Hopf Bifurcation and Sliding Mode Control of Chaotic Vibrations in a Four-dimensional Hyperchaotic System

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1 Hopf Bfurcaon an Slng Moe Conrol of Chaoc Vbraons n a Four-mensonal Hyperchaoc Sysem Wen-ju Du, Jan-gang Zhang an Shuang Qn Absrac he basc ynamc properes of a four mensonal hyperchaoc sysem are nvesgae n hs paper More precsely, he sably of equlbrum pon of hyperchaoc sysem s sue by means of nonlnear ynamcs heory We analyses he exsence an sably of Hopf bfurcaon, an he formulas for eermnng he recon of Hopf bfurcaon an he sably of bfurcang peroc soluons are erve In aon, a slng moe conroller s esgne an conrolle he hyperchaoc sysem o any fxe pon o elmnae he chaoc vbraon by means of slng moe meho Fnally, he numercal smulaons were presene o confrm he effecveness of he conroller Inex erms Sably, Lyapunov exponens, Hopf bfurcaon, Slng moe conrol I INRODUCION he scovery of he emnen Lorenz sysem [1] has le o an exensve suy of chaoc behavors n nonlnear sysems ue o many possble applcaons n scence an echnology here s a huge volume of leraure evoe o he suy of he nonlnear characerscs an basc ynamc properes of chaoc sysem [] An he nonlnear ynamcs an chaos heory has been n-eph researche urng he las ecaes [] Despe he smplcy of four-mensonal auonomous sysems, hese sysems have a rch ynamcal behavor, rangng from sable equlbrum pons o peroc an even chaoc oscllaons, epenng on he parameer values Moreover, he research an applcaon on bfurcaon of auonomous sysems has become a very popular opc [-9] Over he pas few ecaes, more an more chaoc phenomena have been foun n many research fels an can be wely use n secure communcaon, nformaon processng, nonlnear crcus, bologcal sysems, an chemcal reacons Many scholars pa grea effor o generae chaos an analyze s ynamc characerscs Das an Mello [1] sue he Manuscrp receve Ocober, 15; revse December 6, 15 hs work s suppore by he Naonal Naural Scence Founaon (No111617, No6161) Wen-ju Du s wh he School of raffc an ransporaon, Lanzhou Jaoong Unversy, Lanzhou, Chna (phone: ; fax: ; e-mal: uwenjuok@16com) Jan-gang Zhang, Shuang Qn was wh Deparmen of Mahemacs, Lanzhou Jaoong Unversy, Lanzhou, Chna (e-mal: zhangjg @16com, qnshuangok@16com) nonlnear ynamcs of a Lorenz-lke sysem Soomayor e al [11] use he projecon meho escrbe n [1] o calculaon of he fs an secon Lyapunov coeffcens assocae o Hopf bfurcaons of he Wa governor sysem, an was exene o he calculaon of he hr an fourh Lyapunov coeffcens Zhang e al [1] presene a new hree-mensonal auonomous chaoc sysem an nvesgae s basc ynamc properes va heorecal analyss an numercal smulaon Jana e al [1] sue he sably an Hopf bfurcaon for a harvese preaor-prey sysem whch ncorporaes feeback elay n prey growh rae In recen years, he research of robus conrol sysem has mae conserable progress an evelopmen n heory an praccal applcaon As a represenave of he nonlnear robus conrol heory, varable srucure conrol heory has been wely researche aroun he worl, an also has an ncreasng number of nusral applcaons Lee e al [15] presene a slng-moe conroller wh negral compensaon for a magnec suspenson balance beam sysem, an he conrol scheme comprses an negral conroller whch s esgne for achevng zero seay-sae error uner sep surbances akuro e al [16] apple he slng moe conrol o acheve he robus conrol of space robo n capurng operaon of he arge an conrollng he spacecraf moon uner unknown parameers, lke mass an nera ensor Chen e al [17] propose a no-chaerng slng moe conrol sraegy for a class of fraconal-orer chaoc sysems, an he esgne conrol scheme guaranees he asympocal sably of an unceran fraconal-orer chaoc sysem o ensure he robusness of he sysem conrol, Chen e al sablze he chaoc orbs o arbrary chosen fxe pons an peroc orbs by means of slng moe meho an hey presene numercal smulaons o confrm he valy of he conroller [18] Chen e al [19] elmnae he chaoc vbraon of hyro-urbne governng sysem by usng he slng moe meho, an conrolle he sysem o any fxe pon an any peroc orb In hs paper, we conser a novel four-mensonal hyperchaoc sysem whch propose by Gao [] He jus analyze he sably of equlbrum, such as he phase agram of aracors, he bfurcaon agram an Lyapunov exponen However, he Hopf bfurcaon an chaos conrol of he four-mensonal hyperchaoc sysem has no been clarfe ye So, n hs paper we nvesgae he bfurcaons an slng moe conrol of chaoc vbraons of he novel four-mensonal hyperchaoc sysem he res of hs paper s organze as follows In secon, he escrpon of he moel s presene he lnear analyss of equlbra an he exsence of Hopf bfurcaon a equlbrum are nvesgae n secon In secon, we

