Conservative Chaos Generators with CCII+ Based on Mathematical Model of Nonlinear Oscillator

Transcription

1 RADIOENGINEERING, VOL. 7, NO., SEPTEMBER 8 9 Conservtive Chos Genertors with CCII+ Bsed on Mthemticl Model of Nonliner Oscilltor Jiří PETRŽELA, Josef SLEZÁK Dept. of Rdio Electronics, Brno University of Technology, Purkyňov 8, 6 Brno, Czech Republic petrzelj@feec.vutbr.cz Abstrct. In this detiled pper, severl novel oscilltor s configurtions which consist only of five positive second genertion current conveyors (CCII+) re presented nd experimentlly verified. Ech network is ble to generte the conservtive chotic ttrctors with the certin degree of the structurl stbility. It represents clss of the utonomous deterministic dynmicl systems with two-segment piecewise liner (PWL) vector fields suitble lso for the theoreticl nlysis. Route to chos cn be trced nd observed by simple chnge of the externl dc voltge. Advntges nd other possible improvements re briefly discussed in the text. Keywords Anlog oscilltor, ttrctor, conservtive dynmics, chos, stte spce.. Introduction A huge pile of the journl rticles solving problems with chos nd other types of complex behvior hve been published from its erly discovery bout forty yers go. The relible prts of these publictions re bout methods how to crete chotic wveform by n oscilltor. With the ltest progress in the nlog integrted circuit design the new electronic devices re developed nd thus used inside vrious pplictions including chos genertors. It is lso the cse of CCII+ which is modern functionl block with three ports cpble to work t high frequencies. For ny circuit structure nd purpose, hving certin knowledge bout the bsic properties of the generted wveforms in time nd frequency domin is essentil. Understnding complex dynmics is useful lso from the theoreticl stndpoint. Chos cn be considered s the universl phenomenon due to the normliztion of system prmeters nd rbitrry interprettion of stte vribles. Tht is the reson why chos hs been experimentlly confirmed nd reported in mny distinct scientific fields, for exmple in power electronics, digitl circuits, ecology, chemicl rections, popultion growth, optics, etc. The overll informtion bout conservtive dynmics cn be found in []. Severl conservtive systems tken over from the technicl prctice re lso provided in this publiction. The very simple exmple of driven dynmicl system is mentioned in [] where PWL type nonlinerity nd periodic driving force is ssumed. The integrtor bsed conservtive chotic oscilltor with two modifictions is presented nd verified in []. Both presented systems model dynmics of the Nosé-Hoover s thermostt. This pper is orgnized s follows. Section two gives the necessry mthemticl bckground together with the numericl integrtion of the studied systems. The third section is focused on the individul circuits. They re synthesized in order to obtin different structures with qulittively equivlent conservtive dynmics. The next section is imed on the experimentl verifiction of finl oscilltors.. Numericl Anlysis Let us consider mthemticl model of the nonliner oscilltor with two-segment PWL vector field (, ) + I & x&& + && x + x& + x = b mx () x where dots denote the time derivtives, is the brekpoint nd I represents the offset of the nonliner function. Note tht we cn ssume = without loss of generlity. Stright-forwrd nlysis of this differentil eqution following the rules of liner lgebr leds to the three possibilities for the number of fixed points. There re no equilibri if I> nd one fixed point locted on the boundry plne occurs when I=. We get two equilibri per region only in the prticulr cse I< nd they re in positions x =[I/ ] T nd x =[(b -I)/(b - ) ] T, respectively. Only the ltter cse is importnt from the viewpoint of the conservtive chos genertion. It is not hrd to obtin the chrcteristic eqution for the left segment of the vector field in the form x < =, () nd similrly for the right segment x > b =. () It is well known tht the eigenvlues re roots of these polynomils nd determine the locl behvior of the system

