Bifurcations and synchronization using an integrated programmable chaotic circuit

In e na ional Jou nal o Bi u ca ion and Chaos 1 BIFURCATIONS AND SYNCHRONIZATION USING AN INTEGRATED PROGRAMMABLE CHAOTIC CIRCUIT M. DELGADO-RESTITUTO, M. LIÑÁN, J. CEBALLOS and A. RODRÍGUEZ-VÁZQUEZ Cen o Nacional de Mic oelec ónica (CNM) Ed. CICA, A da. Reina Me cedes s/n 41012 - Se ille, SPAIN. This pape p esen s a CMOS chip which can ac as an au onomous s and-alone uni o gene a e di e en eal- ime chao ic beha io s by changing a ew ex e nal bias cu en s. In pa icula , by changing one o hese bias cu en s, he chip p o- ides di e en examples o a pe iod-doubling ou e o chaos. We p esen expe - imen al o bi s and a ac o s, ime wa e o ms and powe spec a measu ed om he chip. By using wo chip uni s, expe imen s on synch oniza ion can be ca - ied ou as well in eal- ime. Measu emen s a e p esen ed o he ollowing syn- ch oniza ion schemes: linea coupling, d i e- esponse and in e se sys em. Expe imen al s a is ical cha ac e iza ions associa ed o hese schemes a e also p esen ed. We also ou line he possible use o he chip o chao ic enc yp ion o audio signals. Finally, o comple eness, he pape includes also a b ie desc ip- ion o he chip design p ocedu e and i s in e nal ci cui y. Running Ti le: A Chip o Real-Time Gene a ion o Chao ic Beha io s Con ac Au ho : Angel Rod íguez-Vázquez Cen o Nacional de Mic oelec ónica (CNM) Ed. CICA, A da. Reina Me cedes s/n 41012 Se illa, SPAIN Phone: +34 5 423 99 23 Fax: +34 5 423 18 32 E-Mail: [email p o ec ed] 2 Bi u ca ions and Synch oniza ion using an In eg a ed P og ammable Chao ic Ci cui BIFURCATIONS AND SYNCHRONIZATION USING AN INTEGRATED PROGRAMMABLE CHAOTIC CIRCUIT M. DELGADO-RESTITUTO, M. LIÑÁN, J. CEBALLOS and A. RODRÍGUEZ-VÁZQUEZ Cen o Nacional de Mic oelec ónica (CNM) Ed. CICA, A da. Reina Me cedes s/n 41012 - Se ille, SPAIN. This pape p esen s a CMOS chip which can ac as an au onomous s and-alone uni o gene a e di e en eal- ime chao ic beha io s by changing a ew ex e nal bias cu en s. In pa icula , by changing one o hese bias cu en s, he chip p o- ides di e en examples o a pe iod-doubling ou e o chaos. We p esen expe - imen al o bi s and a ac o s, ime wa e o ms and powe spec a measu ed om he chip. By using wo chip uni s, expe imen s on synch oniza ion can be ca - ied ou as well in eal- ime. Measu emen s a e p esen ed o he ollowing syn- ch oniza ion schemes: linea coupling, d i e- esponse and in e se sys em. Expe imen al s a is ical cha ac e iza ions associa ed o hese schemes a e also p esen ed. We also ou line he possible use o he chip o chao ic enc yp ion o audio signals. Finally, o comple eness, he pape includes also a b ie desc ip- ion o he chip design p ocedu e and i s in e nal ci cui y. 1. In oduc ion Chaos in elec ical ci cui s has d awn s ong a en ion du ing he las decade [Chua, 1987; Chua & Hasle , 1993]. This opic is o e iden heo e ical in e es since ci cui s p o ide e y simple ehicles o he expe imen al obse a ion o chao ic phenomena (ins ead o only h ough compu e simula ion). Chaos is also o p ac ical enginee ing in e es . Fo ins ance, he inhe en unp edic abili y o de e minis ic chaos has been used o design imp o ed whi e and colo ed noise gene a o s [McGonigal & Elmas y,1987; Rod íguez- Vázquez e al., 1991; Mu ch & Ba es, 1990; Delgado-Res i u o e al., 1992], as well as o he gene a ion o secu e andom numbe ime-se ies [Be ns ein & Liebe man, 1990; Rod íguez-Vázquez e al., 1991]. The andom-like appea ance o chaos has also p o