Method for simulation of the fractional order chaotic systems

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1 Aca Monanisica Slovaca Ročník (26), číslo 4, Mehod fo siulaion of he facional ode chaoic syses Ivo Peáš Absac This pape deals wih he ehod of siulaion of facional ode chaoic syses. We pesen a bief suvey of he exising facional ode chaoic syses. These syses ae descibed by hee facional diffeenial equaions whee ode of deivaives is a non-inege, abiay ode. The oal ode of chaoic syse is less han hee. We pesen an appoach whee we deonsae by an illusaive exaple he ehod fo deiving he odel of such a kind of facional ode chaoic syse and ehod fo is siulaion. This exaple is well known chaoic syse, he so called Chua s oscillao. We have deonsaed he eal easueens and he siulaion in he Malab/Siulink as well. Key wods: facional calculus, Chua s syse, facional-ode chaoic syse, chaos, sange aaco. Inoducion A nube of applicaions whee facional calculus has been used apidly gows. These aheaical phenoena allow o descibe a eal objec oe accuae han he classical inege ehods. The eal objecs ae geneally facional (Ousaloup, 995; Podlubny, 999; Weselund, 22) howeve, fo any of he he facionaliy is vey low. A ypical exaple of a non-inege (facional) ode syse is he volagecuen elaion of a sei-infinie lossy ansission line (Wang, 987) o diffusion of hea ino a seiinfinie solid, whee he hea flow is equal o he half-deivaive of epeaue accoding o ie (Podlubny, 999). The ain eason fo using he inege-ode odels was he absence of soluion ehods fo facional diffeenial equaions. We have o idenify and descibe he eal objec by he facional ode odels. The fis advanage is ha we have oe degees of feedo in he odel. The second advanage is ha we have a "eoy" in he odel. Facional-ode syses have an unliied eoy, being inege-ode syses cases in which he eoy is liied. I is well-known ha a chaos canno occu in coninuous syses of oal ode less han hee. This asseion is based on he usual conceps of ode, such as he nube of saes in a syse o he oal nube of sepaae diffeeniaions o inegaions in he syse. The odel of syse can be eaanged o hee single diffeenial equaions, whee he equaions conain he non-inege (facional) ode deivaive. The oal ode of syse is changed fo 3 o he su of each paicula ode. To pu his fac ino conex, we can conside he facional-ode dynaical odel of he syse. Haley e al. conside he facional-ode Chua s syse; in he wok by Aena e al., 998, (he facional ode cellula neual newok was consideed, in he wok by Gao e al., 25 he facional Duffing s syses was pesened. Ohe facional ode chaoic syses wee descibed in he any ohes woks (e.g. Chunguang e al. 24, Deng e al. 25, Lu 25, Nio e al., 999, ec.). In all hese cases he chaos was exhibied in a syse wih he oal ode less han hee. Facional calculus The idea of facional calculus has been known since he developen of he egula calculus, wih he fis efeence pobably being associaed wih Leibniz and L Hospial in 695. The facional calculus is a genealizaion of inegaion and diffeeniaion o non-inege ode fundaenal opeao (), whee a and ae he liis of he opeaion. The coninuous diffeenial opeao is defined as d R () >, d () ad = R( ) =, ( dτ ) R ( ) <. a Doc. Ing. Ivo Peáš, PhD., Depaen of Applied Infoaics and Pocess Conol, ÚRVP - BERG - Technical Univesiy of Košice, B. Něcovej 3, Košice, Slovak Republic, ivo.peas@uke.sk (Recenzovaná a evidovaná vezia dodaná ) 273

2 Ivo Peáš: Mehod fo siulaion of he facional ode chaoic syses The wo definiions used fo he geneal facional diffeinegal ae he Günwald-Lenikov (GL) definiion and he Rieann-Liouville (RL) definiion (Oldha 974, Podlubny 999). The GL is given hee: [ a] h j ad f() = li h ( ) f( jh), (2) h j= j whee [.] eans he inege pa. The RL definiion is given as n d f( τ ) ad f() = dτ Γ n n ( n ) d, (3) a ( τ ) + fo (n - < < n) and whee Γ(.) is he Gaa funcion. Facional-ode chaoic syse Fo an illusaion we have chosen he classical Chua s oscillao which can be ealized by elecical eleens accoding o he schee shown in Fig., which is a siple eleconic cicui (Kennedy, 992) ha exhibis a nonlinea dynaical phenoenon such as he bifucaion and chaos. Fig.. Cicui of Chua s oscillao. This cicui behavio can be descibed by hee diffeenial equaions. Weselund e al. in 994 poposed a new linea capacio odel. I is based on Cuie s epiical law of 889 which saes ha he cuen hough a capacio is U i =, (4) () h whee h and ae consan, U is he dc volage applied a =, and < <. Fo a geneal inpu volage u() he cuen is d u() i () = C, (5) d whee C is capaciance of he capacio. I is elaed o he kind of dielecics. Anohe consan (ode) is elaed o he losses of he capacio. Weselund povided in his wok he able of vaious capacio dielecic wih an appopiae consan obained by easueens. Weselund in his wok (Weselund, 22) also descibed he behavio of a eal induco. Fo a geneal cuen in he induco, he volage is d i() u () = L, (6) d whee L is inducance of he induco and he consan is elaed o he poxiiy effec. Applying he Kichhoff laws and elaion (5) and (6) ino he cicui depiced in Fig., we ge he following aheaical odel of Chua s syse (Peáš, 26): q d v C = G( v2 v) f( q d v ), d v (7) 2 C2 = G( v v2) + i, d q3 d i L = v q 2, 3 d whee v is volage on he capacio C, v 2 is volage on he capacio C 2, i is cuen in he induco L, G = /R 2, q is he eal ode of he capacio C, q 2 is he eal ode of he capacio C 2, q 3 is he eal ode of he induco L, and f (v ) is he piecewise linea v - i chaaceisic of he nonlinea Chua s diode, which can be descibed as 274

