Mathematical Models of the Twin!T\ Wien!bridge and Family of Minimum Component Electronic Chaos Generators with Demonstrative Recurrence Plots

Transcription

1 Pergamon Chaos\ Solitons + Fractals Vol[ 09\ No[ 7\ pp[ 0288Ð0301\ 0888 Þ 0888 Elsevier Science Ltd[ All rights reserved 9859Ð9668:88:,! see front matter PII] S9859!9668"87#99098!X Mathematical Models of the Twin!T\ Wien!bridge and Family of Minimum Component Electronic Chaos enerators with Demonstrative Recurrence Plots A[ S[ ELWAKIL$ and A[ M[ SOLIMAN Electronics and Communications Engineering Department\ Cairo University\ Cairo\ Egypt "Accepted 03 April 0887# Abstract*Mathematical models describing the chaotic behaviours in the recently reported Twin!T\ Wien! bridge and family of minimum component electronic chaos generators are derived[ Nonideal e}ects of the active element in these circuits are integrated into analysis where necessary while a two segment piece!wise! linear approximation of the passive nonlinear voltage controlled resistor characteristics is adopted[ The chaotic behaviour is shown to extend to the case where an active nonlinear resistor with odd symmetrical characteristics is used[ Three dimensional chaotic attractors obtained from numerical integrations of the proposed mathematical models are constructed[ Demonstrative recurrence plots are included[ Þ 0888 Elsevier Science Ltd[ All rights reserved 0[ INTRODUCTION Several new electronic chaos generators have been recently introduced in literature ð0ð7ł[ The fact that chaos can be controlled ð8ł and that chaotic oscillators can synchronize ð09ł has promising applications especially in communication systems[ At the heart of any proposed chaos based communication system lies the chaos generator\ hence\ developing interest has been directed towards introducing new architectures for these generators[ A collection of _ve new chaotic oscillators have been recently reported in ð0\ 1Ł using a single current feedback operational ampli_er "CFOA# as the active element and a junction _eld e}ect transistor "JFET# as a nonlinear voltage controlled resistor[ These oscillators have the basic advantage of requiring no inductors[ In addition the family of oscillators in ð1ł require the minimum number of passive components "one resistor and three capacitors#[ In this work\ mathematical models of the chaotic oscillators presented in ð0\ 1Ł are derived[ Contributions of the nonideal e}ects of the CFOA to the chaotic behaviour are studied and integrated into analysis where necessary[ A two segment piece!wise!linear approximation of the JFET characteristics is found to su.ciently model its behaviour[ The chaotic dynamics of some of these oscillators were found to extend to the case where the nonlinear resistor possesses odd symmetrical characteristics and is demonstrated using a cubic nonlinearity[ Chaotic attractors in a three dimensional space and demonstrative recurrence plots are constructed[ 1[ DERIVED MATHEMATICAL MODELS Figure 0 represents the recently reported Twin!T\ Wien!bridge and family of minimum com! ponent electronic chaos generators ð0\ 1Ł[ A complete macro model of the CFOA including all $Author to whom correspondence should be addressed[ 0288

2 0399 A[ S[ ELWAKIL and A[ M[ SOLIMAN Fig[ 0[ The Twin!T\ Wien!bridge and family of minimum component chaotic oscillators[ sources of nonideality can be found in ð00ł from which the major sources of nonideality are found to be] 0[ A small resistance "R X # associated with the CFOA inverting input terminal "around the value of 54V#[ 1[ The CFOA output resistance "R o # "approximately equal to 04V#[ 2[ A capacitor "C Z # "typically equal to 4pF# and a resistance "R Z # "around 1MV# associated with the CFOA compensating terminal[ The contributions of these elements as well as the parasitics of the JFET to the circuits chaotic behaviours were studied[ It was evident that the very small resistors R X and R o contribute signi_cantly whereas the rest of the parasitics are of negligible e}ect[ These two resistors are integrated into the following analysis where necessary[ 1[0[ Model of the Twin!T chaotic oscillator It was found that non of the above mentioned sources of nonideality a}ects the chaotic nature of the Twin!T oscillator[ The circuit is thus described by the following set of di}erential equations] C 0 V¾ C0 I C 1 V¾ C1 R 1 C 1 V¾ C1 "K 0#"V C0 V C1 # V C2 "0a# R 0 C 2 V¾ C2 K"V C0 V C1 # V C2 R 0 C 1 V¾ C1 where\