en In e na ional Jou nal o Bi u ca ion and Chaos 3 use ul o imp o e he noise pe o mance o swi ched-capaci o Σ∆ modula o s, making hese ci cui s ope a e in chao ic egimes [Sch eie , 1991; Hein, 1993]. Chao ic ci cui s also exhibi po en ial applica ions in nonlinea signal p ocessing and neu al compu a ion. On one hand, he possibili y o wo o mo e chao ic sys ems oscilla ing in a cohe en , synch o- nized way can be exploi ed o signal enc yp ion and secu e communica ions [Ca oll & Peco a, 1991; Oppenheim e al., 1992; Koca e e al., 1992]. On he o he , he ac ha chaos has been iden i ied o be behind he senso y in o ma ion p ocessing pe o med by na u al ne ous sys ems [Ma sumo o e al., 1987; F eeman, 1992], mo i a es looking o a i icial neu al ne wo k pa adigms based upon chao ic neu ons, in an a emp o be e emula e li ing beings [Aiha a e al., 1990, Nozawa, 1992]. In oday’s elec onic sys ems, economic easons dic a e he con enience o ha ing all componen pa s in eg a ed on common silicon subs a es, ins ead o b eadboa ded using o - he-shel componen s. In his scena io, and be o e he po en ials o chao ic ci cui s can be exploi ed in o u u e ma ke able ins umen a ion, communica ion, o compu ing sys ems, i mus be demons a ed ha chaos can be gene a ed in a con ollable and obus o m using monoli hic ci cui s,p e e ably in s anda d VLSI echnologies. Up o da e, only ew o he p e iously epo ed chao ic ci cui s ha e been ealized as monoli hic†1 in eg a ed ci cui s. In 1987 [Rod íguez-Vázquez e al., 1987], he au ho s s a ed a esea ch line in his di ec ion which has esul ed in a numbe o CMOS chips. Some o hem a e desc ibed by ini e-di e ence equa ions (FDE’s), while o he s a e desc ibed by o dina y di e en ial equa ions (ODE’s). In 1991 a p og ammable in eg a ed noise sou ce was p esen ed based on he Be noulli shi [Rod íguez-Vázquez e al., 1991]. I uses swi ched-capaci o echniques, he same as in he licke noise gene a o p esen ed in 1992 [Delgado-Res i u o e al., 1992]. In 1993, an in eg a ed ci cui o whi e noise gene a- ion was p esen ed [Delgado-Res i u o e al., 1993] which uses nonlinea swi ched-cu en echniques [Rod íguez-Vázquez & Delgado-Res i u o, 1994]. Al hough all hese ICs a e simple and obus , hei sampled-da a na u e es ic s he maximum equency a ainable. In 1993 an in eg a ed chao ic gene a o was p esen ed which o e comes his p oblem h ough he use o con inuous- ime ci cui y o ealize ODE’s [Rod íguez-Vázquez & Delgado-Res- i u o, 1993]. O he wo king†2 ICs in ended o be used as pa s ( oge he wi h o -chip com- ponen s) o chao ic elec onic sys ems a e ound in [C uz & Chua, 1993], [Delgado- Res i u o & Rod íguez-Vázquez, 1994] and [Ho io & Suyama, 1995]. Howe e , hey a e basically in ended o be used as modules o la ge b eadboa ded chao ic ci cui s. 1. By monoli hic we mean all he needed componen s a e ab ica ed on he same silicon subs a e. 2. Chips demons a ed only h ough simula ion esul s a e no included. 4 Bi u ca ions and Synch oniza ion using an In eg a ed P og ammable Chao ic Ci cui The chip p esen ed he e is an upda ed e sion o ha in [Rod íguez-Vázquez & Del- gado-Res i u o, 1993]. The o iginal one was basically aimed o p o e he possibili y o build an ODE-based chao ic gene a o in a ully monoli hic manne . Al hough his goal was achie ed, he ci cui su e ed om he p oblems o such demons a ion IC uni s: a he icky con ollabili y and di icul o use by o he s excep he designe s. The new chip o e - comes hese p oblems. I is easy o use and con ol, and i s obus ness has been