A MODAL SERIES REPRESENTATION OF GENESIO CHAOTIC SYSTEM

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1 International Journal of Instrumentation and Control Sstems (IJICS) Vol., o., Jul A MODAL SERIES REPRESETATIO OF GEESIO CHAOTIC SYSTEM H. Ramezanour, B. Razeghi, G. Darmani, S. oei 4, A. Sargolzaei 5 Tübingen Universit, Tübingen, German hamidreza.ramezanour@gmail.com Sadad Institute of Higher Education, Mashhad, Iran b.razeghi68@sadad.ac.ir Tübingen Universit, Tübingen, German hamidreza.ramezanour@gmail.com 4 Tabriz Universit, Tabriz, Iran noei.shirin@gmail.com 5 Florida International Universit, Miami, Florida a.sargolzaei@gmail.com ABSTRACT In is aer an analtic aroach is devised to reresent, and stud e behavior of, nonlinear dnamic chaotic Genesio sstem using general nonlinear modal reresentation. In is aroach, e original nonlinear ordinar differential equations (ODEs) of model transforms to a sequence of linea r timeinvariant ODEs. B solving e roosed linear ODEs sequence, e exact solution of e original nonlinear roblem is determined in terms of uniforml convergent series. Also an efficient algorim wi low comutational comlexit and high accurac is resented to find e aroximate solution. Simulation results indicate e effectiveness of e roosed meod. KEYWORDS Chaotic Sstems, Genesio Sstem, Modal Series, Ordinar Differential Equations. ITRODUCTIO Generall, it is difficult to obtain exact solutions for nonlinear ordinar differential equations. In most cases, onl aroximate solutions (eier numerical solutions or analtical solutions) can be exected. There are man techniques available for e numerical solution of ODEs. Tical of em are various shooting and multi-shooting aroaches [-], various versions of finite difference or collocation []. Meods for solving ODEs usuall require users to rovide an initial guess for e unknown initial states and/or arameters. A common limitation to all of ese aroaches is at e can, at best, achieve convergence to a local solution of e ODEs, which means oer solutions of interest ma be missed. In recent ears, much attention has been devoted to e newl develoed meods to construct an analtic solutions of nonlinear equation, such meods include Adomian decomosition meod (ADM) [4,5,6] and e Variational Iteration Meod (VIM) [7-8,9], e homoto analsis meod (HAM)[,] and e homoto - erturbation meod (HPM) which is combination of erturbation meod and homoto meod DOI :.5/iics..

2 International Journal of Instrumentation and Control Sstems (IJICS) Vol., o., Jul [,,4,5]. All ese meod have eir own limitations and advantages. Traditional erturbation meods [] strongl deend on e existence of small/large arameters. In nonerturbation meods sometimes e series solutions obtained diverge or e convergence region of eir series solution is generall small and ere is no roof of convergence series which obtain b ese meods. In some cases eir series diverge wi altering some unknown constant arameters in e roblem or wi exanding e time san of e roblem. Initial guesses and choosing linear and nonlinear oerators in HPM and HAM can be ver challenging and e convergence of series solutions is ver sensitive to em. Modal series [6,7], a roosed aroach in nonlinear sstem analsis, extends e concets of linear sstem eor to gain better understanding of e nonlinear sstems. This aroach rovides e solution of autonomous nonlinear sstems in terms of e fundamental and interacting modes and ields a good deal of hsical insight into e sstem behavior [8]. Different from e linearization meod, validit and accurac of e Modal series is not restricted to a small neighborhood of e oerating oint. Also, in contrast to e erturbation techniques, Modal series meod is indeendent uon small/large hsical arameters in sstem s model. Alough e oer traditional non-erturbation techniques are formall indeendent of small/large hsical arameters, e can not ensure e convergence of solution series. Unlike all erturbation and traditional non-erturbation meods, solution obtained in e Modal series form converges uniforml to e exact solution. However, it has limitations wi hard nonlinearities which do not have an analtic model. The solution of Genesio sstem was considered b different researchers such as Goh et al [9], used VIM and MVIM and Bataineh et al [] obtained e solution using homoto analsis meod. The state sace model of sstem is as follows: x = ( = z( z = cx( b( az( x subect to e initial conditions: x( ) =., () =., z() = where x,, z are state variables and a, b, c, are ositive constants satisfing, () () ab < c. Bifurcation stud shows at when a =., b =. 9 and c = 6, e above sstem is chaotic. In is stud for e first time an algorim based on Modal series is roosed to solve Genesio chaotic sstem and to find an aroximate solution for it, analticall. The rest of e aer is organized as follows. In section rincile concets of Modal series is discussed. Section elaborates how to find an aroximate solution rough an eas handling iterative algorim. In section 4 we resent e simulation results, and finall conclusions are given in e last section.. MODAL SERIES A wide class of nonlinear dnamic sstems can be modeled b e differential equations in e form of: X = G( X ) where X is e -dimensional state vector and X ini G : R R is a smoo vector field and is e vector of initial conditions. Exansion of G in e talor series around e origin and