2 analyze he recon of Hopf bfurcaon an he sably of bfurcang peroc soluons he numercal smulaons are gven o llusrae he heorecal analyss n secon 5 An n secon 6, we conrolle he sysem o any fxe pon an any peroc orb o elmnae he chaoc vbraon by means of slng moe meho Secon 7 conclues he paper II DESCRIPION OF HE MODEL In hs paper, we nvesgae a four-mensonal hyperchaoc sysem as follows: x ax by, y ax xz y u, (1) z xy c( x z), u mx, where ( x, y, z, u) R are sae varables, a,b,c, m are real consans he sysem (1) has a hyperchaoc aracor when he real consans a, b 5, c 5, m, as show n Fg 1 (a) Moreover, he ynamcs of he sysem (1) can be characerze wh s Lyapunov exponens whch are compue numercally by Wolf algorhm propose n [1], where he Lyapunov exponens: 1 =77, =198, =, = 6, as show n Fg 1 (b), an he Lyapunov menson D KY =65 Fg 1 (c) an Fg 1 () shows he me hsory an frequency specrum of hyperchaoc aracor, respecvely Fg 1 (a)phase rajecory n -D space, (b)lyapunov-exponen specrum, (c)me hsory, () Frequency specrum III SABILIY ANALYSIS In hs secon, we suy he sably of equlbrum an he exsence of Hopf bfurcaon In a vecoral noaon whch wll be useful n he calculaons, sysem (1) can be wren as x ( x, ζ ), where x ( x, y, z, u) x f ( x, ζ ) ( ax by, ax xz y u, xy c( x z), mx), R an ζ ( a, b, c, ) R By solvng he followng equaons smulaneously ax by, ax xz y u, xy c( x z), mx, () we ge he sysem has a unque equlbrum E (,,, ) Lemma 1 he polynomal L( ) p p p wh 1 real coeffcens has all roos wh negave real pars f an only f he numbers p, p, p are posve an he 1 nequaly p 1 p p s sasfe We have he followng proposon Proposon 1 he equlbrum E s unsable f m m If a 1, mb, a(1 b), c an a( a 1)(1 b) mm, () b hen he equlbrum E s asympocally sable Proof he Jacoban marx a he fxe pon E s gven by a b a 1 1 A, (5) c c m an s characersc polynomal s p( ) ( c)[ +( a 1) +( a ab) mb], (6) Accorng o Lemma 1, he equlbrum E s unsable f m m An f he real pars of all he roos of equaon (6) are negave f an only f c, a 1, mb, a(1 b), m m, So he proposon follows ()