signi ican ly enhanced h ough sys em-le el and ci cui -le el op imiza ion. I has been ab ica ed in a 2.4µm double-poly double-me al CMOS echnology, and occupies 5mm2 wi h a powe con- sump ion o 1.8mW o a 5V ol age supply. A ema kable ea u e o he new p o o ype is i s e sa ili y o he obse a ion o bi u ca ion and synch oniza ion phenomena by jus con- olling a ew ex e nal bias cu en s. The ou line o he pape is as ollows. Sec ion 2 in oduces he s a e equa ions o he oscilla o , de ails he ou pu pins desc ip ion o he chip as well as hei elec ical cha ac e - is ics, and iden i ies which e minals se e as p og amming a iables o he dynamic beha - io . Sec ions 3 and 4 a e ailo ed o illus a e he pe o mance o he p o o ype h ough expe imen al measu emen s o bi u ca ion and synch oniza ion phenomena, espec i ely. Finally, Sec. 5 gi es a heo e ical basis o he unc ional desc ip ion in oduced in Sec. 2 and p esen s he in e nal block diag am o he chao ic oscilla o , igno ing as much as possi- ble mic oelec onic- ela ed de ails. 2. Chip Te minals and In e connec ions Fig.1(a) shows he pin connec ions and in e nal s uc u e o he in eg a ed chao ic gene a o and Fig.1(b) shows he expe imen al se up. The chip a chi ec u e comp ises a co e chao ic oscilla o and some auxilia y ci cui y ( h ee ol age bu e s and a ime cons an e e ence uni ) o inc ease he e sa ili y o he p o o ype. The chip has 16 ex e nal pins. The mos impo an block in he a chi ec u e o Fig.1 is he co e chao ic oscilla o . I implemen s a hi d o de au onomous con inuous- ime sys em, which includes an odd-sym- me ic, h ee- egion piecewise-linea (PWL) nonlinea i y, (8) whe e (see Fig.2) is gi en by, (9) τ d dx1hx 1 ()αx2 += τ d dx2αx1x3 –()γx2 –= τ d dx3βx2 = h() hx 1 () m1x1 m0m1 – 2 ------------------- x1Bp +x1Bp ––{}+= In e na ional Jou nal o Bi u ca ion and Chaos 5 The beha io is de e mined by se en pa ame e s. Fou o hem, , a e ex e nally p og ammable. The o he h ee, , ha e ixed alues. The p og ammable pa ame e s a e con olled h ough he low impedance inpu s , , and . They ha e DC le els a ound −0.5V, and he con olling τm0m1and Bp ,, αβand γ,               Fig. 1. (a) Chip a chi ec u e; (b) Expe imen al se up showing oscilloscope, chip wi h ou uning esis o s, and he ba e y pack. Co e Chao ic Oscilla o Re e ence Uni x1 x2 x3 x1,bu x2,bu x3,bu con 4 con 3 con 2 in ou VSS VDD con 1 o Bu e ed Ou pu Pins Ou pu        Tuning          Con ol con iPins Pins Pins Icon i, Rci, (a) (b) con 1con 2con 3con 4 6 Bi u ca ions and Synch oniza ion using an In eg a ed P og ammable Chao ic Ci cui a iables a e he cu en s en e ing he e minals. Because o he low-impedance ea u e, each cu en can be gene a ed using a simple esis ance (see inse o Fig.1). se s he ime cons an o he chao ic oscilla o ( ) which hus can a y app oxi- ma ely be ween and . and se espec i ely he cen al and ou e slopes o he nonlinea i y. Achie able anges a e be ween 0 and 5 o , and be ween -1 and -3 o . Finally, , oge he wi h , con ols he b eakpoin s o he nonlinea i y ( ). Table I shows he elec ical cha ac e is ics o F ig. 2. Nonlinea i y o he chao ic oscilla o . h(x1) x1 Bp −Bp m1 m0 m1 Cha ac e is ic Symbol Min Typ Max Uni Posi i e Powe Supply Vol age 2.0 3.0 5.0 Vdc Nega i e Powe Supply Vol age -2.0 -3.0 -5.0 Vdc Tuning Pa ame e , () 1.0 1.5 5.0 µA Bi u ca ion Pa ame e , () 0.0 1.5 10.5 µA Bi u ca ion Pa ame e , () 1.0 2.5 4.5 µA Ampli ude Pa ame e , () 0.2 0.3 0.7 µA Table I: Elec ical cha ac e is ics ( ypical condi ions a e o ep oducing he Chua’s double-sc