From Fold to Fold-Hopf Bifurcation in IFOC Induction Motor: a Computational Algorithm

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1 Amrican Journal of Numrical Analysis,, Vol., No., Availabl onlin at Scinc and Education Publishing DOI:.69/ajna--- From Fold to Fold-Hopf Bifurcation in IFOC Induction Motor: a Computational Algorithm Nizar Jabli,*, Hdi Khammari, Mohamd Faouzi Mimouni Dpartmnt of Elctronics, Highr institut of applid scincs and tchnology, Souss, Tunisia Dpartmnt of computr scinc, Faculty of computr scinc, Taif, Saudi Arabia Dpartmnt of lctrical nginring, National nginring school of, Monastir, Tunisia *Corrsponding author: jabli.nizar@yahoo.fr Rcivd Sptmbr, ; Rvisd March 8, ; Accptd March, Abstract Analytical dtrmination of th bifurcation thrsholds is of important intrst for practical lctrical machin dsign and analysis control. This papr prsnts mathmatical invstigation of th qualitativ bhavior of indirct fild orintd control induction motor basd on bifurcation thory. In this contxt, stady-stat rsponss analysis of th motor modl is discussd and an analytical study of gnric bifurcations was mad. Particular attntion is paid to th codimnsion two bifurcation namly Fold-Hopf bifurcation. Th papr introducs som lmntary mchanisms of transit from Fold to Fold-Hopf paramtr singularity, to driv som analytical rigorous xistnc conditions and to dvlop an algorithm for Fold-Hopf bifurcation dtction. Som numrical rsults of quilibrium proprtis and bifurcation diagrams ar thn prformd to outlin our mthodology. Kywords: IFOC Induction Motor, stady-stat rsponss, bifurcation, Fold, Hopf, Fold-Hopf Cit This Articl: Nizar Jabli, Hdi Khammari, and Mohamd Faouzi Mimouni, From Fold to Fold-Hopf Bifurcation in IFOC Induction Motor: a Computational Algorithm. Amrican Journal of Numrical Analysis, vol., no. (): doi:.69/ajna---.. Introduction Applid mathmatics is an intrdisciplinary subjct, which provids a broad qualitativ and quantitativ background for us in lctrical nginring fild. In this study w focus on aspcts of thortical and numrical analysis which is rlatd to th bifurcation thory applid to th lctrical motors. Th hallmark of th nonlinar bhavior is th bifurcation, which is an abrupt qualitativ chang in th phas spac motion which is rfrs to th small variations of on or mor of systm control paramtrs. Th analysis of th bhavior of dynamical systms can b carrid out globally through bifurcation analysis, which dscribs th rang of paramtr valus at which qualitativ proprity changs occurs. Crossing a bifurcation point, xistnc and uniqunss of solutions is not guarantd and a chang in stability and/or ordr and/or th numbr of solutions occurs. In formr studis [,] thr typs of gnric codimnsion on bifurcations of priodic solutions of nonlinar ODE wr dscribd: tangnt, priod doubling and Hopf bifurcation. If an quilibrium point satisfis two bifurcation conditions, w call th singularity as codimnsion two bifurcations. A typical cas of codimnsion- bifurcation is th Fold-Hopf bifurcation (ZH) which is a transvrsal intrsction of fold and hopf bifurcation curvs at a bifurcation valu. Aiming to study th xistnc of such particular bifurcation point in induction motor submittd to an Indirct Fild-orintd control, analytical analysis ar usd to put into vidnc th mathmatical condition of thir xistnc. Such paramtric singularitis can b causd by th paramtr fluctuations namly th rrors in th stimat of th rotor tim constant which changs widly with tmpratur []. Fild-orintd controllrs FOC, frquntly usd as nonlinar controllrs for induction machins, prforms asymptotic linarization and dcoupling []. Stability of FOC is gnrally invstigatd rgarding rrors in th stimat of th rotor rsistanc. It