Nonlinear element of a chaotic generator

INVESTIGACIÓN REVISTA MEXICANA DE FÍSICA 53 (3) 159 163 JUNIO 2007 Nonliner element of chotic genertor E Cmpos-Cntón JS Murguí I Cmpos Cntón b nd M Chvir-Rodríguez Deprtmento de Físico Mtemátics b Fc de Ciencis Universidd Autónom de Sn Luis Potosí Alvro Obregón 64 78000 Sn Luis Potosí SLP México e-mil: ecmp@uslpmx ondeleto@uslpmx icmpos@glifcuslpmx mchvir@uslpmx Recibido el 17 de febrero de 2006; ceptdo el 4 de myo de 2007 A mthemticl model of nonliner electronic circuit which is the core of the electronic chotic oscilltor is presented This mthemticl model or nonliner function hs direct reltionship with the vlues of the components used to build n experimentl electronic system In order to get good pproximtion to the chrcteristic response curve of the nonliner circuit the mthemtic model considers the current-voltge curve of nonliner elements Keywords: Chotic oscilltor; bifurction prmeter; nonliner converter Se present un modelo mtemático de un circuito electrónico no linel el cul es el corzón del oscildor cótico Este modelo mtemático o función no linel tiene un relción direct con los vlores de los componentes usdos pr construir el sistem electrónico experimentl de tl form que pr obtener un buen proximción l curv de respuest crcterístic del circuito no linel el modelo mtemático consider l curv de corriente-voltje de los elementos no lineles Descriptores: Oscildor cótico; prámetro de bifurcción; convertidor no linel PACS: 0545-; 0545Pq 1 Introduction The employment of electronic circuits tht disply vrious forms of nonliner behvior re useful in mny physicl engineering biologicl nd other systems In generl electronic circuits with chotic behvior use nonliner elements; for instnce the Chu s electronic circuit employs nonliner negtive resistnce [1] nd the Rössler electronic circuit proposed in Ref 2 lso includes nonliner element in order to generte chos In these cses the diode produces the nonliner behvior At present the most positive ppliction seems to be in improving the performnce of existing nonliner circuits: frequency multipliers mixers chotic communiction systems etc In order to study chos synchroniztion in experiments some rticles [1 4] hve employed chotic circuits nonliner element is the core of the chotic behvior In Refs 3 nd 4 chotic oscilltor is employed which consists of nonliner circuit clled nonliner converter (NC for short) nd liner feedbck The min functionlity of the NC is to trnsform n input voltge into output voltge The output voltge bers resemblnce to n odd function with nonliner dependence Rulkov et l [4] used n empiricl nonliner function in order to model the NC the prmeters of this function do not hve ny reltionship with the vlues of the electronic components used to build the NC In similr wy in Ref 5 different model of the output voltge is considered but gin without ny reltionship with vlues of electronic components These models hve been employed in severl numericl simultions The gol of this pper is to present mthemticl model for the NC hving direct reltionship with the electronic components used to build it This pper is orgnized in the following wy Section 2 describes the model of the NC s in Sec 3 we describe the chotic genertor using the NC Finlly Sec 4 presents the conclusions 2 Nonliner converter In this section we shll discuss how to obtin direct reltionship between the electronic components of the NC nd the mthemticl model The schemtic digrm of the NC is shown in Fig 1 FIGURE 1 Schemtic digrm of the nonliner converter NC