2 J. PETRŽELA, J. SLEZÁK, CONSERVATIVE CHAOS GENERATORS WITH CCII+ BASED ON MATHEMATICAL MODEL in the neighborhood of the fixed points. The eigenvlues cn be computed symboliclly by mens of the Crdn formuls. This kind of pproch cn give us n ide when eigenvlues re complex conjugted forming composition of the eigenplne nd eigenvector or if the geometry is more likely combintion of three eigenvectors. Also the question bout the stbility indexes cn be esily nswered. In spite of this, the existence of chos will remin uncovered. Due to this, it is time to leve the symbols nd work with numbers. The bsic requirement for no dissiption tell us tht =. Other vlues re given in the publiction [4] =.7,, b, (4) = = or cn be obtined using the stochstic optimiztion wy described in [5] = 8., 5.5, b 8.7. (5) = = In both cses the prmeters of the nonliner function were the sme, in detil I=- nd =. The set of prmeters (4) leds to the following rel nd pir of complex conjugted eigenvlues x < x >,, =.77 ±.67 j =.77 ±.67 j =.54 (6) =.54 The sme geometry of the vector field holds for the second cse (5) s expected, i.e. x < x >,, The geometry of the vector field is u =. ±.9 j =.646 (7) =.94 ±.866 j =.88 x < R R x > R R. (8) s The numericl integrtion using the fourth-order Runge- Kutt method with finl time t mx =4, time step t step =.4 nd zero initil condition is shown in Fig. for the first set of prmeters (4) with the stte vribles rnges x (-,), y (-,), z (-,). Anlogiclly, Fig. corresponds to the ttrctor when (5) is set up. In this cse, the xes re fixed t x (-,), y (-5,5), z (-,). Note tht such rnges of the individul stte vribles re nrrow enough for the prcticl implementtion. This is indeed importnt becuse of current nd/or voltge sturtion of the ctive devices. The numericl nlysis involving the computtion of the spectrum of Lypunov exponents s well s the dimensions of the stte spce ttrctors revels tht the chotic regions re significntly surrounded by the regions with unbounded solution. The system prmeters I nd do not ffect the eigenvlues so it cn be hrdly considered s nturl bifurction prmeters. Also thin bsin of ttrction cn cuse serious problems, especilly in cse of experimentl verifiction. Chos is often destructed simply by switching on/off the power supply. u s Fig.. D stte spce trjectories for the st set of prmeters. Fig.. D stte spce trjectories for the nd set of prmeters.. Circuit Reliztion Before speking bout prticulr circuit reliztions, the universl ctive element CCII should be introduced by mens of the hybrid three-port mtrix V =, I =, I z = ± I. (9) x x V y y The orienttions of ll currents in this pper follow stndrd conventions, i.e. inwrds the three-port. Severl integrted circuits cn be used s CCII where the sign in equlities (9) determines the type of the conveyor. The negtive vrint cn be constructed by mens of EL8C while positive CCII is commercilly vilble under the nottion AD844. The best wy how to crete n oscilltor for modeling the dynmicl behvior of the differentil eqution () is prllel connection of some third-order dmittnce function with the shifted pproximtion of the AV chrcteristic of the idel diode. Ech ctive block (CCII+) in the oscilltor is supposed to be in the liner regime of opertion for ll possible vlues of node voltges or brnch currents. This proposition hs the consequence while choosing impednce normliztion fctor.. Nonliner Function Synthesis The constnt current source (CCS) includes one ctive block X5, resistor R x nd externl dc voltge V x providing the output current I out =-I x =-V x /R x. It is obvious tht the direction of the output current cn be reversed nd the