has bn prviously shown that th spd control of induction motors through undr fild-orintd control (IFOC) is globally asymptotically stabl for any constant load torqu. An analysis of paramtr plan singularitis (saddlnod and Hopf bifurcations) in IFOC drivs with rspct to th rotor tim constant variation provids a guidlin for stting proprly th motor paramtrs in ordr to avoid such bifurcations [5,6]. In th rcnt yar, th qualitativ approach has bcom incrasingly usful tool in th analysis of nonlinar powr systms. Aiming to undrstand th bifurcation mchanism and thir associatd rsponss, on should idntify phas plan singularitis (quilibrium, limit cycls, attraction basins) and paramtr plan singularitis (bifurcations, chaos) [7]. In formr studis [8,9,], it has bn shown th occurrnc of ithr codimnsion on bifurcation such as saddl nod bifurcation and Hopf bifurcation and codimnsion two such as Bogdanov-Takns or zro-hopf bifurcation in IFOC induction motors. Th canclation of sustaind oscillations which ar, in gnral undsirabl was th purpos of othr studis which proposd th

2 7 Amrican Journal of Numrical Analysis 'oscillation killr' in ordr to adjust th systm and control paramtrs so that on can gt rid of limit cycls []. Othr rsults [] prmit to promot fficincy or improv dynamic charactristics of drivs. An adquat combination btwn analytical and numrical tools may provid a dp comprhnsion of som nontrivial dynamical bhavior rlatd to bifurcation phnomna in a slf-sustaind oscillator []. Th robustnss margins for IFOC of induction motors can b dducd from th analysis of th bifurcation structurs idntifid in paramtr plan []. Sinc th slf-sustaind oscillations in IFOC for induction motors may b du to th apparanc of a Hopf bifurcation [5], an xhaustiv study of th bifurcation structurs is mainly ddicatd to prsrv th local stability of th dsird quilibrium point. An outlin of th papr is organizd as follows. Th govrning quations of th IFOC induction motor and a gnral rmindr will b dscribd in sction. Th dtaild analytical analysis of fold and zro-hopf bifurcations will b proposd in sction, togthr with th numrical algorithm of ZH-bifurcation dtction. Som numrical simulations will b givn in sction and th conclusion in th sction 5.. Systm Equation Formulations.. Mathmatical Modl of IFOC Motor Mathmatically, th govrning quations usd for th modling of indirct fild-orintd control of induction motor can b dscribd as [9] by th following st of four nonlinar autonomous diffrntial quations: =. (. / ).. +. x c x kc u x x c u () =. + (. / ).. +. x c x k c u x x c x () 5 x = c. x c.( c ( x. x x. u ) T L ) () = i p p 5 L x ( k k. c ). x k. c.( c ( x. x x. u ) T ) () For this st of quations, x, x, x and x dnot th variabl stats. Prciously, x and x ar th dirct and th quadratic componnt of th rotor flux, rspctivly. x bing th diffrnc btwn rfrnc and th ral mchanical spd. Th last variabl x prsnt s th quadratic componnt of th stator currnt rsults from th outr loop PI. Th paramtr bifurcation k = τr τ is th ratio of th rotor tim constant τ r to its stimat τ and u is a dsign paramtr. k p and ki ar th proportional and th intgral PI controllr gains, rspctivly. W can also dfin th following constants c, c, c and c, whr: c = τ r = Lr. Rr is th rotor flux tim constant. c = Lm. τ r, L m is th mutual inductanc c = fc. τ r, f c is th friction constant. c = np. J, whr J is th momnt of inrtia and n p is th pol pair numbr. Finally, lt Γ= c TL + rf c ω b a constant whr T L dnots th load torqu... Analytical Stady- Stat Solution In numrical mthods or bifurcation problms, th phas plan analysis is an important tchniqu for studying th bhavior of nonlinar systms. Sinc, a twoparamtr plan can b considrd as mad up of shts as