3 Mathematical models of the Twin!T 0390 F"K 0#V C0 KV C1 I j R J J V T f R J ð"k 0#V C0 KV C1 ŁrV T ð"k 0#V C0 KV C1 Ł³V T and K"R B :R A #[ For the J1N3227 JFET\ which was used in experiments and PSpice simulations ð0\ 1Ł\ the parameters R J "small signal resistance# and V T "threshold voltage# are approximately equal to 649V and 9[6 V respectively[ For the design set] C 0 C 1 C\ C 2 1C\ R 0 R 1 R and by introducing the following dimensionless quantities] V C0 X\ V C1 Y\ V C2 t Z\ V T V T V T RC t n equation set "0# becomes] and R R J a\ X¾ Y¾ a "K 0#X KY²0 6"K 0#X KY 0 "K 0#X KY 0 "0b# Y¾ "K 0#"X Y# Z "1# Z¾ 0 ð"1k 0#"X Y# 1ZŁ 1 Numerical integration of equation set "1# was carried out using a RungeÐKutta fourth order algorithm with a 9[994 step size[ "Fig[ 1"a## represents the X!Y!Z phase space trajectory given for a 1[02 and K 0[13[ Note that the dimensionless variables are related to circuit state variables by the division of a negative threshold voltage[ Equation set "1# can be rewritten in the form] &X¾ K 0 ak 0 Y¾ K 0 0 Yr' &b Z¾'&"K 0#"0 a# 9' 9 "2a# Z K 0 1 K 0 1 0'&X where\ 6 aa+b9if"k 0#X KY²0 a9 +ba if "K 0#X KY 0 The following eigenvalues are calculated for a 1[02 and K0[13] 8 0[9204\ 9[400232j 9[767 aa 9\ 9[272j 9[4852 a9 For this mathematical model\ the behaviour using an odd symmetrical nonlinearity is dem! onstrated with a cubic polynomial[ In this case\ equation set "1# is modi_ed such that] "2b# X¾ Y¾ b 0 ð"k 0#X KYŁ b 1 ð"k 0#X KYŁ 2 "3# Numerical integration of the modi_ed model was carried out for the following two cases] I# b 0 9[8\ b 1 9[1 and K 0[3\ implying a nonlinearity with positive slope at the origin[ The chaotic attractor corresponding to this case is shown in "Fig[ 1"b##[ II# b 0 9[4\ b 1 9[2 and K 0[15\ implying a nonlinearity with negative slope at the origin[ The attractor corresponding to this case is shown in "Fig[ 1"c##[

4 0391 A[ S[ ELWAKIL and A[ M[ SOLIMAN "a# "b# "c# Fig[ 1[ Space trajectories obtained by integrating "1# and "3#[

5 Mathematical models of the Twin!T 0392 Realization of physical nonlinear resistors that can be approximated by a cubic polynomial have been reported in ð01ł[ Simple ~ip\ mirror and merge operations on two of the attractors shown in "Fig[ 1"a## result in the attractors shown in "Fig[ 1"b## and "Fig[ 1"c##[ This is due to the fact that a cubic polynomial can be approximated using three segment piece!wise!linear charac! teristics[ Hence\ studying the case of a two segment piece!wise!linear nonlinearity in "1# develops greater understanding of the more complicated behaviour with "3#[ 1[1 Model of the Wien!brid`e chaotic oscillator Although the CFOA output resistance R o is very small\ it was found to contribute signi_cantly to the chaotic nature of the Wien!bridge oscillator[ With its inclusion into analysis\ the circuit is described by the following equation set] R o C 0 V¾ C0 0K 0 R o R 01 V C0 V C2 C 1 V¾ C1 I "4a# R 0 C 2 V¾ C2 R 0 C 0 V¾ C0 R 0 C 1 V¾ C1 V C0 where\ I J f F"V C2 V C1 # j R J V T R J "V C2 V C1 #rv T "V C2 V C1 #³V T "4b# By setting] V C0 X\ V C1 Y\ V C2 t Z\ C 0 t n \ R 0 a\ R 0 b\ C 0 o V T V T V T R 0 R J R o C 1 and with the choice of C 1 C 2 C 0 \ the dimensionless form of equation set "4# becomes] X¾ b $0K 0 0 b1 X Z % Y¾ ao "Z Y#²0 6"Z Y# 0 "Z Y# 0 "5# Z¾ ob o 0 ð"k 0#X ZŁ 0 o 0 Y ¾ Numerical integration of equation set "5# was carried out using a RungeÐKutta fourth order algorithm with a 9[994 step size[ "Fig[ 2"a## represents the X!Y!Z trajectory given for a 1 2 \ e 1\ K 1[85 and b 14[ Equation set "5# can also be rewritten in the form]

