INTERNATIONAL JOURNAL OF ELECTRICAL ENGINEERING & TECHNOLOGY (IJEET)

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1 INTERNATIONAL JOURNAL OF ELECTRICAL ENGINEERING & TECHNOLOG (IJEET) Internatonal Journal of Electrcal Engneerng and Technology (IJEET), ISSN 976 6(Prnt), ISSN 976 6(Onlne) Volume, Issue, February (), pp. -9 IAEME ISSN 976 6(Prnt) ISSN 976 6(Onlne) Volume, Issue, February (), pp. -9 IAEME: Journal Impact Factor ():.9 (Calculated by GISI) IJEET I A E M E A NEW CHAOTIC ATTRACTOR GENERATED FROM A -D AUTONOMOUS SSTEM WITH ONE EQUILIBRIUM AND ITS FRACTIONAL ORDER FORM Kshore Bng, Susy Thomas M.Tech Student, Electrcal Engneerng Department, Natonal Insttute of Technology, Calcut, Kerala, Inda Professor & Head, Electrcal Engneerng Department, Natonal Insttute of Technology, Calcut, Kerala, Inda ABSTRACT In ths paper, a novel three-dmensonal autonomous chaotc system s proposed. The proposed system contans four varatonal parameters, a cubc nonlnearty term (.e. product of all the three states) and exhbts a chaotc attractor n numercal smulatons. The basc dynamc propertes of the system are analyzed by means of equlbrum ponts, Egen values and Lyapunov exponents. Fnally, the commensurate and non-commensurate fractonal order form of the system whch exhbts chaotc attractor s also analyzed. Keywords: Chaos, Chaotc Systems, Chaotc Attractors, Commensurate Order System, Lyapunov Exponents, Non-Commensurate Order System.. INTRODUCTION Chaotc behavor of dynamc systems can be utlzed n a varety of dscplnes, such as algorthmc tradng, bology, computer scence, cvl engneerng, economcs, fnance, geology, mathematcs, mcrobology, meteorology, physcs, phlosophy, and robotcs and so on. In 98, G. Duffng ntroduced a duffng equaton whch can be extended to complex doman n order to study strange attractors and chaotc behavor of forced vbratons of ndustral machnery []. In 9, Van der Pol ntroduced a model known as VPO model to study oscllatons n vacuum tube crcuts. The Van der Pol oscllator (VPO) represents a nonlnear system wth an nterestng behavor that exhbts naturally n several applcatons, such as heartbeat, neurons, acoustc models etc. []. In 9, Alfred J. Lotka and Vto Volterra proposed predator-prey equatons to descrbe the dynamcs of bologcal systems n whch two speces nteract on each other, one s a predator and the other s ts prey []. In

2 Internatonal Journal of Electrcal Engneerng and Technology (IJEET), ISSN 976 6(Prnt), ISSN 976 6(Onlne) Volume, Issue, February (), pp. -9 IAEME 96, Lorenz found the chaotc attractor n a three-dmensonal autonomous system whle studyng atmospherc convecton []. In 976, Otto Rossler proposed Rossler s system wth strange attractor whch s useful n modelng equlbrum n chemcal reactons []. In 98, Newton and Lepnk obtaned the set of dfferental equatons from Euler rgd equatons whch are modfed wth the addton of a lnear feedback. Two strange attractors startng from dfferent ntal condtons and same parameter condtons were obtaned [6]. In 98, the chaotc phenomenon n macroeconomcs was found. The contnuous economcal system was descrbed and analyzed by Ma and Chen n [7]. In 988, the basc crcut unt of the Cellular Neural Network (CNN) was ntroduced by L.O.Chua whch contans lnear and non-lnear elements. Such types of CNN are able to show chaotc behavor [8]. In 999, Chen found a smple three-dmensonal autonomous system, whch s not topologcally equvalent to Lorenz s system and whch has a chaotc attractor as well [9]. In, Lu ntroduced a system whch s known as a brdge between the Lorenz system and Chen s system []. In, a new system was ntroduced whch contan two varatonal parameters and exhbts Lorenz lke attractor []. In the same year a new type of four wng chaotc attractor was generated from a smooth canoncal -D contnuous system []. Motvated by such prevous work, ths paper ntroduces another smple three-dmensonal autonomous system whch contans four varatonal parameters and one cubc nonlnearty term whch s a product of all the three states.e. dsplacement, velocty and acceleraton. Secton explans the basc defntons. In secton the new system s brefly ntroduced. In secton the dynamc behavors of the proposed system are dscussed. The fractonal order form of the system s dscussed n secton. Fnally, some concludng comments are gven n secton 6.. BASIC DEFINITIONS. CHAOS There s no unversally accepted defnton for chaos, but the followng characterstcs are nearly always dsplayed by the soluton of chaotc system.. Aperodc (non-perodc) behavor.. Bounded structure.. Senstvty to ntal condtons.. FRACTIONAL DERIVATIVE AND INTEGRAL The contnuous ntegral-dfferental operator s defned as d, > dt D f ( t), a t t ( dτ ), < a (). GRUNWALD-LETNIKOV FRACTIONAL DERIVATIVE The Grunwald-Letnkov fractonal order dervatve defnton of order s defned as t a h j D f ( t) lt ( ) f a t h h j j ( t jh) ()