mntiond in prvious papr [], ach on bing associatd with a wll dfind bhaviour such as a fixd point, or an quilibrium or a priodic orbit. With chang in tmpratur or frquncy thr will b chang in th valus of th motor paramtrs. As th variation of ths paramtrs incrass, so will thir ffcts on systm dynamic bhavior. In particular, th rotor tim constant which may vary considrably ovr th oprational rang of th rotor rsistanc which changs widly with tmpratur. So, w choos in this study k = τr τ as th bifurcation paramtr. This paramtr is dfind as th dgr of tuning, i.., if k = th systm is considrd to b tund, othrwis it is said to b dtund [9]. Th purpos of th prsnt work is to giv an analytical invstigation of th influnc of small variation of th parmtr k on th dynamical bhavior of motor modl. A closd form analytical solution is dvlopd for th tund cas charactristic of th motor ( k = ). In addition to this tund phas, a considrabl simplification of quation modl ()-() may b achivd in th magntization phas of IFOC obtaind by putting T L = and ω rf =. It is asy to vrify that th quilibrium point t c c t x = ( x, x, x, x) = (, u,,) is rachd whil th c machin is in th standstill conditions. In prvious studis (s [5,6,7]) th quilibrium solutions of th systm of quations ()-() hav bn dtrmind undr gnral conditions. Mor importantly, it has bn provn that an quilibrium point is paramtrizd in trm of two dimnsionlss paramtrs ( r and ˆr ) and it has th following analytical xprssion: x r c cu k r x c x c cu kr = = c x x ur (5) Analytically, r prsnts any ral solution of th following rd ordr polynomial quation: kr rk ˆ r + kr rˆ = (6) In fact, r is a mathmatically trm rlatd to physical proprtis of th systm loading. This trm is dfind by th following xprssion: kr rk ˆ r + kr rˆ = (7)

3 Amrican Journal of Numrical Analysis 7 cccu 5 whr β is a constant dnotd by β =. c On th othr hand, th trm r is rlatd to th quadratur axis componnt of th stator currnt. This paramtr is dfind by th following simpl xprssion: c x r = (8) u In th prvious papr [8], it has bn provn that for any numrical paramtr k which vrifis th following proprty k, this polynomial has a uniqu ral solution. Whn k >, it has bn dmonstratd also that th numbr of quilibrium points varis undr th following two critria: rˆ rˆ = kr a a rˆ rˆ = kr Th xprsssions of rspctivly: b b ra a rb b r a and r b c (9) () ar dfind by, ra = k + ( k 9)( k ) () k rb = k ( k 9)( k ) () k Whn k >, th IFOC quations ()-() possss thr, two or on quilibrium point according to th fact that critria (9) and () ar strictly vrifid or not. If ithr (9) and () ar both vrifid with quality, thn th systm has quilibrium points. In any othr cas th systm has a uniqu quilibrium point... Mathmatic Rmindrs in Bifurcation Thory A composit pictur of th phas spac bhaviour can b gaind by studying th th ordr autonomous ODEs (- ) dscribing th IFOC induction motor (Equilibrium points, limit cycls, chaotic orbits). Each phasingularity involvs four ignvalus dscribing its stability. Gnrally, a local bifurcation at an quilibrium happns whn som ignvalus of th paramtrizd linar approximating diffrntial quation cross som critical valus such us th origin or th imaginary axis. Slfsustaind oscillations in IFOC of induction motors can b originatd by a codimnsion on bifurcation namly th Hopf bifurcation (H). Such kind of bifurcation can b computd from diffrntial systm ()-(), whn a pair of complx conjugat ignvalus among th ignvalus st of th associat linarizd systm chang from ngativ to positiv ral parts or vic vrsa. Thrfor th Hopf bifurcation rsults from th transvrsal crossing of th imaginary axis by a pair of complx conjugat ignvalus. Such bifurcation is said to b suprcritical if th priodic