6 0393 A[ S[ ELWAKIL and A[ M[ SOLIMAN "a# "b# Fig[ 2[ Space trajectories obtained by integrating "5# and "7#[ K &X¾ bk b 0 9 b L K 9 L 9 a a b Y¾ ob"k 0# a a ob Y b Z¾' Z' k o 0 o 0 o 0 l&x ko 0l where\ 6aao +b9 if "Z Y#²0 a9+bao if "Z Y# 0 The following eigenvalues were calculated with a 1 \ e 1\K1[85 and b 14] 2 "6a# "6b# 8 6[3091\ 0[26062j 1[5563 aao 9\ 02j 6 a9 The chaotic behaviour of this model also extends to the case where an active nonlinear resistor

7 Mathematical models of the Twin!T 0394 with odd symmetrical characteristics is employed[ This is demonstrated using a simple sinusoidal nonlinearity with which equation set "5# is modi_ed such that] Y¾ ao sin "Z Y# "7# The state space trajectory observed in this case is shown in "Fig[ 2"b## for the same e\ K\ b and with a 9[7[ 1[2[ Model of the family of minimum component chaotic oscillators The family of minimum component chaotic oscillators ð1ł constitutes the circuits of "Fig[ 0"c##\ "Fig[ 0"d## and "Fig[ 0"e##[ For this family the contribution of the CFOA inverting input resistance R X is signi_cant[ Including R X into analysis\ the circuit of "Fig[ 0"c## is described by the following model] C 0 V¾ C0 I R 1 C 1 V¾ C1 R 1 C 0 V¾ C0 R 1 C 2 V¾ C2 V C1 "8a# R X C 2 V¾ C2 V C1 V C2 R X C 0 V¾ C0 where\ I J f FV C2 V C0 j R J V T R J "V C2 V C0 #rv T "V C2 V C0 #³V T "8b# For the choice of C 0 C 1 C 2 C and with the following settings] the dimensionless form of "8# becomes] Which can be written as] V C0 X\ V C1 Y\ V C2 t Z\ V T V T V T R 1 C t n\ R 1 a\ R 1 b\ R J R X X¾ a "Z X#²0 6"Z X# 0 "Z X# 0 Y¾ "b 0#Y bz "09# Z¾ b"y Z# X¾ &X¾ 9 a Y¾ 9 b 0 Y Z¾'& a a b b a'&x Z' & b 9 b' "00a# where\

8 0395 A[ S[ ELWAKIL and A[ M[ SOLIMAN 6aa+b9 if "Z X#²0 a9+ba if "Z X# 0 "00b# Numerical integration of "09# was carried out taking a 0[3 and b 05[ The X!Y!Z attractor is constructed in "Fig[ 3"a## and the following eigenvalues are calculated] 8 4[1726\ 9[63072j 0[8196 aa 9\ 9[42j 2[8576 a9 The chaotic behaviour of this circuit model also persists with odd symmetrical nonlinearities[ The circuit model of the chaotic oscillator in "Fig[ 0"d## is given by] C 0 V¾ C0 I C 1 V¾ C1 R X C 1 V¾ C1 R X C 0 V¾ C0 R X C 2 V¾ C2 V C2 V C1 V C0 "01# R 1 C 2 V¾ C2 V C0 V C1 1R 1 C 1 V¾ C1 R 1 C 0 V¾ C0 where I is the same as given by "8b#[ Using the same settings as to obtain "09# in addition to "C 1 :C 0 #o and for the choice of C 0 C 2 C 1 C\ "01# transforms into] X¾ o"b 0#"X Y# obz oa "Z X#²0 6"Z X# 0 "Z X# 0 Y¾ "b 0#"X Y# bz "02# Z¾ 0 o 0 ðo"1b 0#"X Y# 1obZ X ¾ Ł In a matrix form "02# is written as] K &X¾ ob o a ob o a obl K b L b 0 b 0 b Y¾ 9 Y ob a ob ob a Z¾' Z' k o 0 o 0 o 0 l&x b k o 0l "03a# where\ 6aoa +b9 if "Z X#²0 "03b# a9+boa if "Z X# 0 Numerical integration of "02# was carried out taking a 0[71\ e 0[54 and b 04 and the X! Y!Z attractor is constructed in "Fig[ 3"b##[ The system was found to be sensitive to the value of e[ The following eigenvalues have been calculated] 8 8[508 9[40342j2[3965 aea 3[9263e 96 9[382j09[923 a9 Finally\ the model of the chaotic oscillator in "Fig[ 0"e## is derived[ Although this oscillator