160 E CAMPOS-CANTÓN JS MURGUÍA I CAMPOS CANTÓN AND M CHAVIRA-RODRÍGUEZ TABLE I The component vlues employed for the NC shown in Fig 1 Components R 1 R 2 R 4 R 3 R 5 R 6 Nonliner converter Vlues 27 K Ω 75 K Ω 50 Ω 177 K Ω 2 K Ω A1 A2 Op Amp TL082 A3 Op Amp LF356N D1 D2 Silicon diode 1N4148 i x (V + ) is supposed to be known function for the moment 3 By voltge superposition V is given s nd V = f + bx (2) R 2 R 4 = R 5 + R 2 R 4 b = R 5 R 4 R 2 + R 5 R 4 In stndrd conditions of opertion the Op Amp djusts its input voltge in order to get V (x) := V + = V (3) From Eqs (2) nd (3) we hve tht f(x) = V (x) bx (4) According to Eq (1) V (x) is solution of the following eqution: i x (V ) = x V R 1 (5) Now we need to compute V in terms of x; this is ccomplished considering two cses: ()Sudden Commuttion Model In this cse we consider the on-off commuttion of the diodes nd (b)smooth Commuttion Model This cse consider the current-voltge reltion using Shockley s diode lw FIGURE 2 The circuit digrm of the pir of diodes of the NC This converter trnsforms the input voltge x(t) into the output voltge which is expressed by the nonliner function F (x) = kf(x) the prmeter k corresponds to the gin of the converter t x = 0 The vlues of electronic components for the NC re shown in Tble I The nonliner behvior of the NC circuit is due to the functionlity of turn on-off of the pir of diodes In Fig 1 we cn see tht the k prmeter belongs to intervl [0 1] (potentiometer R 6 ) Now we shll focus on finding n expression of f in terms of the input voltge x The procedure for computing f(x) is the following: 1 Let i x (V ) be the current-voltge reltion of the nonliner module shown in Fig 2 2 Applying KVL(Kirchhoff s voltge lw) in the A 1 R 1 D 1 /D 2 R 3 network elements (see Fig 1) we cn obtin the voltge V + which is expressed s V + = x i x (V + )R 1 (1) FIGURE 3 () Chrcteristic response curve of the nonliner converter (circuit NC) (b) The solid line is obtined with the mthemticl model (7) with k = 1/3 nd dotted line is obtined from the experimentl dt

21 Sudden Commuttion Model If we suppose tht the set of diodes long with pssive liner components of Fig 2 hs sudden commuttion nd tht it occurs in voltge V D then the current-voltge reltion is given by V V D if V > V D (1 w)r 3 i x (V ) = 0 if V V D (6) NONLINEAR ELEMENT OF A CHAOTIC GENERATOR 161 V + V D wr 3 if V < V D w is the blnce prmeter given by potentiometer R 3 The sudden commuttion model for f(x) is obtined by solving (5) for V nd considering expression (6) for i x (V ) For this sitution there exist the following three cses I: If V V D then i x (V ) = 0 As consequence the voltge in R 1 is zero nd therefore V (x) = x From (4) we hve tht ( ) 1 b f(x) = x with x V D II: If V > V D then the current flows through (1 w)r 3 From Eqs (5)-(6) we find solution for V (x) nd solving for f in (4) f(x) = [(1 b)(1 w)r 3 br 1 ]x + R 1 V D [R 1 + (1 w)r 3 ] with x > V D III: If V < V D then the current flows through wr 3 In similr wy to the bove we find f(x) = [(1 b)wr 3 br 1 ]x R 1 V D (R 1 + wr 3 ) with x < V D Finlly we hve the output voltge function F in terms of the input voltge x s the following piecewise function: F (x) = kf(x) (7) [(1 b)(1 w)r 3 br 1 ]x + R 1 V D (R 1 + (1 w)r 3) if x > V D ( ) f(x) = 1 b x if x V D [(1 b)wr 3 br 1 ]x R 1 V D if x < V D (R 1 + wr 3 ) FIGURE 4 () The chrcteristic response curve of the nonliner converter NC (b) The solid line is obtined from smooth commuttion given by Eqs (11) nd (9) The dotted line shows the dt cquired from the experimentl circuit Figure 3 shows the chrcteristic response curve of the NC Prmeters V D nd w were fixed to 1/2 V nd w = 1/2 respectively Figure 3b shows zoom of this response curve the dotted line corresponds to dt obtined from the experimentl mesurements nd the solid line corresponds to the mthemticl model of Eq (7) This mthemticl model of the trnsfer function is directly relted to the components employed to build the NC This trnsfer function F (x) ws obtined using the on-off commuttion in diodes In the next subsection Shockley s lw is used 22 Smooth Commuttion Model On the other hnd if we consider the following current voltge reltion (Shockley s lw): ) i x (V ) = i st (e [(V V D)/V T ] 1 (8) from Eq (8) nd the nonliner block of the schemtic digrm shown in Fig 2 we hve tht the current voltge reltion cn be written s follows: V (i x ) = ( i x R 3 + sign(i ( x ) )) V D + V T ln 1 + i x i st V D = 1/3 V V T = 1/10 V nd i st = 004242A In order to compute the trnsference function F (x) we hve to solve (5) which cn be written s follows: (9) V (i x ) = x i x R 1 (10)