branch is initially stabl and subcritical if th priodic branch is initially unstabl. Th singular curvs of th paramtr plan corrsponding to codimnsion- bifurcations may contain singular points of highr codimnsion [8,9]. Th simplst on locatd on a Hopf curv has th codimnsion- singular point, a Fold-Hopf (ZH). This bifurcation, also calld th Zro-Hopf (ZH), is a codimnsion bifurcation of an quilibrium point at which th critical quilibrium lis at a tangntial intrsction of curvs of fold bifurcation curv and Andronov-Hopf bifurcation curv within th two paramtr family. Dfinition: Considr an autonomous systm of ordinary diffrntial quations (ODEs), n x = f( x, µ ), x () whr x dnots th stat vctor, µ = ( µ, µ ) is a - dimnsional paramtrs vctor and f is smooth. Suppos that at µ th systm has quilibrium x *. In th n-dimnsional cas with n, th Jacobian matrix J has: f. a simpl zro ignvalu = and a simpl pair of purly imaginary ignvalus, = ± iq, as wll as. ns ignvalus with R( j ) <, and. nu ignvalus with R( j ) >, with + ns + nu = n.. Analytical Bifurcation Analysis Local phas portrait nar singularitis can b vry complicatd. This phas-plan pictur is dvlopd by dtrmining th form of th solution of th th ordr autonomous ODEs ()-() dscribing th IFOC induction motor nar ach of its singularitis (quilibrium points EP, limit cycls LC). Each on of ths solutions involvs four ignvalus dscribing its stability. Evaluating th xprssion of Jacobian J F at th quilibrium point (5), w gt th following form: J F c ckr cδ ckr c cδkr c c = β βr c βδ c c c c c βk p βrk p ki k pc βδk p c ( k) ( + kr) whr γ = and γ =. Lt P dfin th charactristic quation of J F : () P = dt( J. I ) (5) F bing th ignvalu paramtr and I dnots a unit matrix of ordr. It is asy to vrify that P is a th dgr polynomial function:

4 7 Amrican Journal of Numrical Analysis P i= i i = ω (6) Th cofficints ω i ( i =,..., ) ar dfind as follows: c kk i ω = β σ (7) cc ( kr) c kk p c ki ω = + + β σ + β σ (8) c( ) c kp ki cc ω = + + β σ + βδ + (9) ω = βδ k + c + c () p ω = () whr σ and σ hav th following xprssions: + ( k ) r + σ = k( kr ) + k+ σ = () () W nxt giv th appropriat analytical conditions charactrising th occurnc of fold and fold-hopf bifurcations... Analytical Condition of Fold Dtction A Fold (F) bifurcation of quilibrium point, calld also Saddl-Nod or a limit point (LP) is a codimnsion on bifurcation which occurs whn a singl ignvalu of th charactristic polynomial is qual to zro. To chck whthr = is a solution of P =, simply vrify that th cofficint ω is qual to zro. Thus, announcd th following thorm using th quations (6) and (): Thorm : If thr xists a ral k ( k > ) and a ral r, solution of (6), for which th following condition is satisfid: + ( k ) r + = Thn, th Jacobian of systm ()-() prsnts a singl zro ignvalu and fold bifurcation can occur... Analytical Condition of Fold-Hopf Bifurcation (ZH) In th gnral cas, th ncssary analytical condition for th xistnc of a Zro-Hopf bifurcation (ZH) for th systm ()-(), at th quilibrium point (5), is that th quation P = has a simpl zro ignvalu = and on pair of purly imaginary roots,. Lt's solv th following four dgrs quation: P = ω + ω + ω + ω + ω = () Lt ω =, th condition of fold bifurcation occurnc. Thn th quation () bcoms: P =.( + ω + ω + ω) =. Q (5) Lt's considr th following third dgr polynomial function Q = a + a + a + a (6) Wr a =, a = ω, a = ωand a = ωar th ral cofficints. Examing th zro of -dimnsional polynomial Q. Lt th roots of Q b dnotd µ, µ and µ, thn th factord form of th polynomial is givn by: Q = a.( µ ).( µ ).