3 Aca Monanisica Slovaca Ročník (26), číslo 4, f ( v) = Gbv+ ( Ga Gb)( v+ E v E ) 2 (8) wih E being he beakpoin volage of a diode, and G a < and G b < being soe appopiae consans (he slope of he piecewise linea esisance). Chua s diode (8) negaive ipedance convee was ealized by opeaing he aplifie LM 358 and he esisos R, R 7 and R 8 (R 7 = R 8 ). Illusaive exaple Expeienal easueens Fo an expeienal veificaion of Chua s syse depiced in Fig. and descibed by equaions (7) and (8) wee chosen he following values of elecical eleens: C = 4.7 nf, C = 48 nf, L = 4.64 H, R =897 Ω, R = 998 kω, R = R = 39Ω (9) We used a ceaic capacios C and C 2 wih he eal ode q = q 2 =.98 and we assue he eal ode of induco q 3 =.94 (Weselund, 22). Toal ode of he syse is q = 2.9. Assuing he hee-segen piecewise-linea volage ansfe chaaceisic of negaive ipedance convee (8), we have he slopes G = a, (fo R7 = R8 ), G R = () b R7 The beakpoin of he non-linea chaaceisic (8) is E 9 [V] (expeienally found). The esisos R 3, R 4, R 5, R 6, and he diodes D and D 2 geneae he posiive and negaive half of he non-lineaiy. In Fig. 2 is depiced a phoo of he oscilloscope sceen. I is a eal easueen of volages v - v 2 of he cicui pesened in Fig., fo he elecical coponens (9) and paaees (), pojeced ono he v - v 2 plane. Fig. 2. Sange aaco of Chua s syse (7). The esul shown in Fig. 2 is he double-scoll aaco of facional ode Chua s syse descibed by he equaions (7) and (8). The easueens wee done by he digial oscilloscope Tekonix TDS2, 6 Mhz. Siulaion in Malab/Siulink Fo a siulaion in he Malab/Siulink envionen we will use a diensionless fo of he facional ode Chua s syse which can be descibed by he following hee diffeenial equaions (x v /E, y v 2 /E, z i/eg, A C 2 /C, Β C 2 /(L G 2 ), G a /G, G b /G): D x () = Ay ( () x () f( x)), q D y() = x() y() + z(), D z() = By(), q3 () 275

4 Ivo Peáš: Mehod fo siulaion of he facional ode chaoic syses whee he nonlinea funcion f(x) is descibed as f( x) = x() +.5( ) ( x() + x() ) (2) We have used he Malab/Siulink appoach. The block diaga of he siulaion is depiced in Fig. 3. Fo he siulaion of he facional deivaive (inegal) we used a Siulink block nid ceaed by Duae Valeio wih he cobinaion of he classical inegao using he popey of couaion by wo opeaos. In he enioned block fo he facional ode opeao, we used a coninuous facion expansion ehod of he geneaing funcion. The ode of such an appoxiaion in he fo of a aional funcion was n = (Vinage e al., 23). Fig. 3. Block diaga of Chua s syse in he Malab/Siulink. In Fig. 4 is depiced he siulaion esul obained by he nueical siulaion in he Malab/Siulink fo he following values of paaees: A =.9, B =.3, q = q 2 =.98 and q 3 =.94, fo he iniial condiions: (x, y, z) = [.6,., -.6] and fo he slopes of Chua s diode (2) chaaceisics: = -. and = Fig. 4. Siulaion esul (x vs. y) of Chua s syse (). Conclusion We have consideed an exaple of chaoic facional-ode Chua s cicui, which exhibis a chaoic behavio wih he oal ode less han hee. As has been deonsaed, he idea of facional calculus 276

