Dynamics in a discrete fractional order Lorenz system

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1 Available online a Advances in Applied Science Research, 206, 7():89-95 Dynamics in a discree fracional order Lorenz sysem A. George Maria Selvam and R. Janagaraj 2 ISSN: CODEN (USA): AASRFC Sacred Hear College, Tirupaur, S. India 2 Kongunadu College of Engineering and Technology, Thoiam, S. India Dedicaed o Dr. Major. M. Reni Sagayaraj, (Head, Deparmen of Mahemaics, Scared Hear College, Tirupaur, S.India) ABSTRACT In his paper, we are ineresed in he discree version fracional order Lorenz sysem. A discreizaion process is applied o obain a discree version. Fixed poins are compued and he sabiliy properies are analyzed. Bifurcaion and chaos for differen values of he fracional order parameer are presened. Key words: Fracional order, discreizaion, Lorenz Sysem, phase porrais. INTRODUCTION The concep of fracional order differeniaion and inegraion is nearly as old as calculus iself he fracional calculus is he generalizaion of ineger calculus. In recen years fracional order differenial equaions and sysems have araced many researchers because of heir applicaions in many areas of science and engineering; see, for example,[, 3, 4. Weprovide he basic definiions (Capuo) and properies of fracional order differeniaion and inegraion.le us recall he following definiions. Definiion. The fracional order inegral of a funcion, R of order R 0,+ is defined by, where. is he gamma funcion. Definiion. 2 For a funcion given on he inerval,, he order Riemann-Liouville fracional derivaive of is defined by ' D f Γn $d d & s ' fsds, n < < *,* N, where N,,2,3, 0. Definiion.3 For a funcion given on he inerval,, he order Capuo fracional derivaive of is defined by c D f Γ' s' f ' sds, n < < *, f f ', n,n N. c D 89

2 A. George Maria Selvam and R. Janagaraj Adv. Appl. Sci. Res., 206, 7(): DISCRETIZATION PROCESS In [5, 6, 7, 8, a discreizaion process is inroduced o discreize he fracional order differenial equaions/sysems. We noiced ha when he fracional order parameer, Euler's discreizaion mehod is obained. The discreizaion mehod is applied o he logisic fracional-order differenial equaion, Riccai's fracional order differenial equaionand Chua's sysem [8. Here, we are ineresed in applying he discreizaion mehod o Lorenz sysem of differenial equaions which is capable of generaing chaoic behavior. In he early 960s, Lorenz discovered he chaoic behavior of a simplified 3 dimensional sysem. This sysem is well known and has been sudied widely [2. Le (0,) and consider he differenial equaion of fracional order , > , 0. () The corresponding equaion wih a piecewise consan argumen is $9: ;&<, > , 0. (2) Le 0,9, hen 0,.We ge2 3 3, 0,. Thus > 3?. Le 9,29, hen,2. We ge , 9,29. Thus >? Le 29,39,hB* 2,3. So we ge , 29,39 Thus @ 295.? Repeaing he process, we ge when *9,* + 9, hen *,* +. So we ge C *95, *9,* + 9 Thus *9 3 C 3 C * C* FRACTIONAL ORDER LORENZ SYSTEM AND DISCRETIZATION The Lorenz equaion is a model of hermally induced fluid convecion in he amosphere and published [9,0 by E.N Lorenz (an amospheric scienis) of M.I.T. in 963. In Lorenz's mahemaical model of convecion, hree sae variables are used (x, y, z). They are no spaial variables bu are more absrac. The variable x is proporional o he ampliude of he fluid velociy circulaion in he fluid ring, posiive represening clockwise and negaive represening counerclockwise moion. The variable y is emperaure difference beween up and down fluids and z is he disorion from lineariy of he verical emperaure profile. The model is coninuous in ime, bu a modificaion of he coninuous equaion oa discree quadraic recurrence equaion known as he Lorenz map is also widely used. 3 D3 + DE E F3 3G G 3E HG The Lorenz sysem includes hree equaions and hree parameers wih he following properies. (i) Nonlineariy he wo nonlineariies are 3E and 3G. (ii) Symmery Equaions are invarian under 3 ;E 3; E. Hence if 43;E;G5 is a soluion, so is 4 3; E; G5; (iii) Volume conracion The Lorenz sysem is dissipaive i.e. volumes in phase space conrac under he flow. The dimensionless parameers: a Prandl number is aken o be 0, b is relaed o he horizonal wave number of he 90