( µ ) (7) Expanding th plynomial factors, w asily obtain th following rlations: µ + µ + µ = ω a µ. µ + µ. µ + µ. µ = = ω a a µ. µ. µ = = ω a (8) Th gnral form of th discriminant formula $\dlta$ of a n-dimnsional polynomial is givn by: δ = a µ µ (9) n n. ( i j) i< j Whr µ i, j( i < j n) ar th roots of th polynomial. For n =, th discriminant of Q is dfind as follows:.( ).( ).( ) δ = a µ µ µ µ µ µ () Th discriminant δ can b xprssd in trm of th cofficints µ i ( i = : ) as follows: 8 7 δ = aa + aaaa aa aa aa () Finally, 8 7 a δ= ω + ωωω ω ω ω () Whn dos a ral polynomial hav a pair of complx conjugat roots?... Root Typs Computation of th purly imaginary ignvalus basd on lmntary proprtis of cubic polynomials, w rport on thr cass of root typs: Cas : µ, µ and µ ar distinct ral roots. In which cas, δ can b writtn as: δ = ( a.( µ µ ).( µ µ ).( µ µ )) It is asy to vrify that th discriminant δ is positiv. Cas : Existnc of two qual ral roots. W can find thr possibl doubl roots. µ µ =, µ µ =, µ µ = Sinc th discriminant is zro: δ = ( a.( µ µ ).( µ µ ).( µ µ )) =

5 Amrican Journal of Numrical Analysis 7 Cas : Two complx conjugat roots. By considring th symmtric proprtis, lt ( µ, µ ) dnot th two complx conjugat roots and µ dnots th ral root of Q. Roots of th polynomial can b writtn in th following forms: µ = r + iq µ = r iq () µ = s Thn th discriminant δ can b calculatd as:.( ).( ).( ) δ = a µ µ µ µ µ µ thn, δ = qa (( r s) + q ). In this cas, th discriminant δ is ngativ. W rport all rsults on root typs in th following Lmma: Lmma : If δ >, thn Q has thr ral roots. If δ =, thn Q has a doubl root. If δ <, thn Q has has a pair of complx conjugat roots.... Computation of th Purly Imaginary Eignvalus In th cas of δ <, Q has on pair of complx conjugat roots. Th Hopf bifurcation point can b dtctd whn only on pair of th roots of Q hav a zro ral parts. This condition can b dducd gnrally from th gnralizd Routh-Hurwitz critrion applid to th charactristic polynomial (6) [9]: ω ω ω ω. ω ω γ = ω ω. ω ω if γ = = thn on pair of purly imaginary ω roots can b dtctd. Proof: Lt th purly imaginary roots of Q b dnotd µ = iq and µ = iq, wr q is a non-zro ral numbrs. Each of th two roots ( µ, µ ) satisfis th givn rlation: and As a rsult, Q ( µ ) = Q ( µ ) = () Q ( iq ) = iq ω q + iω q + = (5) Q ( iq ) = iq ω q iω q + = (6) Subtraction of th quation (5) from th quation (6) givs: iqω iq = (7) Thn q ω = (8) Addition of two quations (5) and (6) givs: Thn ωq ω = (9) q ω = () ω Th quality btwn two xprssions (8) and () ω givs: ω ω =, w thn obtain ω. ω ω =. Th particular roots of Q ar:. ω is ral positiv numbr.. Two purly imaginary roots µ = µ = i ω. On ral root µ = ω.... Th conditions of Fold-Hopf Bifurcation Now, our main rsults, with rspct to th abov conditions, ar announcd in th following thorm. Thorm: If thr xists a ral k ( k > ), r and µ for which th following conditions ar satisfid: (ZH. ) Occurrnc tst: δ <, (ZH. ) On zro root for: + ( k ) r + =, (ZH. ) On pair of purly imaginary roots for: c ( + ) + cβkpσ + +. βδki cc + + ( βδkp c c) ( cc kr cβkk p σ cβkiσ) ( + ) + + = Th Jacobian of systm ()-() prsnts on zro ignvalu in addition of on pair of purly imaginary ignvalus and a Fold-Hopf bifurcation can b dtctd... Fold-Hopf Bifurcation Dtction Algorithm Lt s now dfin th following numrical algorithm of ZH dtction basd on th announcd thorm and in light of th obtaind rsults discussd abov. Th proposd algorithm includs 7 stps dtaild as follows: Stp : St appropriat initial paramtr valus starting out from th tund cas k = and th quilibrium point (,,, c x x x x ) = (, u c,,). c Stp : Idntify th cofficints ωi of th polynomials xprssions P. Stp : Computing th discriminant δ of Q, tsting if δ <. Stp : vrify th condition ω =. Stp 5: Vrify th condition ωω ω =. Stp 6: Th Zro-Hopf bifurcation is dtctd with ignvalus = = i. ω, = and is a non zro ral numbr.

