Feedback Control and Synchronization of Chaos for the Coupled Dynamos Dynamical System *

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1 ISSN England UK Jornal of Informaion and Comping Scinc Vol. No. 6 pp. 9- Fdbac Conrol and Snchroniaion of Chaos for h Copld Dnamos Dnamical Ssm * Xdi Wang Liin Tian Shmin Jiang Liqin Y Nonlinar Scinific Cnr facl of scinc Jiangs Univrsi Zhnjiang Jiangs P.R.China Rcivd Ocobr 6 5 accpd Fbrar 6 6 Absrac: In his papr b h simpl linar conrollr h copld dnamos dnamical ssm can b conrolld o a sabl priodic orbi and a sabl fid poin. Manwhil h sabili of h priod orbi and fid poin is provd b h vals of Lapnov ponn. Frhrmor b h nonlinar fdbac conrollr h fficin compl Snchroniaion of h copld dnamos dnamical ssm is compld. Nmrical simlaion rsls show h ffcivnss and fasibili of h simpl linar conrollr and nonlinar conrollr. Kwords: chaos fdbac conrol fficin compl Snchroniaion copld dnamos dnamical ssm Lapnov ponn.. Inrodcion In h las dcad conrolling chaos has bn h focs of h rsarch of nonlinar chaoic ssms. Rcnl hr has bn incrasing inrs in h rsarch of chaos conrol and snchroniaion. Varios mhods of chaos conrol and snchroniaion hav bn proposd in rcn ars. Th crrn conrol algorihms can b classifid ino wo main cagoris: fdbac conrol and non-fdbac conrol. Among hs linar fdbac conrol is an imporan and ffciv mhod basd on h dsign of diffrn conrollr has achivd a lo of saisfacor rsls s []-[5]. A h sam im chaos snchroniaion also is an imporan opic and has obaind a lo of availabili rsls s [6] - [7]. Th ssm ha w will sd in his papr coms from H.N. Agia s []. Th ssm consiss of wo dnamos ssms which ar conncd wih ach ohr so ha h crrn gnrad b an on of hm prodcs h magnic fild for h ohr. W dno h anglar vlociis of hir roors bω ω and h crrn gnrad b rspcivl. Thn wih appropria normaliaion of variabls h mahmaical modl qaions for his ssm ar: ω ω ω q εω ω q ε ω Whr q and q ar h orqs applid o h roors and ε ε ar posiiv consans rprsning dissipaiv ffcs. B singε ε q q h abov ssm can b simplifid as h following dnamical ssm: α α * Rsarch was sppord b h Naional Nar Scinc Fondaion of China No: 94 and Edcaion Fondaion of Jiangs Provinc No: 5KJB8 wd959@js.d.cn Pblishd b World Acadmic Prss World Acadmic Union

2 94 X. Wang: Fdbac conrol and snchroniaion of chaos for h copld dnamos dnamical ssm This ssm is diffrnc from h Lorn ssm and L ssm and Chn ssm. For his ssm in [] H.N.Agi sd h mhod of linar fdbac conrol and bond fdbac conrol and chosn h following conrol inp And < < < < Sch ha h chaos of h copld dnamos dnamical ssm can b conrolld o h qilibrim poin. Frhrmor h abov mhod was b sd and siabl conrol inp is chosn o sabili h nsabl qilibrim poins E E. A h sam im in ordr o spprss h chaoic bhaviors H.N.Agi s [] choic h following conrol inp f f sin ω. Addd i o h scond qaion so ha h chaos of h dnamical ssm is conrolld o h limi ccls. Th nmrical simlaions showd h ffc of h conrol. Th olin of his papr is as follows: Scion inrodcs h copld dnamos dnamical ssm and givs is propris; Scion inrodcs h modls and mahmaical srcr of h linar conrol for chaos; Scion b h simpl linar conrollr h copld dnamos dnamical ssm can b conrolld o a sabl priodic orbi and a sabl fid poin. Manwhil h sabili of h priod orbi and fid poin is provd b h vals of Lapnov ponn; Scion 4 compls h fficin Snchroniaion of h copld dnamos dnamical ssm b h nonlinar conrollr. Th nmrical simlaion rsls will prov h corrcnss of h simpl linar conrollr and nonlinar conrollr.. Th copld dnamos dnamical ssm Th copld dnamos dnamical ssm inrodcd b H.N.Agia s [] is a nonlinar dnamical ssm. Th ssm consiss of wo dnamos ssms conncd oghr so ha h crrn gnrad b an on of hm prodcs h magnic fild for h ohr. B simplificaion finall h copld dnamos dnamical ssm can b wrin as follows α α. Whr α and ar consan of h moion. Whn α i has a chaoic aracor as shown in Fig..Th drivaiv of h flow. is givn b F F F F <. Whr F F F F α α.thn ssm. is a forcd dissipaiv ssm similar o Lorn ssm. B h ar diffrn from. Ths h solions of h ssm. ar bondd as for posiiv vals of α and. B in a sns dfind b Vanc and Cliovs s [8] h Lorn ssm saisfis h condiion a a > whil h copld dnamos dnamical ssm saisfis h condiion a a < Hnc h copld dnamos ssm and h Lorn ssm ar diffrn ps of ssm. JIC mail for conribion: dior@jic.org.