JS MURGU IA I CAMPOS CANTON AND M CHAVIRA-RODR IGUEZ E CAMPOS-CANTON 162 TABLE II Component vlues employed to build the experimentl chotic oscilltor shown in Fig 5 model given by Eq (7) The chotic genertor tht uses the NC is discussed in the next section Chotic Oscilltor Components 200 nf C 998 nf L 261 mh r 70 Ω R 1139 Ω C 3 Vlues 0 Chotic Genertor The electronic circuit of Fig 5 is clled Chotic Genertor(CG) This CG is chep nd esily implemented experimentlly wy Some vlues of the electronic components which hve been employed to build the CG experimentlly re given in Tble II The liner feedbck in the schemtic digrm of the CG consists of low-pss filter RC 0 nd resontor circuit rlc Depending on prmeters of the liner feedbck nd the prmeter k the behvior of the CG cn be in regimes of periodic or chotic oscilltions F IGURE 5 The circuit digrm of nonliner nd chotic oscilltor The only difference from (5) is tht here the currentvoltge reltion of the nonliner block shown in Fig 2 is expressed s function V (ix ) Once solution ix (x) hs been obtined from (10) nd (9) it is possible to compute f (x) with (4) We hve tht F (x) = kf (x) (11) V (ix (x)) bx Figure 4 shows the response of the smooth commuttion model given by Eq (11) nd experimentl dt of the NC Figure 4b is zoom in the rupture voltge of the diodes the solid line is given by Eq (11) nd the dotted line is the curve generted experimentlly by the NC The k prmeter were fixed to 1/3 nd prmeters = 017 nd b = 0489 ccording to the vlues of the Tble I With this model it is possible to find the response of the nonliner converter if the vlues of the components chnge Unfortuntely the mthemticl model given by Eq (11) needs more computtion thn the f (x) = F IGURE 6 The chotic ttrctors of the CG projected on the (x y) plne for different vlues of k:() 04167 (b) 03125 nd (c) 02083 Rev Mex F ıs 53 (3) (2007) 159 163

NONLINEAR ELEMENT OF A CHAOTIC GENERATOR 163 The dynmic of the CG shown in Fig 5 is very well modelled by the following set of differentil equtions: ẋ = y ẏ = z x δy ż = γ [kf(x) z] σy (12) x(t) nd z(t) re voltges cross the cpcitors C nd C respectively nd y(t) = J(t)(L/C) 1/2 is the current through the coil The prmeter k is the gin of the nonliner converter t x = 0 The unit time is given by τ = LC The other prmeters of the model depend on the physicl vlues of the circuit elements: γ = LC/RC δ = r C/L nd σ = C/C Figure 6 shows different projections of the ttrctors on the plne (x y) generted by system (12) nd the nonliner function given by Eq (7) We used fourth order Runge- Kutt nd step equl to 001 These ttrctors cn be obtined just by chnging the k gin prmeter which is the bifurction prmeter We choose the vlue of the gin to be equl to k = 04167 nd 03125 for the ttrctors of Figs 6-6b respectively These ttrctors tke the shpe of fmily of double scroll oscilltions For the ttrctors of Fig 6c we select the vlue of k = 02083 but with different initil conditions These ttrctors tke the shpe of fmily of Rössler systems The initil conditions re (-01-01 -01) for the left ttrctor of Fig 6c nd (01 01 01) for the right ttrctor 4 Conclusions We presented mthemticl model for the nonliner function F in terms of the input voltge x This mthemticl model F (x) hs direct reltionship with the components used to build the nonliner electronic circuit experimentlly Also different projections of the chotic genertor onto the (x y) plne hve been presented for different vlues of k prmeter is bifurction prmeter k inside the nonliner function The mthemticl model is in good greement with the experimentl mesured curves Acknowledgements This work ws prtilly supported by FAI-UASLP 2006 under contrcts C06-FAI-03-1114 nd C06-FAI-03-1619 1 LO Chu L kocrev K Eckert nd M Itoh Int J Bif nd Chos 2 (1992) 705 2 TL Crroll Am J Phys 63 (1995) 377 3 NF Rulkov Chos 6 (1996) 262 4 NF Rulkov nd MM Sushchik Int J Bif nd Chos 7 (1996) 625 5 HDI Abrbnel TW Frison nd LS Tsimring IEEE Signl Processing Mgzine 3 (1998) 49