6 7 Amrican Journal of Numrical Analysis Stp 7: Updat paramtr, chang k by k + h whr h is an appropriat small numbr. Go to stp.. Numrical Simulations Th Zro-Hopf bifurcation mchanism s gnration from an quilibrium point, is illustratd by th graph of Figur. For th paramtrs k =, k =, k p and T =.5 an quilibrium point (EP) is idntifid. L Th EP vctor is ( x *, x *, x *, x * ) = (.6, -.76,.89,.95), solution of th diffrntial systm () for th initial conditions st ( x, x, x, x ) =(,.5,, ). i Th continuation of th Hopf point (H) lads to trac th Hopf bifurcation curv shown in Figur. Such curv includs a -codimnsion Zro-Hopf bifurcation point in (.57, -.57,,.8) for th paramtrs k =.779 and T L =.965. Th corrsponding ignvalus of this singularity ar = (.86, i.859, - i.859, ). In fact, th ZH-bifurcation point in th paramtrs ( kt, L ) -plan is a critical point at which th critical quilibrium lis at a tangntial intrsction of fold bifurcation curv ζ F and Hopf bifurcation curv ζ within th two paramtr family (s Figur ). H Figur. Equilibrium Point: bifurcation procss Starting from this locatd initial quilibrium, a continuation mthod prmits to obtain th volution of th dirct compound flux x vrsus th valus of bifurcation paramtr k (s Figur ). On Hopf bifurcation is obtaind in such curv. This singularity has on pair of purly imaginary ignvalus and th following coordinats in phas spac: (.69, -.7,, and.) for k =.5. Th corrsponding ignvalus ar (-.6, -.96, i.96, - i.96). Figur. Fold-Hopf bifurcation dtction For th paramtrs valus k =. and T L =.9, th Fold curv ζ F includs two branchs joining in a codimnsion two bifurcation points, namly cuspidal point (CP) having th following phas spac coordinats ( x, x, x, x ) = (., -.57,,.577). Bsids, such curv prsnts a Bogdanov-Takn bifurcation (BT) in ( x, x, x, x ) = (.6, -.77,,.89) for th paramtrs valus k =.585 and T L =.67, with th associats ignvalus = ( i.77, i.77,, ) Figur. Bifurcation diagram x x Figur. Hopf bifurcation curv Figur 5. Phas plan trajctory at Fold-Hopf bifurcation Th procss of convrgnc of phas trajctory in th ( x, x ) plan shows a critical quilibrium at th Zro-

7 Amrican Journal of Numrical Analysis 75 Hopf bifurcation point (s Figur 5). Th rd points prsnt th gnratd limit cycl. 5. Conclusion This papr applis bifurcation and singularity analysis to study th complx dynamic bhavior of IFOC induction motor. Th computd stady stat is shown to b adquat for low codimnsion bifurcation studis. Particularly, th papr clarly proposs an analytical analysis of fold and fold-hopf (ZH) bifurcations. This mathmatical approach yilds th thortical tst conditions undr which th xistnc of such bifurcation singularitis is vrifid and that can b solvd analytically. Th papr also introducs a computational algorithm for th dtction of fold-hopf bifurcation using dvlopd conditions. Finally, bifurcation curvs that xhibit fold (F), hopf (H) and foldhopf (ZH) bifurcations for various paramtrs ar tracd through numrical continuation mthods. Acknowldgmnt This work was supportd by ESIER-US, Industrial systms and Rnwabl Enrgy Unit of Sarch, Tunisia. Rfrncs [] H. Kawakami. Bifurcations of priodic rsponss in forcd dynamic non linar circuits, computation of bifurcation valus of th systm paramtr, IEEE trans. circuits and systms, cas-, pp 8-6, 98. [] H. Khammari, C. Mira and J.P. Carcass` s. Bhaviour of harmonics gnratd by a