3 International Journal of Instrumentation and Control Sstems (IJICS) Vol., o., Jul using again X and reresentation: x i = Ai X x i as e new state vector and state variable resectivel, ields e following i H xk xl k= l= 6 = q= r= E i qr x x q x r Where x belongs to e convergence domain of e talor series; v R, row of Jacobian matrix; i Gi H =, k l x= x= (4) G Ai = is e i i Gi E =, and so on and Gi is e q r x= element of vector field G. Assuming e sstem has distinct eigenvalues,, =,,...,, and denoting b U and V e matrices of e right and left eigenvectors of A, resectivel, e transformation X = UY ield e following equivalent sstem for (4): = C k l k = l = = q= r = where Y belongs to e linear maing of v denoted b transformation, here and from now on =,,...,, T T C = V [ U H U ] = [ C qr = 6 D = P And V P P= Q= R= is e PQR V P V Q q ] V R r element of e P contains e main concet of modal series meod. Proosition.. D qr q r v C (5) i under defined linear (6) (7) left eigenvector and so for. The following roosition Let e solution of (5) an arbitrar initial conditions such as T Y = VX = U X = [,,,,...,,,...,, ], can be written as follows: = for =,,,n ( ) in which m, contains e terms at deends on an k -states multiles of initial () conditions. For examle for m =, (8) contains e terms deend on an combination such as, ) for k, l =,,...,. ow b substituting (8) into (5) we have: ( k, l, = C k l k= l= (9) D qr q r = q= r= en e solution of (9)(and equivalentl (5)), can be found b solving e following sequence of differential equations:

4 International Journal of Instrumentation and Control Sstems (IJICS) Vol., o., Jul () () = = = () () k= l= C () () ( ) k l () () () () () () () ( k l l k ) Dqr ( q r ) C k= l= = q= r= wi e initial conditions: (a) (b) (c) () () =, () =, m >, =,,..., () Proof. The solution of (5) for some interval t [, T ) R for arbitrar initial conditions T Y = [,,,,...,,,...,, ] for =,,..., can be exressed as: = Λ ( Y, () Where Λ : C R C, is a smoo analtic function wi resect to Y and t. ow we can exand () as Maclaurin series wi resect to Y which ields; ( Y, Where, i i, k, l, qr, q, r, i= k= l= 6 = q= r= = () Λ i =, i, Y = ( ) Λ = k, l, Y = and qr =, Λ q, r, Y =. Since Λ is analtic function, existence and uniforml convergence of Maclaurin series in is guaranteed. ow let e initial condition be T Y = [,,...,, ] where is an arbitrar scalar arameter. This arameter onl simlifies e calculations and its value doesn't have an significant. One can similarl to write: () () ( Y, = (4) Since ( Y,) =,, it follows: () (), = () () () (5) 4

5 International Journal of Instrumentation and Control Sstems (IJICS) Vol., o., Jul Substituting (4) into (5) and rearranging wi resect to e order of ields: () () = k= l= C () ( ) () k= l= () () () () () () () ( ) k l k l Dqr q r = q= r= C () () k l (6) Since (6) must be hold for an, terms wi e same order of on each side must be equal and e roosition is us roved. Remark.. It should be noted at (a) ields linear aroximate solution to e sstem, (b) ields correction terms to linear aroximate solution b considering second order nonlinearit, (c) considers e ird order nonlinearit and so on. It is obvious at () is a sequence of inhomogeneous linear time invariant ODE s in which at each ste, inhomogeneous terms are calculated from e revious ste, so e above rocess is a recursive rocess.. APPROXIMATE SOLUTIO In fact, obtaining e exact solution of (5) or equivalentl sstem, as in (8) is imossible since (8) contains infinite series. Therefore in ractical alication b interceting M terms of e series an aroximate closed-form solution can be achieved as follows: [ ] ( M ) = M for =,,,n The integer M is generall determined according to a desirable recision of e aroximate solution. In order to obtain an accurate enough aroximate closed-from solution, we resent an iterative algorim wi low comutational comlexit as follows: Algorim: Ste. Let m =. ( ) Ste. Calculate e m order term m from e resented linear ODEs () wi initial conditions () for Ste. Let M = i =,,...,. and calculate Ste4. If ere was no significant difference between [ ] relace i b i and go to ste. Ste5. Sto e algorim; [ ] 4. SIMULATIO () M order aroximate closed-form solution according to (). ( M ) ( M ) and [ ] ( M ) is e desirable aroximate solution. go to ste5; else B using e above algorim we found ver accurate aroximate solution of e Genesio chaotic sstem ust wi iterations of e roosed meod. In Fig., Fig. and Fig., it can be observed e validit of e Modal series b comaring to e numerical outut of ode45 Matlab software code ode45. umerical Results show e more accurac of Modal series in comarison wi VIM and MVIM [9] and in comarison wi numerical outut from ode45. As e time 5