3 Jornal of Informaion and Comping Scinc 6 pp Fig. Th chaoic aracor of h copld dnamos dnamical ssm. Mahmaical modls of linar fdbac conrol for chaos Considr an N -dimnsional ssm f λ. Whr λ λ λ λ ar h paramrs of ssm.. n m In h following discssion if h linar conrol inp is addd o h righ hand hs h ssm f λ. Can b considrd as a conrolld ssm if h nonlinar ssm. hav bn considrd as a chaoic ssm. B siabl choic h chaoic ssm. can b conrolld o a sabl priod orbi or a sabl fid poin. In his papr wih his ind conrolling modls compl h conrol of h copld dnamos dnamical ssm. 4. Linar fdbac conrol of h copld dnamos dnamical ssm Considr h conrolld copld dnamos dnamical ssm α α Whr is a conrol inp for h copld dnamos dnamical ssm. If h paramr hn h abov ssm is a chaoic ssm. If.65 hn h. can b rwrin as α α.65 dv Whr η.75 < I can b provd ha h ssm is a dissipaion ssm v d and h chaoic ssm. can b conrolld o a sabl priod orbi. Morovr according o h A.Wolf J.SwifH.Swinn and J.Vasanos[9] w can obain h Lapnov ponns λ of h dnamical ssm. ar and -.7rspcl.h Lapnov ponns graph as Fig..Ths h conrolld chaoic copld dnamos dnamical ssm. is a sabl priod orbi. B h nmrical.. JIC mail for sbscripion: info@jic.org.

4 96 X. Wang: Fdbac conrol and snchroniaion of chaos for h copld dnamos dnamical ssm simlaion w can obain h graph of h sabl priod orbi as Fig.. Fig. Th volion of h Lapnov ponn of h conrolld chaoic ssm Fig. conrolld sabl priodic orbi If w choic h conrollr as and hn add i o h firs qaion of ssm. w can obain h conrolld ssm as following Whn paramr α α.66 hn h ssm. can b rwrin as dv Whr η 4.94 < v d α.66 α Li h abov h ssm is a dissipaion ssm and h chaoic ssm. can b conrolld o a..4 JIC mail for conribion: dior@jic.org.

5 Jornal of Informaion and Comping Scinc 6 pp 9-97 sabl fid poin. According o h A.Wolf J.SwifH.Swinn and J.Vasanos[9] w can obain h Lapnov ponns λ of h dnamical ssm.4 ar and -.564rspcl.h Lapnov ponns graph as Fig.4.Ths h conrolld chaoic ssm.4 is a sabl fid poin. B h nmrical simlaion w can obain h graph of h sabl fid poin as Fig.5. Fig.4 Th volion of h Lapnov ponn of h conrolld chaoic ssm Fig.5 sabl fid poin 5. Snchroniaion of h copld dnamos dnamical ssm Considr h ssm of diffrnial qaions f 4. g 4. Whr n n R R f g : R n R n ar assmd o b analic fncions. L b solions o 4. and 4. rspcivl. Th solions ar said o b fficin compl snchroni if JIC mail for sbscripion: info@jic.org.

6 X. Wang: Fdbac conrol and snchroniaion of chaos for h copld dnamos dnamical ssm 98 c lim 4. Whr is a consan s [7]. If hn ar said compl snchroniaion s []. c c If w choic h driv ssm as α α 4.4 Consrc h rspond ssm as α α 4.5 Thn h rror bwn driv ssm and rspond ssm is 4.6 Choic h Lapnov fncion E 4.7 whn choic h conrol inp as 4.8 As long as < > >. Thn h drivaiv < E So h rror ssm is sabl. Fig.6 Tim sris of sa variabl and snchro rror JIC mail for conribion: dior@jic.org.

7 Jornal of Informaion and Comping Scinc 6 pp 9-99 Choic his pariclar choic will lad o h rror sas convrg o consan as im nds o infini and hnc h fficin snchroniaion is achivd. Nmrical simlaion rsls as Fig.6 Fig.7 and Fig.8. Fig.7 Tim sris of sa variabl and snchro rror 6. Conclsion Fig.8 Tim sris of sa variabl and snchro rror In his papr w s h simpl linar conrollr so h copld dnamos dnamical ssm is conrolld o a sabl priodic orbi and a sabl fid poin; b h nonlinar conrollr h fficin snchroniaion is JIC mail for sbscripion: info@jic.org.

8 X. Wang: Fdbac conrol and snchroniaion of chaos for h copld dnamos dnamical ssm achivd. Th Lapnov ponn vals and Lapnov fncion prov h corrcnss of h mhods ha sd in his papr. Acall as long as chos a siabl conrol inp and h compl snchroniaion bwn ssm 4.4 and 4.5 is also achivd. 7. Rfrncs [] H.N.Agia Conrolling chaos for h dnamical ssm of copld dnamos Chaos Solions and Fracals 4-5. [] G. Chn On som conrollabili condiions of chaoic dnamics conrol Chaos solions & Fracals [] G. Chn X. Dong From chaos o ordr: prspcivs mhodologis and applicaions Singapor: World Scini. Prss 998. [4] Xdi Wang Liin Tian Tracing conrol of chaos for h copld dnamos dnamical ssm [J] Chaos solions & Fracals [5] J Lü Conrolling ncrain Lü s ssm sing linar fdbac [J] Chaos solions & Fracals [6] F. Li Y. Rn X Shan Z. Qi A linar fdbac snchroniaion horm for a class of chaoic ssms [J]. Chaos solions & Fracals [7] L. L T. Zho S. Zhang Chaos snchroniaion bwn linarl chaoic ssms [J]. Chaos solions & Fracals [8] A. Van & S. Cliovs_ Conrol Ssms: From Linar Analsis o Snhsis of Chaos London: Prnic- Hall 996. [9] A. Wolf J. Swif H. Swinn and J. Vasano Drmining Lapnov ponns from a im sris [J]. Phsica D [] HN. Agia MT. Yassn Snchroniaion of Rosslr and Chn chaoic dnamical ssms sing aciv conrol Phs. L. A JIC mail for conribion: dior@jic.org.