Duffing typ quation with a nonlinar damping, Part I. Intrnational Journal of Bifurcation and Chaos, Vol. 5, No., 8-, 5. [] K. Yahyia, S. zouzou, F. Bnchaban. Indirct vctor control of induction motor with on lin rotor rsistanc idntification, Asian Journal of Information and Tchnology Volum (5), 6, pp. -5. [] Sok Ho Jon, Dan Baang, and Jin Young Choi. Adaptiv Fdback Linarization Control Basd on Airgap Flux Modl for Induction Motors, Intrnational Journal of Control, Automation, and Systms, vol., no., pp. -7, August 6. [5] A. S. Bazanlla R. Rginatto. Instability Mchanisms in Indirct Fild Orintd Control Drivs: Thory and Exprimntal Rsults, IFAC 5th Trinnial World Congrss, Barclona, Spain,. [6] A.S. Bazanlla, R. Rginatto and R. Valiatil. On Hopf bifurcations in indirct fild orintd control of induction motors: Dsigning a robust PI controllr,, Procdings of th 8th Confrnc on Dcision and Control, Phonix, Arizona USA, Dcmbr, 999. [7] C. Mira, J. P. Carcass` s, C. Simo, J. C. Tatjr. Crossroad araspring ara transition. (II) Foliatd paramtric rprsntation, Int. J. Bifurcation and Chaos (), 9-8, 99. [8] R. Rginatto, F. Salas, F. Gordillo and J. Aracil. Zro-Hopf Bifurcation in Indirct Fild Orintd Control of Induction Motors, First IFAC Confrnc on Analysis and Control of Chaotic Systms: CHAOS 6, 6. [9] F. Salas, R. Rginatto, F. Gordillo and J. Aracil. Bogdanov- Takns Bifurcation in Indirct Fild Orintd Control of Induction Motor Drivs, rd IEEE Confrnc on Dcision and Control, Bahamas,. [] F. Salas, F. Gordillo, J. Aracil and R. Rginatto. Codimnsion-two Bifurcations In Indirct Fild Orintd Control of Induction Motor Drivs, Intrnational Journal of Bifurcation and Chaos, 8 (), , 8. [] C. C. d-wit, J. Aracil, F. Gordillo and F. Salas. Th Oscillations Killr: a Mchanism to Eliminat Undsird Limit Cycls in Nonlinar Systms, CDC-ECC 5, Svill, Spain, 5. [] B. Zhang, Y. Lu and Z. Mao. Bifurcations and chaos in indirct fild orintd control of induction motors, Journal of Control Thory and Applications,, 5-57,. [] A. Algaba, E. Frir, E. Gamro and A. J. Rodriguz-Luis. Analysis of Hopf and Takns Bogdanov Bifurcations in a Modifid van dr PolDuffing Oscillator, Nonlinar Dynamics 6: 69, 998. [] A. S. Bazanlla and R. Rginatto. Robustnss Margins for Indirct Fild- Orintd Control of Induction Motors, IEEE Transaction on Automatic Control, 5, 6-,. [5] F. Gordillo, F. Salas, R. Ortga and J. Aracil. Hopf Bifurcation in indirct fild-orintd control of induction motors, Automatica, 8, 89-85,. [6] N. Jabli, H. Khammari, M.F. Mimouni, An Analytical Study of Low- Codimnsion Bifurcations of Indirct Fild-Orintd Control of Induction Motor. Intrnational Journal of Mathmatical Modls and Mthods in Applid Scincs, Vol., Issu. [7] N. Jabli, H. Khammari, M.F. Mimouni. An analytical study of bifurcation and nonlinar bhavior of indirct fild-orintd control induction motor. 6 th WSEAS Intrnational Confrnc on dynamical systms & control, CONTROL, Souss, May -6,. [8] N. Jabli, H. Khammari, M.F. Mimouni. Bifurcation scnario and chaos dtction undr variation of PI controllr and induction motor paramtrs. 9 th WSEAS Intrnational Confrnc on Non- Linar Analysis, Non-Linar Systm and Chaos, NOLASC, Souss, Tunisia, -6 Mai. [9] N. Jabli, H. Khammari, M.F. Mimouni, Multi-Paramtr Bifurcation Analysis of a Thr-Phass Induction Motor. Intrnational Rviw of Elctrical Enginring, IREE, volum 5, Issu,.