6 International Journal of Instrumentation and Control Sstems (IJICS) Vol., o., Jul rogresses, e accurac of solution of ree deendent variables, x (, ( and z( via Modal series is much more an VIM and MVIM. Just -iterations of Modal series is enough to obtain an aroximate accurate solution in comarison wi 5-iterations of VIM and MVIM. However we should mention at on a longer time frame, all e meods miss eir accurac more and more and it can be due to one maor issue, which is e chaotic nature of sstem..8.6 ode45 iterations of Modal.4. x( t Fig. Comarison of -iterations Modal series wi ode45 for x( ode45 iterations of Modal.5 ( t 6 8 Fig. Comarison of -iterations of Modal series wi ode45 for ( 6

7 International Journal of Instrumentation and Control Sstems (IJICS) Vol., o., Jul.5 ode45 iterations of Modal.5.5 z( t Fig. Comarison of -iterations of Modal series wi ode45 for z( Table. Comarison of umerical Results for x( t VIM MVIM Modal ode45 EVIM E E E E E E E VIM = xvim xode45 = xmvim xode45 = xmodal xode45 Table. Comarison of umerical Results for ( t VIM MVIM Modal ode45 EVIM E E E E E E EVIM = VIM ode45 = MVIM ode45 = Modal ode45 7

8 International Journal of Instrumentation and Control Sstems (IJICS) Vol., o., Jul Table. Comarison of umerical Results for z( t VIM MVIM Modal ode45 EVIM E E E E E E EVIM = zvim zode45 = zmodal zode45 = zmodal zode45 As it can be seen, e recision of -iterations of Modal is close to 5-ietarations of MVIM in comarison wi ode45. As e time rogress VIM is comletel invalid whereas Modal and MVIM agree well to e numerical solution. 5. COCLUSIOS The main obective of is aer was to show e efficienc of e Modal series meod for solving e nonlinear chaotic sstems. B using e Modal series meod e exact solution of nonlinear chaotic sstems can be determined in e form of uniforml convergent series. Besides for obtaining an aroximate solution wi eas comutable terms, a straightforward algorim wi low comutational comlexit was brought. The roosed meod onl requires solving a sequence of linear time-invariant ODEs in a recursive manner. Thus in comarison to oer aroximate meods, is meod is more accurate and more ractical. Also uniforml convergent of is meod is guaranteed unlike oer famous meods such as HPM or VIM. umerical simulations are evidences to is assertion. REFERECES [] S.M. Roberts, J.S. Shiman, Two Point Boundar Value Problems: Shooting Meods, American Elsevier, ew York, 97. [] Morrison, D. D., Rile, J. D., and Zancanaro, J. F., Multile Shooting Meod for Two-Point Boundar Value Problems, Communications of e ACM, 96, [] Holt, J. F., umerical Solution of onlinear Two-Point Boundar Problems b Finite Difference Meods, Communications of e ACM, 964, [4] G. Adomian, Solving Frontier Problems of Phsics: The Decomosition Meod, Kluwer Academic, Dordrecht, 994. [5] G. Adomian, R. Rach, Modified decomosition solution of linear and nonlinear boundar-value roblems, onlinear Anal. (5) (994) [6] Hashim, I., M.S.M. oorani, R. Ahmad, S.A. Bakar, E.S.I. Ismail and A.M. Zakaria, 6. Accurac of e Adomian decomosition meod alied to e Lorenz sstem. Chaos Soliton Fract., 8: DOI:.6/.chaos [7] 6. Batiha, B., M.S.M. oorani and I. Hashim, 7. umerical solution of sine Gordon equation b variational iteration meod. Phs. Lett. A., 7: DOI: org/.6/.hsleta [8] S. Momani, S. Abuasad, Z. Odibat, Variational iteration meod for solving nonlinear boundar value roblems, Al. Ma. Comut. 8 (6) [9] J. Lu, Variational iteration meod for solving two-oint boundar value roblems, J. Comut. Al. Ma. 7 (7)

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