3 Proposon Assume ha a, b, c If equaon (6) has a par of purely magnary roos 1, an Re( ( m )), hen he Hopf bfurcaon occurs a he m pon E when he bfurcaon parameer m pass hrough he crcal value m Proof Le ( ) s a roo of Eq (6), we have ( a 1) ( a ab) mb, (7) hen separang he real an magnary pars of equaon (7), an we ge a ab ( ), (8) ( a a) + mb= hrough calculaon, we have a( a 1)(1 b) a ab, m m, (9) b an he followng four characersc roos, c, ( a c), (1) 1, ake he ervave of boh ses of Eq (6) wh respec o m, we oban b, (11) m ( a 1) ( a ab) an Re b, m (a ab 1) mm Im a1 m (a ab 1) mm (1) Assume ha a, b, c, when m passes hrough he crcal value m, he sysem (1) occurs Hopf bfurcaon a he equlbrum E (,,, ) IV HOPF BIFURCAION ANALYSIS In hs secon, we suy he recon an sably of Hopf bfurcaon uner he conon a, b, c an m m Usng he noon escrbe n [1], he mullnear symmerc funcons corresponng o f can be wren as B x y x y x y x y x y (, ) (, 1 1, 1 1,), C x y z (,, ) (,,,), (1) he egenvalues of A are, c, ( a c), (1) 1, Le p, q C be vecors such ha Aq q, A p= p, p, q pq 1, (15) where A s he ranspose of he marx A, an by calculae we ge a c c q,,, m bm mc ( ) 1 m ( a 1) m( a 1+ ) bm bm( a 1) p,, (16) ( a1) ( a1) bm( a 1) bm, ( a 1) c c ( a ) B( q, q),,,, m ( c ) bm c a B( q, q),,,, m ( c ) bm a c,,,, ( ) 11 bcm m c h h E A B q q 1 ( ) (, ) ( h1 h, h h, h5 h6, h7 ), where k1k kk kk k1k k5 k6 h1, h, h, h, k k k k k k h 6 bm c k k bm c k k h 7 m c a a bm ab h (17) (18) (19) () ( k k k k ) ( k k k k ),, ( )( ) ( )( ) b( c ), ( )( ) k bc (c 1)( ab a) b ( c ), 1 k b ( c )( a ab) b (c 1)( a 1), 5 k m ( c )( r r r r ), k 8 ( r r r r ), k m ( c )( r r r r ), k 8 ( r r r r ), k m ( c )(a b 8a b 16a a 7 8abm ab b m 8bm 6 16 ), 6 k r ( ac ) r ( c a), k r ( ac ) r ( c a), r a ab, r ( a 1) c( ab a ), 1 r ab a, r c ( a 1) ( ab a ), r ( ac ), r a ab, 5 7 r ( c a), r bc bc ( a ab ), 6 9 r a bm, r bc ( ab a) bc ( a 1), hrough rec calculaon, we also has a B( q, h11) (,,,), bcm B( q, h ) (, n n, n n,), 1 a H n n n n bcm (1) 1 (, 1 ( ),,), () bm a G1 ( a 1)( n ) n1 ( a 1) bcm () bm a n 1( a 1) ( n ), ( a 1) bcm where h1c hc h6 ( c ) hb h1 ha n1, n, mc ( ) bm n h c h c h ( c ) h b h a h, n mc ( ) bm heorem 1 Conser he four-parameer famly of fferenal equaons (1) he frs Lyapunov coeffcen assocae o he equlbrum E s gven by ( a 1)( bcm n a ) bcm n l ( a, b, c) 1 1 bcm [( a 1) ] ()

4 If l 1 s fferen from zero, hen sysem (1) has a ransversal Hopf pon a E V NUMERICAL EXAMPLE Nex, we gve a numercal example of Hopf bfurcaon Le a, b 1, an by compue we ge he crcal value m 1 he equlbrum s sable when m 1 m an unsable when m 1 m, as show n Fg From he formulas n prevous secon, we have l hus, he peroc soluon bfurcang from E s supercrcal an sable Fg Nonlnear ynamcs of sysem (1) for specfc values a, b 5, c 5 versus he conrol parameer m (a) bfurcaon agram of x ; (b) Lyapunov exponen specrum Fg he sable regon on he parameer plane ( am, ) Fg Phase agram of sysem (1) wh (a) a, b 1, c 5, m 1, (b) a, b 1, c 5, m 1, (c) a, b 1, c 5, m 1 he bfurcaon phenomenon can be eece by examnng graphs of x versus he conrol parameer m for sysem (1) We fxe a, b 5, c 5 an whle m vares on he nerval[5,7], he bfurcaon agrams an corresponng Lyapunov exponen specrum, as show n Fg Obvously, wh he ncrease of he parameer m, he sysem s unergong some represenave ynamcal roues, such as chaos, pero-oublng bfurcaons an peroc loops Fxe he parameers b5, c 5, an we can ge he characersc polynomal of he Jacoban marx of sysem (1) a E s p( ) ( 5)[ +( a 1) a 5 m], (5) he equlbrum E s asympocally sable f 1 a, m, a( a 1) 5m an he sysem (1) has a ransversal Hopf pon a E f 1 a, m, a( a 1) 5m Le a 1, a, 5m, a( a 1) 5m, an raw he sably regon on he parameer plane am, - as show