oll a ac o ). VDD VSS τ VDD VSS – 3.0 V== Icon 1 m0 VDD VSS – 3.0 V== Icon 2 m1 VDD VSS – 3.0 V== Icon 3 Bp VDD VSS – 3.0 V== Icon 4 Icon 1 ττIcon 1 () 12⁄– ∼ 12µs60µsI con 2Icon 3m0 m1m0 m1Icon 4Icon 1Bp BpIcon 4Icon 1 () 12⁄– ∼ In e na ional Jou nal o Bi u ca ion and Chaos 7 he con ol pins a oom empe a u e, as well as he ange o biasing condi ions o he chip, assuming ha powe supply is symme ical wi h espec o g ound ( ). Fig.3 shows he a ia ion o he ealized nonlinea cha ac e is ic o di e en pa ame- e con igu a ions. They ha e been ob ained by a ying quasi-s a ically om ail o ail he ol age a pin o Fig.1, while ixing he ou pu pins and o g ound. Fig.3(a) illus- a es he e ec o changing he biasing cu en , while keeping he es o con ol a iables cons an ( , and ). No e ha as he alue is inc eased by he e ec o lowe ing , he nonlinea cha ac e is ics su - e s om a b eakpoin displacemen owa ds he powe ails, which may p eclude he exis - ence o chao ic egime. This p oblem can be o e idden by o cing a p ope educ ion on he cu en . Fig.3(b) illus a es he e ec o a ying while keeping he es o con ol inpu s ixed ( and he biasing cu en s and as be o e). Finally, Fig.3(c) and (d) show he a ia ion o he nonlinea cha ac e is ic o di e en slopes and o he cen al and ou e pieces, espec i ely. As p e iously s a ed, hey can be ex e nally con olled h ough biasing cu en s and applied o pins VDD VSS –= Fig. 3. Va ia ion o he PWL cha ac e is ics o he nonlinea i y wi h: (a) ; (b) ; (c) he cen al slope, (con ol a iable ); and (d) he ou e slopes, (con ol a iable ). Icon 1 Icon 4m0Icon 2 m1Icon 3 -2.5 -1.5 -0.5 0.5 1.5 2.5 Inpu Vol age (V) -3.0 0.0 3.0 Ou pu Cu en (µA) (a) 0.5 /di 0.6 /di (b) -2.5 -1.5 -0.5 0.5 1.5 2.5 Inpu Vol age (V) -3.0 0.0 3.0 Ou pu Cu en (µA) 0.5 /di 0.6 /di -2.5 -1.5 -0.5 0.5 1.5 2.5 Inpu Vol age (V) -3.0 0.0 3.0 Ou pu Cu en (µA) (c) 0.5 /di 0.6 /di (d) -2.5 -1.5 -0.5 0.5 1.5 2.5 Inpu Vol age (V) -3.0 0.0 3.0 Ou pu Cu en (µA) 0.5 /di 0.6 /di x1x2x3 Icon 1 Icon 21.12µA= Icon 32.7µA= Icon 40.3µA= τIcon 1 Icon 4Icon 4 Icon 11.4µA= Icon 2Icon 3 m0m1Icon 2Icon 3 8 Bi u ca ions and Synch oniza ion using an In eg a ed P og ammable Chao ic Ci cui and , espec i ely. Ou pu pins , and a e high impedance nodes which co espond o he s a e a iables o he co e chao ic oscilla o . Since hese s a e a iables a e ol ages, and because o he high-impedance ea u e (abou 1.5MΩ unde usual ope a ion condi ions), signi ican loading e o s may appea when measu ing a hese ou pu e minals. These loading p ob- lems a e alle ia ed by using he low-impedance bu e ed ou pu pins , and ( hei ou pu impedances a e below 200Ω unde usual ope a ion condi ions). A ime-cons an e e ence uni has been also included (see Fig.1) o gua an ee p ope pa ame e ma ching among synch onizing chips. Fo synch oniza ion o occu , i is neces- sa y no only o ha e good ela i e pa ame e ma ching inside each chip (gua an eed by ou adop ed design s a egies), bu also good ela i e ma ching among he same pa ame e a di e en chip ins ances. This is di icul o achie e wi hou uning because o uncon olla- ble andom luc ua ions, as well as a ia ions wi h empe a u e and aging. Due o his, designe s ha e o ace a scena io whe e pa ame e s ha e a ound 20% e o s -- in ole able o gua an ee he asymp o ic synch oniza ion o he oscilla o s. Fig.4 shows he block diag am o he au oma ic uning ci cui y. The on-chip e e - ence uni simply consis s o an in eg a o ma ched wi h hose in he co e chao ic oscilla o . The ime cons an o his in eg a o (mas e sys em) is