3 A. George Maria Selvam and R. Janagaraj Adv. Appl. Sci. Res., 206, 7():89-95 convecive moions and o be J. The remaining parameer F is called he Rayleigh number. I is proporional o he K difference in emperaure from he warm base of a convecion cell o he cooler op. (iv) Here we are concerned wih he fracional order Lorenz sysem given by D x ax + ay ; ( ) ( ) ( ) ( ) + γ ( ) ( ) ( ) ( ) ( ) + ( ) ( ) D y y( ) x x z( ); D z b z x y where is he fracional order. Now we are ineresed in discreizing fracional order Lorenz sysem given in he form D x ( ) ax r + ay r r r D y ( ) y r γ x r x r z r r + r r r D z ( ) ( b) z r + x r y r r r r wih iniial condiion30 3,E0 E,G0 G. The proposed discreizaion mehod is explained in he following seps. (.) Le 0,9, hen 0,. So we ge (3) (4) 2 3 L3 + LE ;2 E E + F3 3 G ;2 G MG + 3 E and he soluion of (4) is given by x ( ) x0 + I ax0 + ay0 x0 + ( ax0 + ayo) Γ ( + ) ( ) + + γ + ( + γ ) y y I y x x z y y x x z Γ ( + ) z ( ) z0 + I ( b) z0 + x0 y0 z0 + (( b) z0 + x0 yo) Γ( + ) (2.) Le 9,29, hen,2. We obain 2 3 L3 + LE,2 E E + F3 3 G,2 G MG + 3 E and he soluion of (4) is x2 ( ) x + I ax + ay x + ( ax + ay) Γ ( + ) ( ) + + γ + ( + γ ) y y I y x x z y y x x z 2 Γ ( + ) z2 ( ) z + I ( b) z + x y z + (( b) z + x y) Γ( + ) 9

4 A. George Maria Selvam and R. Janagaraj Adv. Appl. Sci. Res., 206, 7():89-95 Repeaing he process, we deduce he soluion of (4) as ( ) ( ) ( nr) Γ ( + ) ( ) ( ) xn+ xn nr + axn nr + ayn nr ( nr) yn+ ( ) yn + [ yn + γ xn xn zn Γ ( + ) ( nr) zn+ ( ) zn + [( b) zn + xn yn Γ ( + ) Le * + 9, we obain he following discreizaion (( ) ) ( ) ( ) ( ) ( ) xn+ n + r xn nr + axn nr + ayn nr Γ + yn+ (( n + ) r) yn + [ yn + γ xn xn zn Γ ( + ) zn+ (( n + ) r) zn + [( b) zn + xn yn Γ ( + ) which can be expressed as r x x + ax + ay Γ + ( ) [ n+ n n n r yn+ yn + [ yn + γ xn xnzn Γ ( + ) zn+ zn + [( b) zn + xn yn Γ ( + ) (5) 4. FIXED POINTS, STABILITY AND NUMERICAL SIMULATIONS Now we analyze he sabiliy of he fixed poins of he sysem (5) which has he following hree fixed poins. N 0,0,0, Trivial poins. N OM F +,OM F +,F +. OM F +, OM F +,F +. By considering a Jacobian marix for inerior fixed poin and calculaing heir eigen values, we can invesigae he sabiliy of he inerior fixed poin based on he roos of he sysem characerisic equaion [4. Linearizaion of he sysem (5) abou F 0 yield he characerisic equaion: PQ Q K + L + M 3R + QS L M M L LF + RT + L + M 2 M L + L2 + F R + K LF + M R 0 where >. Le? L L + M L M M L LF + R L K L + M 2 M L + L2 + F R + K LF + M From he Jury es, if P > 0,P < 0, and L K < 0, M K > M,V K > where 92

