchenko Konstntin E Strkov Loclition of compct invrint sets of the Loren sstem Phsics Letters A 6 35 (5): [7] Boichenko VA Leonov GA Dimension theor for ordinr differentil equtions Wiesden: Teuner Verlg (5) [8] SHU Yonglu XU Hongxing ZHAO Yunhong Estimting the ultimte ound nd positivel invrint set for new chotic sstem nd its ppliction in chos snchronition[j] Chos Solitons & Frctls 94 (5): [9] SHmmmi KBenSd MBenreje On the snchronition of identicl nd non-identicl 4-D Chotic sstems using rrow form mtrix Chos Solitons nd Frctls 9 4:- [] AVne c ek nd S C elikovský Control Sstems: From Liner Anlsis to Snthesis of Chos Prentice-Hll London (996) [] LJC'PCO KOCAREV Lie Derivtives nd Dnmicl Sstems Chos Solitons & Frctls (8) :