3 RADIOENGINEERING, VOL. 7, NO., SEPTEMBER 8 vlue of resistor R x cn be computed such tht voltge V x cn be tken s positive or negtive supply voltge. CCS cn be connected in prllel with diode-resistor composite. Using such conception, only single CCII+ is needed. The nonliner device (diode) is strictly pssive nd cn be replced with other element with similr AV curve. Such modifiction will not mke the qulittive chnges in the globl behvior of the circuit. The finl circuit cn be simpler, tking dvntge of the ctive dmittnce network with reversed signs of ll coefficients in (). It turns out fter pplying the resonble scling fctors tht the rel vlues of the circuit components (including R y ) mke n effect of the forwrd-bised diode differentil resistnce negligible. Unfortuntely, numericl nlysis s well s the lbortory experiments revel the problems with other non-idel diode property, brekpoint. This voltge is typiclly locted round V p =.5 V (N47, BAT4). To overcome this obstcle, nother CCII+ mrked s X4 hs been used s voltge controlled current switch [6]. In this cse the third-order dmittnce is pssive.. Individul Configurtions Ech finl circuit is composed of few bsic building blocks which work simultneously but their function cn be understood seprtely. The min components re shown in Fig. nd idelly behve like lossy grounded frequency dependent negtive resistors (FDNR) with the possibility of the current inversion (this holds only for the first circuit). The ssocited dmittnce functions re ) Y = I s / V s = ± C C R s + C + C s,() [ ] () () () ( ) s in in i i i i i where V in (s) nd I in (s) re Lplce trnsform of the input voltge nd input current, respectively. The first type of the finl oscilltor rises from () where one liner cpcitor is replced by nother FDNR. Evidently there re lot of possible combintions s shown in Fig. 5. Let ssume tht switching on/off the resistor R y must chnge the sign of the bsolute coefficient of the chrcteristic polynomils s well s the rel eigenvlue. For input voltge higher thn zero the connection X4 nd D behves like negtive impednce converter. Due to this G y =-/R y nd the dmittnce Y(s) must consume the energy from the rest of the circuit. Tking into ccount the gol restriction = we immeditely get ~ ) Y () s = Y () s + Y () s ~ s + ~ s + ~ s + ~ ) ) s s = s + +. () s It mens tht nother energy providing FDNR should be connected in prllel. The second type of third-order dmittnce function originted from the modified connection of two CCII+ used for the filter design nd with the generl impednces of ll brnches. The finl eqution for the input dmittnce is then simplified by substituting resistors, cpcitors or even inductors nd their seril or prllel interconnections s shown in Fig. 6. The third type of the oscilltor is bsed on the fundmentl dmittnce cell shown in Fig. 4. The input dmittnce of such cell is () s [ Y () s + Y () s Y () s Y () s / () s ] Y ˆ = +. () Y As this eqution suggests, ny of the individul dmittnce cn be frequency dependent s well s ny of the used generl impednce cn be substituted by more complicted network leding to the desired liner prt of the vector field. For exmple, the finl structures illustrted in Fig. 7 cn ply this role. Only single FDNR is used compring to the first type of the oscilltor. Note tht ech circuit works in the hybrid voltgecurrent mode. Stte vribles of the originl mthemticl model re mesurble vi their liner combintions. In ccordnce to common form () the coefficients of the dmittnce function for the circuits in Fig. 5 re cccrr c r c + c + c c r c c r c + c c c c b /r /r =, =, =, =. () y b =, ( ) In these terms the normlized vlues of cpcitors nd resistors were used. Serching for concrete numbers is the problem from the re solving the system of the nonliner lgebricl equtions. Such tsk cn cuse the serious problems with finding the rel positive vlues only. This is nother dvntge of the proposed structures of oscilltors. Although the reltive lrge mount of the circuit elements seems to be redundnt it leds to the good overll results (quick convergence with zero error). It is not hrd to lern tht the stisfying performnce lso lst for the second type of the circuit. Fig.. Two reciprocl reliztions of the FDNR using CCII+. Fig. 4. CCII+ bsed elementl dmittnce cell.