5 n Fg In he fgure, he symbol L, 1,,, represens a 1, a,5m an a( a 1) 5m, respecvely he Hopf bfurcaon conons are sasfe on he curve L In regon (Ⅰ), we have m, a( a 1) 5m 1 a, an all pons are sable, bu n oher regons he pons are unsable Fg 5 he sable regon on he parameer plane ( bm, ) Fxe he parameers a, c 5, an he characersc polynomal of he Jacoban marx of sysem (1) a E s p( ) ( 5)[ +1 +(1 b) mb], (6) he equlbrum E s asympocally sable f b 1, mb, (1 b) mb, an he sysem (1) has a ransversal Hopf pon a Ef b 1, mb, (1 b) mb Le b 1, mb, (1 b) mb, an use MALAB o raw he sably regon on he parameer plane bm, - as show n Fg 5 he symbol L,,, represens b 1, mb an (1 b) mb he Hopf bfurcaon conons are sasfe on he curve L In regon (Ⅰ), we have b 1, mb, (1 b) mb an all pons are sable, bu n oher regons he pons are unsable VI SLIDING MODE CONROL OF CHAOIC VIBRAIONS 61 he esgn of he conroller We esgne a slng surface wh goo naure an mae he sysem possess he esre properes when make he sysem lms on he slng surface In orer o faclae conrol, we make he sysem reach he slng surface an keep slng Afer jonng he conroller, he sysem (1) has he followng form x x 5 y 1 u1, y x xz y u u, (7) z xy 5( x z) u, u x u where u 1, u, u an u are conrol npus We can conrol he chaos o he requre range or a fxe pon f we jon a reasonable conroller Defne he followng marx A, B, xz, g, xy where A s he lnear marx of he sysem, B s he conrol marx, s he boune perurbaon marx,an g s he nonlnear marx of he sysem he conrol goal s o le he sysem s sae x x, x, x, x 1 sae x x, x, x, x 1 rackng error rackng a me-varyng So, we can efne he followng e x x, (8) he error sysem can be wren as e x x Ax Bg Bu x, (9) Defne a me-varyng proporonal negral slng moe surface S Ke K(A - BL)e (), () where K R,e( KB ) o faclae he calculaon, we le K ag(1,1,1,1) he aonal marx L R, an A BL s negave efne marx Uner he slng moe, he equaon S S mus be sasfe, where S KBg KBLe KBu K KAx Kx, (1) o mee he slng conons, he followng conroller s esgne 1 u g Le ( KB) KAx Kx () 1 ( KB) KBg sgn ( S), where sgn( S) s sgn funcon Proposon [17] Assume ha he consan sasfe he nequaly 1 1, where 1, are arbrary small posve numbers hen he sysem (7) can reach he slng moe S n a lme me uner he conroller (), an he sae varables an he selece reference sae x are encal Proof Consruc he Lyapunov funcon V S S S, 1 accorng o (), (1) an () one has S S S KBg KBLe KBu K KAx Kx S sgn S S K KBg sgn( S) ( ) S S S S 1 1 By he same oken, we ge, V S S So he proposon follows 1 1 S

6 6 he numercal smulaon In he case ofu 1 u u u, he me-oman chars of he sae varables of sysem (7) as show n Fg 6 Fg 6 llusraes ha he sysem (7) has an aperoc moon sae before conrol In orer o conrol he sysem (7) o he arge sae, we selec he egenvalue of P 5, 5, 5, 5 he A BL are pole-placemen meho s aope o ge he followng marx L () 5 5 () me oman char of u before conrol Fg 6 me oman chars of sae varables before conrol Selec he proporonal negral slng moe surface as follows: S1 e1 5 e 1( ), S e 5 e ( ), () S e 5 e ( ), S e 5 e ( ), (a) me oman char of x before conrol (b) me oman char of y before conrol (a) me oman char of x afer conrol (c) me oman char of z before conrol (b) me oman char of y afer conrol