9 Mathematical models of the Twin!T 0396 "a# "b# "c# Fig[ 3[ Space trajectories obtained by integrating "09#\ "02# and "05#[

10 0397 A[ S[ ELWAKIL and A[ M[ SOLIMAN contains one more resistor than those of "Fig[ 0"c## and "Fig[ 0"d##\ R X still contributes to its chaotic nature[ The circuit is described by the following set of equations] "KR 1 R X #C 0 V¾ C0 "0 K#V C1 V C0 R X C 2 V¾ C2 R 1 C 1 V¾ C1 "0 K#R 1 C 0 V¾ C0 R 1 C 2 V¾ C2 "K 0#V C1 C 2 V¾ C2 I "04a# where FV C0 V C2 V I j R J J V T f R J "V C0 V C2 V#rV T "V C0 V C2 V#³V T K R 0 R 0 R 1 and V K"V C1 R 1 C 0 V¾ C0 #[ "04b# For the special case of C 0 C 1 C 2 C and using the same settings as for "09#\ the dimen! sionless form of equation set "04# becomes] X¾ 0 K 0 b $ "0 K#Y X Z ¾ b% Y¾ "K 0#X¾ Z¾ "K 0#Y "05# ¾ # X Z K"Y X¾ #²0 Z¾ a 6X Z K"Y X 0 X Z K"Y X¾ # 0 The X!Y!Z phase space trajectory is shown in "Fig[ 3"c## obtained by numerically integrating "05# with a 0[3\ b 01\ K 9[34 and the quantity "0 K# slightly increased to 0[54[ It can be seen from "05# that the switching condition depends not only on the space dimensions but on the velocity along the X direction "X¾ #[ Dependency on X¾ can be eliminated by considering the dynamics in the region X Z K"Y X¾ #²0 and substituting for X¾ with its expression[ With this manipulation the matrix form of "05# can be given as] K &X¾ cb a cb"0 K# ak"0 b# L 0 d d &X 0 Z' Y¾ ak"b 0# cb"0 K# "b 0#ðc"0 K# ak c Z¾' 1 "0 b#ł Y ak"b 0# d d k a ak"0 b# a"0 bk# l K b L d K"b 0# b d k b l "06# where

11 ) Mathematical models of the Twin!T Fig[ 4[ "a#\ "b# Recurrence plots of chaotic behaviour from "1# and "02#[ CMYK Page 0398 ) 0398

12 ) 0309 A[ S[ ELWAKIL and A[ M[ SOLIMAN Fig[ 4[ "c#\ "d# Recurrence plots of white noise and a period one data series[ CMYK Page 0309 )