4 J. PETRŽELA, J. SLEZÁK, CONSERVATIVE CHAOS GENERATORS WITH CCII+ BASED ON MATHEMATICAL MODEL Fig. 5. Type I conservtive chotic oscilltors. Fig. 6. Type II conservtive chotic oscilltors. Fig. 7. Type III conservtive chotic oscilltors. Nmely for the left circuit in Fig. 6 we get =, b = /r, y = /r, b cccr r ( c + c ) cc cr r =, = c c c, (4) nd for the right circuit in Fig. 6 the coefficients re cccr r =, b = /r, y = /r, b ( c + c + cb ) cc = cr r, =. (5) c c c Note tht slight modifiction of the circuit (dding single cpcitor) results into chnge of the single eqution. This feture cn ensure convergence in mny prticulr cses. The circuits shown in Fig. 7 differ only in the dmittnces of two brnches. In this cse this interchnge does not ffect the input dmittnce, which is in the form ( r / r ) = cccr r b c +, /rc =, b = /r, y ( c + c )( r r + ) + c c r ( r / r ) = cr b/ c c ( c + c )( r / r ) + c ( r / r ). (6) = c b c +,

5 RADIOENGINEERING, VOL. 7, NO., SEPTEMBER 8 4. Experimentl Results AV chrcteristic of the nonliner two-port cn be visulized by current-sensing resistor Ω connected in series with tringulr wve genertor. The result is shown in Fig. 8, where both input nd output signls re centered. The concrete vlues of circuit components for the first type of configurtion nd (5) re C =C =C = nf, C =47 nf, C = nf, R =8 kω, R =5.5 kω, R =.4 kω, R b = kω nd R y =5 kω. For the sme configurtion nd (6) some vlues chnge C = nf, R =77 kω, R =7 kω, R =6 Ω, R b =.8 kω nd R y =. kω. For the second type of oscilltor with (5) the list of components is C =C = nf, C =4 nf, R =4 kω, R = kω, R b = kω, nd R y =5 kω. Anlogiclly, for the set (6) the following pssive elements chnge C =5 nf, R =.4 kω, R b =.8 kω, nd R y =. kω. Finlly, the third type of oscilltor with set (5) consists of C =6 nf, C = nf, C =5 nf, R =7.5 kω, R =8 Ω, R b = kω, R c = kω, R y =5 kω, nd for (6) we get C =9 nf, C =7 nf, C =9 nf, R =4.7 kω, R =4 Ω, R b =9 kω, R c =.8 kω, R y =. kω. Definitely, other sets of vlues re lso possible for the different initil guess (Mthcd optimizer ws used). The digitl oscilloscope HP546B screenshots from number of lbortory experiments re shown by mens of Fig. 8 (AV chrcteristic of the nonliner element nd chotic wveforms) nd Fig. 9 (plne projections). Also the initil conditions close to zero should be introduced into the circuit before mking ech prticulr mesurement. Fig. 8. Nonliner AV chrcteristic nd chotic wveforms. Fig. 9. Plne projection of chotic ttrctors using XY disply. 5. Conclussion In this rticle severl novel inductorless structures of the simple oscilltors with ssocited volume conserving dynmics hve been suggested nd experimentlly verified. The good greement between theory nd prctice cn be considered since the orbits closely relted to conservtive ones hve been found. It is worth nothing tht Pspice circuit simultor gives the sme results. Moreover, single externl dc source cn be used to turn the oscilltor inside single-equilibrium regime or even into the equilibriumless one. The integrted circuit AD844 provides n extr node which cts s the output voltge follower. These buffered outputs llow us to observe other combintions of stte vribles without ffecting the proper function of the oscilltor. The higher order dmittnces presented in this pper cn be directly used for the genertion of the multi-spirl chotic ttrctors. On the other hnd, mthemticl model of the studied systems is not suited for D or D grid ttrctors. More detils bout such oscilltors cn be found in some recent publictions [7]. The symmetry of the vector field nd the possibility of mpping bsolute vlue onto the AV chrcteristic of the diode llow synthesized oscilltors to model ny shpe of the stte spce ttrctors ssocited with two segment PWL vector field. The uthors believe tht much simpler oscilltors with the sme dynmics nd employing only three CCII± cn be discovered. This is the topic for further study. Acknowledgements The reserch described in this pper ws finncilly supported by the reserch progrm MSM 65.

6 4 J. PETRŽELA, J. SLEZÁK, CONSERVATIVE CHAOS GENERATORS WITH CCII+ BASED ON MATHEMATICAL MODEL References [] SPROTT, J. C. Chos nd Time Series Anlysis. Oxford University Press,. [] GOTTLIEB, H. P. W., SPROTT, J. C., Simplest driven conservtive chotic oscilltor. Physics Letters A,, vol. 9, p [] PETRŽELA, J. Widebnd signl genertor bsed on chos. In Proceedings of the th Electrotechnicl nd Computer Conference ERK 4. Portorož (Sloveni), 4, p [4] SPROTT, J. C., LINZ, S. J. Algebriclly simple chotic flows. Interntionl Journl of Chos Theory nd Applictions,, vol. 5, no., p. -. [5] PETRŽELA, J., HANUS, S. On the optimiztion of the specific clss of chotic oscilltors. In Proceedings of the th Electrotechnicl nd Computer Conference ERK 5. Portorož (Sloveni), 5, p. to 4. [6] LIU, S. I., WU, D. S., TSAO, H. W., WU, J., TSAY, J. H. Nonliner circuit pplictions with current conveyors. In IEE Proceedings-G, 99, vol. 4, no., p. 6. [7] LU, J., CHEN, G. Generting multi-scroll chotic ttrctors: theories, methods nd pplictions. Interntionl Journl of Bifurction nd Chos, 6, vol. 6, no. 4, p About Authors... Jiří PETRŽELA ws born in Brno, Czech Republic, in 978. His reserch interest is nonliner dynmics, chos theory nd circuit design. He received the MSc. nd PhD. degrees t the Brno University of Technology in nd 7 respectively. Now he is ssistnt professor t Deprtment of Rdio Electronics, Brno University of Technology. Josef SLEZÁK ws born in Zlín, Czech Republic, in 98. His reserch interest is circuit theory, computer ided design of nlog filters. He received the MSc. Degree t the Brno University of Technology in 7. Now he is working towrds PhD. degree t Deprtment of Rdio Electronics, Brno University of Technology. Cll for Ppers Advnced Electronic Circuits from nlog to digitl from low frequencies to millimeter wves A Specil Issue of October 5, 8: Submission of pper A pper is requested being submitted vi e-mil to editor@rdioeng.cz. The mnuscript hs to be prepred in MS Word for Windows in Rdioengineering Publishing Style. The exmple document with proper styles is vilble t: Due to the very short publiction time, extreme cre hs to be devoted to the forml processing of the pper. A pper of n indequte form might be rejected from tht reson. An uthor is encourged to submit PDF version of the pper together with the MS Word document in order to simplify hndling the pper. October, 8: Notifiction of cceptnce Two independent reviewers review pper submitted. Reviews re distributed to the uthors vi e-mil. If pper is ccepted for publiction, uthors re sked to revise the pper in ccordnce with review comments. November 5, 8: Submission of revised pper The revised pper hs to be delivered to the editors vi e-mil in the form of the MS Word document s described in "Submission of pper".