5 Aca Monanisica Slovaca Ročník (26), číslo 4, equies one o econside dynaic syse conceps ha ae ofen aken fo ganed. So, changing he ode of a syse fo inege o eal, we also ove fo a hee-diensional syse o he infinie diension. The conclusion of his wok confis he conclusions of he woks (Haley e al., 995; Lu, 25; Podlubny, 999; Weselund; 22), ha hee is a need o efine he noion of he ode of a syse which can no be consideed only by he oal nube of diffeeniaion. Fo facional-ode diffeenial equaions he nube of es is oe ipoan han he ode of diffeeniaion. The facional-ode odel fo chaoic Chua s syse was diecly deived because elecical eleens used in he cicui ae no ideal. As was enioned in he wok by Weselund (22), he eal elecical eleens (e.g. capacios, inducos, ec.) have a facional ode and should be descibed by facional-ode odels. As has been deonsaed, he idea of facional calculus equies o econside he dynaical syse conceps. Soe of he have been noed in his aicle and also in he wok by Peáš a al. (26). We also pesened an appoach o a siulaion of he facional ode chaoic syse in he Malab/Siulink envionen. The esuls obained fo he siulaion ae copaable wih he esuls fo a eal easueen. Refeences Acknowledgeen: This wok was suppoed in pa by he Slovak Gan Agency fo Science unde gans VEGA /279/5, VEGA /26/5 and VEGA /332/6. Aena, P., Caponeo, R., Founa, L., Poo, D.: Bifucaion and chaos in non-inege ode cellula neual newoks, In. J. Bifucaion and Chaos, vol. 8, no. 7, pp , 998. Bai-Lin, H.: Eleenay Sybolic Dynaics and Chaos in Dissipaive Syses, Wold Scienific, Singapoe, 989. Chunguang, L., Guanong, Ch.: Chaos and hypechaos in he facional-ode Rossle equaions, Physica A, vol. 34, pp. 55 6, 24. Dočák, Ľ.: Nueical Models fo Siulaion he Facional-Ode Conol Syses, The Acadey of Sciences, Insiue of Expeienal Physic, Košice, Slovak Republic, 994. Deng, W. H., Li, C. P.: Chaos synchonizaion of he facional Lu syse, Physica A, vol. 353, pp. 6-72, 25. Duae Pedo Maa De Oliveia Valeio.: Toolbox ninege fo Malab, v. 2.3 (Sepebe 25). hp://ega.is.ul.p/~dov/ninege/ninege.h Gao, X., Yu, J.: Chaos in he facional ode peiodically foced coplex Duffing s oscillaos, Chaos, Solions and Facals, vol. 24, pp. 97 4, 25. Haley, T. T., Loenzo, C. F., Qae, H. K.: Chaos on a facional Chua s syse, IEEE Tans. on Cicuis and Syses. Theoy and Applicaions, vol. 42, no. 8, pp , 995. Kennedy, M. P.: Robus OP AMP ealizaion of Chua s cicui, Fequenz, vol. 46, no. 3-4, pp. 66-8, 992. Lu, J.G.: Chaoic dynaics and synchonizaion of facional-ode Aneodo s syses, Chaos, Solions and Facals, vol. 26, pp , 25. Nio, S., Evans, A. K.: The Effecs of Coninuously Vaying he Facional Diffeenial Ode of Chaoic Nonlinea Syses, Chaos, Solions & Facals, vol., no. 7, pp. 8, 999. Oldha, K. B., Spanie, J.: The Facional Calculus, Acadeic Pess, New Yok, 974. Ousaloup, A.: La Déivaion Non Eniee: Theoie, Synhese e Applicaions, Pais, Fance: Hees, 995. Peáš, I.: The facional-ode conolles: Mehods fo hei synhesis and applicaion, J. Elecical Eng., vol. 5, no. 9-, pp , 999. Peáš, I., Bednáová, D., Dočák, Ľ., Tepák, J.: Mehods fo design of facional chaoic syses. Poc. of he ICCC26, Beskydy, May 28-3, Czech Republic, pp Peáš, I.: A noe on he facional ode Chua s syse. Chaos, Solions & Facals, 26, doi:.6/j.chaos Podlubny, I.: Facional Diffeenial Equaions, Acadeic Pess, San Diego, 999. Vinage, B. M., Chen, Y. Q., Peáš, I.. Two diec Tusin disceizaion ehods fo facional-ode diffeeniao/inegao. Jounal of Fanklin Insiue, vol. 34, pp , 23. Weselund, S., Eksa, L.: Capacio heoy, IEEE Tansacions on Dielecics and Elecical Insulaion, vol., no. 5, pp , 994. Weselund, S.: Dead Mae Has Meoy!, Kala, Sweden: Causal Consuling, 22. Wang, J. C.: Realizaions of genealized Wabug ipedance wih RC ladde newoks and ansission lines, Jounal of Elecocheical Soc., vol. 34, no. 8, pp ,

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