3 Internatonal Journal of Electrcal Engneerng and Technology (IJEET), ISSN 976 6(Prnt), ISSN 976 6(Onlne) Volume, Issue, February (), pp. -9 IAEME For bnomal coeffcents calculaton we can use the relaton between Euler s Gamma Γ( + ) functon, defned as for j Γ( j + ) Γ( j + ) Γ Is Euler s Gamma functon and a, t are the bounds of operaton for f (t). ( ). RIEMANN-LIOUVILLE FRACTIONAL DERIVATIVE The Remann-Louvlle fractonal order dervatve defnton of order s defned as a D t a D t n d t f ( τ ) f ( t) n dτ () + Γ a n ( n ) dt ( t τ ) Γ( ) Is Euler s Gamma functon and t a, are the bounds of operaton for f (t).. STABILIT OF FRACTIONAL NONLINEAR SSTEMS Accordng to stablty theorem, the fractonal order system that m s the LCM of the denomnators we setγ m u ' s of q ' s, where. The fractonal order system s asymptotcally stable f π arg( λ ) > γ For all roots λ of the followng equaton mq mq mq ( dag( [ λ... λ n ] ) v q a D t q... q n and suppose + q, v u u, for,,... n and det λ ().6 CONDITION FOR MINIMUM COMMENSURATE ORDER Suppose that the unstable Egen values of scroll focus ponts are λ,, ± jβ,. The necessary condton to exhbt double scroll attractor of fractonal order system s the Egen values λ remanng n the unstable regon. The condton for commensurate order s, β q > a tan,, π () Ths condton can be used to determne the mnmum order for whch a nonlnear system can generate a chaos.. THE PROPOSED -D DNAMICAL SSTEM Consder the followng smple -D autonomous system: x& y y& z z& az by cx + dxyz (6)

4 Internatonal Journal of Electrcal Engneerng and Technology (IJEET), ISSN 976 6(Prnt), ISSN 976 6(Onlne) Volume, Issue, February (), pp. -9 IAEME Where [ x, y, z] T R the state vector, and a, b, c and d are postve constant parameters of the system (6). In the followng, some basc propertes of system (6) are analyzed.. EQUILIBRIA The equlbra of system (6) can be found by solvng the followng algebrac equatons: x& y y& z z& az by cx + dxyz (7) From the frst and second equatons of (7), y, z Substtutng ths nto the thrd equaton of (7), x Therefore O(,,) s the only equlbrum pont of the system (6).. STABILIT AND EISTANCE OF ATTRACTOR By lnearzng the system (6), one obtans the Jacoban J dyz c dxz b dxy a Therefore J O (,, ) c b a So, the Egen values of the lnearzed system are obtaned as follows: λ I J λ + aλ + bλ + c O Case : If a b c d the Egen values are, ± j, the crtcal case. Case : If a >, b c d the equlbrum O s stable, ensures that system (6) s not chaotc. Case : To ensure that system (6) s chaotc mplyng that the equlbrum O s saddle pont, the condton on the postve constant parameters of system should be consdered,.e. a <, b., c and d any value.. DNAMICAL BEHAVIOR OF THE PROPOSED SSTEM When a., b., c and d, the Egen values of the lnearzed system are.79,. ±. j. Therefore, based on the Egen values we know that equlbrum O s a saddle pont. In ths secton the fourth and ffth order Range-Kutta ntegraton algorthm was performed to solve the dfferental equatons. Settng the ntal condton to [...], the chaotc attractor s shown n fgure.

5 Internatonal Journal of Electrcal Engneerng and Technology (IJEET), ISSN 976 6(Prnt), ISSN 976 6(Onlne) Volume, Issue, February (), pp. -9 IAEME The Lyapunov spectrum of the system (6) versus tme s shown n fgure wth parameters a., b., c and d. When a., b, c and d, the chaotc attractor of the system (6) wth ntal condton [...] s shown n fgure. -D vew Projecton on - plane Projecton on - plane Projecton on - plane Fg : Chaotc attractor the system (6) wth parameters a., b., c and d, ntal condton [...]. Dynamcs of Lyapunov exponents Lyapunov exponents λ.98 λ -.8 λ -.9 tme Fg : Lyapunov spectrum of the system (6) wth parameters a., b., c and d, ntal condton [...]