5 A. George Maria Selvam and R. Janagaraj Adv. Appl. Sci. Res., 206, 7():89-95 M K L L K L K L,V K M M K M hen he roosofpq saisfy WQ X W) and hus N is asympoically sable. Suppose P)0 hen N is unsable.linearizing sysem (5) abou N or yields he characerisic equaion: PQQ K MLFR ML K LFMR0 where >. Le? L LM3 MLF L K ML K LFM From he Jury es, if P60,P)0, and L K )0, M K 6M,V K 6 where M K L L K L K L,V K M M K M hen he roosofpq saisfy WQ X W) and hus N or is asympoically sable. Suppose P)0 hen N or is unsable. Numerical simulaions are useful in undersanding he dynamical behavior of he sysem. Numerical sudy of fracional order discree dynamical sysems provides an insigh in o he dynamical characerisics. In his secion, we presen he ime plos for 3,E,G, phase porrais and bifurcaion diagrams for he sysem (5). Dynamic behavior of he sysem (5) abou he inerior fixed poins under differen ses of parameer values are presened below. Example. Le us consider he parameers wih values F 0.02 ;L.95;M.98, he iniial condiions are ; E 0.4 ; G 0.2 and he fracional derivaive order Applying Jury es we ge P ,P.9788)0 and L K ), hus F is asympoically sable see Fig. FIGURE. Time series and Phase diagram of fixed poin _` wih Sabiliy Bifurcaion diagrams provide informaion abou abrup changes in he qualiaive behavior in he dynamics of he sysem. The parameer values a which hese changes occur are called bifurcaion poins. They provide informaion abou he dependence of he dynamics on a cerain parameer. If he qualiaive change occurs in a neighborhood of an equilibrium poin or periodic soluion, i is called a local bifurcaion. Any oher qualiaive change ha occurs is considered as a global bifurcaion. 93

6 A. George Maria Selvam and R. Janagaraj Adv. Appl. Sci. Res., 206, 7():89-95 Figure 2. Differen Phase diagrams of Fixed poin _` wih various values of a Figure 3. Bifurcaion diagram for X wih differen values of a and he numerical values of a 0 2, γ 0.02, b

7 A. George Maria Selvam and R. Janagaraj Adv. Appl. Sci. Res., 206, 7():89-95 Figure 4. Bifurcaion diagram for Y wih differen values of a and he numerical values of a 0 2, γ 0.02, b 0.98 Figure 5. Bifurcaion diagram for Z wih differen values of a and he numerical values of a 0 2, γ 0.02, b 0.98 REFERENCES [ Bhalekara, S, Dafardar-Gejjib, V, Baleanuc, D, Magine, R: Compu. Mah.Appl. 64, (202) [2 Eienne Ghys, The Lorenz Aracor, a Paradigm for Chaos, Chaos, 54, 203 Springer Basel AG. [3 Faieghi, M, Kunanapreeda, S, Delavari, H, Baleanu, D: Nonlinear Dyn. 72, (203) [4 Alireza K. Golmankhaneh, Roohiyeh Arefi, and Dumiru Baleanu, Adv. Mah. Phys (203) [5 El-Sayed, AMA, Salman,.J. Frac. Calc.Appl. 4, (203) [6 El-Sayed, AMA, Salman, SM: On a discreizaion process of fracional-order Logisic differenial equaion. J. Egyp.Mah. Soc. (acceped) [7 K. S. Miller and B. Ross, An Inroducion o he Fracional Calculus and Fracional Differenial Equaions, John Wiley & Sons, New York, NY, USA, 993. [8 Ravi P Agarwal, Ahmed MA El-Sayed and Sanaa M Salman, Advances in Difference Equaions 203, 203:320. [9 Francis C. Moon, Chaoic and Fracal Dynamics, Wiley-Inerscience Publicaion, New York (2008). 95