13 Mathematical models of the Twin!T 0300 F aa+b9 if 0 c0 K"a b#\ d0 Kb and j c ðx K"0 b#y dzł²0 J a9+ba if 0 f c ðx K"0 b#y dzł 0 The following eigenvalues are calculated with a 0[3\ b 01\ K 9[34] 6 0[6421\ 9[19842j9[8184 aa 9\ 9[353742j0[9852 a9 By looking at the calculated eigenvalues from all of the proposed models\ a similarity can be noticed[ In one region of operation a real negative eigenvalue along with a complex conjugate pair with positive real part are located while in the other region of operation a zero eigenvalue and a complex conjugate pair with negative real part are located[ The exponentially decaying sinusoid in the region where a zero eigenvalue is located can be clearly identi_ed in the phase space trajectories[ 2[ RECURRENCE PLOTS Recurrence plots have been advocated as a useful diagnostic tool for the assessment of dynamical time series ð02\ 03Ł[ The basic idea is that after choosing an embedding dimension\ dots are plotted "i\j# on an NN array whenever point x"j# is su.ciently close to point x"i# ofan orbit for a given embedding and delay[ The de_nition for su.ciently close means that x"j# falls within a ball of radius r centred at x"i# ð02ł[ Examination of these plots for chaotic systems revealed the existence of short line segments parallel to the diagonal of the recurrence plot\ which are related to the inverse of the largest positive Lyapunov exponent "the de_nition of a line is at least two adjacent points#[ Random data do not show these short line segments while for a periodic data serious continuous long lines parallel to the diagonal can be seen[ Hence\ by visual inspection of a recurrence plot\ random\ chaotic or periodic data sets can be identi_ed[ In order to construct a recurrence plot\ a suitable embedding dimension and delay should be estimated[ The _rst zero crossing of the autocorrelation function can be used to estimate delay[ For visual quali_cation of a dynamical system\ the choice of the radius r is relatively ~exible whereas for quanti_cation analysis it should not be greater than 09) of the normalized mean distance ð02ł[ Recurrence plots were constructed from data series representing the X state space variable of the dynamical systems modelled by "1# and "02#[ An embedding dimension of 6 was chosen while the delay was estimated to be 04 for "1# and 28 for "02#[ An Euclidean norm was used for calculation of distances[ "Fig[ 4"a## and "Fig[ 4"b## represent the two recurrence plots when both systems operated in a chaotic mode[ For the sake of comparison\ "Fig[ 4"c## and "Fig[ 4"d## are recurrence plots for a sample white noise data series and a period one data series "obtained from "02# with b 8#[ The short lines segments are clear for chaotic dynamics[ 3[ CONCLUSION Mathematical models of the chaotic Twin!T\ Wien!bridge and family of minimum component electronic chaos generators have been derived using an approximate two segment piece!wise! linear model of the JFET current!voltage characteristics[ Numerical simulations of the models con_rm their validity leading to results identical to those observed experimentally and using the

14 0301 A[ S[ ELWAKIL and A[ M[ SOLIMAN PSpice circuit simulator[ The chaotic behaviour with odd symmetrical nonlinearities has been demonstrated[ REFERENCES 0[ Elwakil\ A[ S[\ Soliman\ A[ M[\ Chaos from two modi_ed oscillator con_gurations using a current feedback op amp[ Chaos Solitons + Fractals\ 0886\ 7\ 278[ 1[ Elwakil\ A[ S[\ Soliman\ A[ M[\ Chaos from a family of minimum component oscillators[ Chaos Solitons + Fractals\ 0886\ 7\ 224[ 2[ Morgul\ O[\ Wien bridge RC chaos generator[ Electronics Letters\ 0884\ 20\ 1947[ 3[ Namajunas\ A[\ Tamasevicius\ A[\ Modi_ed Wien!bridge oscillator for chaos[ Electronics Letters\ 0884\ 20\ 224[ 4[ Elwakil\ A[ S[\ Soliman\ A[ M[\ New current mode chaos generator[ Electronics Letters\ 0886\ 22\ 0550[ 5[ Elwakil\ A[ S[\ Soliman\ A[ M[\ A family of Wien!type oscillators modi_ed for chaos[ Int[ J[ Circuit Theory + Applications\ 0886\ 14\ 450[ 6[ Elwakil\ A[ S[\ Soliman\ A[ M[\ Two Twin!T based op amp oscillators modi_ed for chaos[ J[ Franklin Institute\ 0887\ 224B\ 660[ 7[ Namajunas\ A[\ Tamasevicius\ A[\ Simple RC chaotic oscillator[ Electronics Letters\ 0885\ 21\ 834[ 8[ Special issue on controlling chaos[ Chaos Solitons + Fractals\ 0886\ 7\ 8[ 09[ Special issue on chaos synchronization and control] Theory and applications[ IEEE Trans[ Circuits + Syst[!I\ 0886\ 33\ 09[ 00[ Toumazou\ C[\ Lidgey\ J[ and Payne\ A[\ Emerging techniques for high frequency BJT ampli_er design] A current mode perspective\ Parchment Press\ Oxford\ 0883[ 01[ Moro\ S[\ Nishio\ Y[\ Mori\ S[\ Synchronization phenomena in oscillators coupled by one resistor[ IEICE Trans[ Fundamentals\ 0884\ E67!A\ 133[ 02[ Zbilut\ J[ P[\ Webber\ C[ L[\ Embeddings and delays as derived from quanti_cation of recurrence plots[ Physics Letters A\ 0881\ 060\ 088[ 03[ Trulla\ L[ L[\ iuliani\ A[\ Zbilut\ J[ P[\ Webber\ C[ L[\ Recurrence quanti_cation analysis of the logistic equation with transients[ Physics Letters A\ 0885\ 112\ 144[