6 Internatonal Journal of Electrcal Engneerng and Technology (IJEET), ISSN 976 6(Prnt), ISSN 976 6(Onlne) Volume, Issue, February (), pp. -9 IAEME -D vew Projecton on - plane Projecton on - plane Projecton on - plane Fg : Chaotc attractor the system (6) wth parameters a., b, c and d, ntal condton [...]. FRACTIONAL ORDER FORM OF THE PROPOSED SSTEM The fractonal order form of the system (6) s defned as follows q d x y q dt q d y z q dt q d z az by cx + dxyz q dt (8) Where q, q and q are the dervatve orders. For numercal smulaton of the fractonal order system (8), we have consdered the two cases: frst, commensurate order system and second, non-commensurate order system. Case : Commensurate order system From equaton () the commensurate order of the system s gven by q > a π. tan.96. 6

7 Internatonal Journal of Electrcal Engneerng and Technology (IJEET), ISSN 976 6(Prnt), ISSN 976 6(Onlne) Volume, Issue, February (), pp. -9 IAEME In fgure s depcted the chaotc attractor of the commensurate fractonal order (8) wth parameters a., b., c, d, dervatve orders q q q. 97 wth ntal condton [...] for smulaton tme T sm s and step tme h.. Case : Non-commensurate order system We consder non-commensurate order system wth parameters a., b., c, d, dervatve orders q.97, q.98 and q.99. Therefore γ m From equaton () the characterstc equaton of the lnearzed system s λ +.λ +.λ + π The unstable roots are λ,.89 ±.67j, because arg( λ, ).6 < γ In fgure s depcted the chaotc attractor of the non-commensurate fractonal order (8) wth parameters a., b., c, d, dervatve orders q.97, q.98, q. 99 wth ntal condton [...] for smulaton tme T sm s and step tme h.. In fgure 6 s depcted the chaotc attractor of the non-commensurate fractonal order (8) wth parameters a., b., c, d, dervatve orders q., q., q. wth ntal condton [...] for smulaton tme T sm s and step tme h.. -D vew Projecton on - plane Projecton on - plane Projecton on - plane Fg : Chaotc attractor of the commensurate fractonal order (8) wth parameters a., b., c, d, dervatve orders q q q. 97 wth ntal condton [...] 7

8 Internatonal Journal of Electrcal Engneerng and Technology (IJEET), ISSN 976 6(Prnt), ISSN 976 6(Onlne) Volume, Issue, February (), pp. -9 IAEME -D vew Projecton on - plane Projecton on - plane Projecton on - plane Fg : Chaotc attractor of the non-commensurate fractonal order (8) wth parameters a., b., c, d, dervatve orders q.97, q.98, q. 99 wth ntal condton [...] -D vew Projecton on - plane Projecton on - plane Projecton on - plane Fg 6: Chaotc attractor of the non-commensurate fractonal order (8) wth parameters a., b., c, d, dervatve orders q., q., q. wth ntal condton [...] 8

9 Internatonal Journal of Electrcal Engneerng and Technology (IJEET), ISSN 976 6(Prnt), ISSN 976 6(Onlne) Volume, Issue, February (), pp. -9 IAEME 6. CONCLUSION In ths paper, a new novel three-dmensonal chaotc system s proposed. The proposed system has only one equlbrum pont for any arbtrary set of parameters and also some dynamc propertes of the system have been nvestgated. The dynamcs of the proposed system wth commensurate and non-commensurate fractonal order forms was studed based on stablty theorems. On the other hand chaos control and synchronzaton of ths system are nterestng problems to be nvestgated and should also be consdered n the future work. REFERENCES [] Ivana Kovacc, Mchael J. Brennan, The Duffng equaton-nonlnear oscllators and ther behavor, A John Wley and sons, Ltd. Publcatons, [] B. Van der Pol, A theory of the ampltude of free and forced trode vbratons, Rado Revew, 9, 7-7, [] Alfred J. Lotka, Contrbuton to the Theory of Perodc Reactons, J. Phys.Chem.,, 9, 7-7. [] E.N. Lorenz, Determnstc non perodc flow, J. Atmos. Sc.,, 96,. [] O.E. Rossler, An equaton for contnuous chaos, Physcs Letters A, 7, 976, [6] Lepnk R. B. and Newton T. A., Double strange attractors n rgd body moton wth lnear feedback control, Physcs Letters, 86, 98, [7] Ma J. H. and Chen. S., Study for the bfurcaton topologcal structure and the global complcated character of a knd of nonlnear fnance system, Appled Mathematcs and Mechancs,,,. [8] L.O. Chua and L. ang, "Cellular Neural Networks: Theory," IEEE Trans. on Crcuts and Systems,, 988, 7-7. [9] G. Chen, T. Ueta. et another chaotc attractor. Int. J. Bfurcaton and Chaos, 9, 999, 666. [] Deng W. H. and L C. P., Chaos synchronzaton of the fractonal Lu system, Physca A,,, 6 7. [] Ihsan P, lmaz U, A chaotc attractor from General Lorenz system famly and ts electronc expermental mplementaton, Turk J Elec Eng & Comp Sc, 8,, 7-8. [] enghu Wang, Guoyuan Q, anxa Sun, Barend Jacobus van Wyk, A new type of fourwng chaotc attractors n -D quadratc autonomous systems, Nonlnear Dyn, 6,, 7. 9