7 (c) me oman char of z afer conrol (b) me oman char of S afer conrol () me oman char of u afer conrol Fg 7 me oman chars of sae varables afer conrol Se he nal value (), (), (), () 1,1,1,1 x x x x, 1 an he reference sae x1 x x x x Followng s he conrol sgnal u1 15e1 5e 15 x x sgn( S1), u xz e1 e e 18x x xz sgn( S), u xy 5e1 1 x x xy sgn( S), u e1 5e x x sgn( S) (5) (c) me oman char of S afer conrol (a) me oman char of S1 afer conrol () me oman char of S afer conrol Fg 8me oman chars of slng surfaces afer conrol 6 Conrol o he fxe pon We can sablze he sysem (7), an le he sysem s sae o reach any pon by hs meho In hs paper, we selec he fxe pon 1, 1, 1, 1, reference sae x 1, small parameer an he nal value of slng moe S (), S (), S (), S () 1,1,1,1 We surface 1 acvae he conroller u() a 1s, an ge he me oman chars of sae varables an slng surfaces as show n Fg 7 an Fg 8, respecvely

8 he Fg 7 an Fg 8 ncae ha he sysem (7) rack o he reference sae 1,1,1,1 ulmaely an he slng moe surface S become zero afer jon he conroller I s proves ha he sysem (7) reache he slng moe 6 Conrol o he peroc orb We can also sablze he sysem (7), an le he sysem s sae o reach a peroc orb We selec he reference sae x sn( ) hen acvae he conroller u() a 1s, an we ge he me oman chars of sae varables as show n Fg 9 Obvously, he sysem (7) racks o reference sae x sn( ) o he peroc orb ulmaely () me oman char of u afer conrol Fg 9 me oman chars of sae varables afer conrol (a) me oman char of x afer conrol (b) me oman char of y afer conrol (c) me oman char of z afer conrol VII CONCLUSION he paper nvesgae he basc ynamc characerscs of a new hyperchaoc sysem Frs, he exsence an local sably of he equlbrum are scusse hen, we choose m as he bfurcaon parameer an sue he exsence an sably of Hopf bfurcaon of he sysem by usng he cener manfol heorem an bfurcaon heory In aon, n orer o elmnae he chaoc vbraon, we use slng moe meho an conrolle he sysem o any fxe pon an any peroc orb Numercal smulaon resuls show ha he hyperchaoc sysem (1) occurs Hopf bfurcaon when he bfurcaon parameer m passes hrough he crcal value, an he recon an sably of Hopf bfurcaon can be eermne by he sgn of l 1 hen he slng moe meho can make he sysem rack arge orb srcly an smoohly wh shor ranson me Apparenly here are more neresng problems abou hs chaoc sysem n erms of complexy, conrol an synchronzaon, whch eserve furher nvesgaon REFERENCES [1] E N Lorenz, Deermnsc nonperoc flow, Journal of he amospherc scences, vol, no, pp 1-11, Nov 196 [] C Sparrow, he Lorenz Equaons: Bfurcaon, Chaos, an Srange Aracors, n Sprnger Scence & Busness Mea, frs e vol 1, Sprnger-Verlag, 175Ffh Avenue, New York, New York 11, USA, 1, pp 6-9 [] E O, Chaos n ynamcal sysems, n Cambrge unversy press, n e, he P Bulng, rumpngon Sree, Cambrge, Une Kngom,, pp [] H L, Dynamcal analyss n a D hyperchaoc sysem, Nonlnear Dynamcs, vol 7, no, pp 17-1, Oc 1 [5] K Zhang, Q Yang, Hopf bfurcaon analyss n a D-hyperchaoc sysem, Journal of Sysems Scence an Complexy, vol, no, pp , Aug 1 [6] X F L, K E Chlouveraks, D L Xu, Nonlnear ynamcs an crcu realzaon of a new chaoc flow: A varan of Lorenz, Chen an Lü, Nonlnear Analyss: Real Worl Applcaons, vol 1, no, pp 57-68, Aug 9 [7] L F Mello, M Messas, DC Braga, Bfurcaon analyss of a new Lorenz-lke chaoc sysem, Chaos, Solons & Fracals, vol 7, no, pp 1-155, Aug 8 [8] F M Amaral, L F C Albero, Sably regon bfurcaons of nonlnear auonomous ynamcal sysems: ype-zero sale-noe bfurcaons, Inernaonal Journal of Robus an Nonlnear

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