uned o an accu a ely de ined ex e - nal e e ence equency. I all he in eg a o s included on-chip a e simul aneously uned, he con 2con 3 x1x2x3 x1bu ,x2bu , x3bu , 1 τ ∫ F ig. 4. Au oma ic Tuning Mechanism. LPF C ys al Oscilla o Ampli ude De ec o Ampli ude De ec o + −k A/ωτ A A sin ω Vo VIC Icon 1 V eq Co e Chao ic Oscilla o x1 x2 x3 x1,bu x2,bu x3,bu con 4 con 3 con 2 in ou VSS VDD con 1 o Re e ence Uni In e na ional Jou nal o Bi u ca ion and Chaos 9 ime cons an o he oscilla o (sla e sys em) is ela ed o he e e ence equency as well. The accu acy o he uning mechanism is de e mined by he ma ching o on-chip compo- nen alues (absolu e e o s o abou 1-2% can be ob ained). No e ha uning is based on ampli ude de ec ion. Pins and in Fig.1 ep esen espec i ely he inpu and ou pu nodes o he in eg a o . A ol age-mode c ys al oscilla o is applied o and he changes in he ou pu ampli ude (measu ed a pin ) wi h he equency o he e e ence signal, a e de ec ed and used o une he sys em. The con ol signal gene a ed by he sys em in closed loop is con e ed o a cu en and hen applied o pin so ha he ime con- s an o he ci cui becomes locked o ha o he ex e nal c ys al oscilla o . P ope ope a ion o he p oposed uning mechanism elies on he in eg a o be o se - ee. O he wise, he ou - pu ampli ude will change linea ly wi h ime ega dless o he signal p o ided by he c ys al oscilla o . To a oid his si ua ion, an o se co ec ion e minal (pin in Fig.1) is added o he scheme, so ha any de ia ion can be ex e nally compensa ed. 3. Expe imen al Bi u ca ions Nex , we p esen a pic u e book o bi u ca ion sequences, chao ic a ac o s and pe i- odic windows which has been measu ed on he silicon p o o ype by changing he bias cu - en s and . The o he p og ammable pa ame e s we e se o and . The book comp ises Fig.5 h ough Fig.21. Among hem, he i s se en igu es illus a e co esponding ins ances o a ypical pe iod-doubling ou e o chaos which ha e been ob ained by only a ying he biasing cu en while ixing . Fo each alue o and (indica ed in he associa ed igu e cap ions) along he pic u e book we show he phase po ai s o he a ac o , he powe spec um o he ol age a pin , and he ime wa e o ms o he h ee s a e a iables. In bo h he Lissajous igu es and ime wa e o ms, he ep esen a ion scale o he s a e a iable is se o . Co esponding oscilloscope scales o he and a iables a e and , espec i ely. The wa e o m empo al basis is o Figs.5-8, and o Figs.9-21. Finally, o he ho izon al scale o he spec- um, he le side o he display is nea ly DC, wi h , while he e ical scale is . The expe imen al esul s ob ained om he p o o ype a e in ull acco dance wi h mea- su emen s p e iously epo ed om disc e e componen ealiza ions [Chua e al., 1993]. in ou in ou V eq con 1 o Icon 2Icon 3Icon 11.4µA= Icon 40.3µA= Icon 2 Icon 32.35 µA= Icon 2Icon 3 x1x1 350mV di ⁄x2x3 200mV di ⁄400mVdi ⁄0.2ms di ⁄ 0.5ms di ⁄ 2kHz di ⁄ 10dB di ⁄ 16 Bi u ca ions and Synch oniza ion using an In eg a ed P og ammable Chao ic Ci cui Fig. 11. Expe imen al Lissajous igu es, s a e wa e o ms, and powe spec um o he a iable o , .x1Icon 21.15 µA=Icon 32.35 µA= Double Sc oll Chao ic A ac o P ojec ion x1 - x2P ojec ion x1 - x3P ojec ion x2 - x3 Wa e o m x1Wa e o m x2Wa e o m x3 Spec um x1 In e na ional Jou nal o Bi u ca ion and Chaos 17 Fig. 12. Expe imen al Lissajous igu es, s a e wa e o ms, and powe spec um o he a iable o , .x1Icon 21.24 µA=Icon 32.47 µA= 3-3 Pe iodic Window P ojec ion x1 - x2P ojec ion x1 - x3P ojec ion x2 - x3 Wa e o m x1Wa e o m x2Wa e o m x3 Spec um x1 18 Bi u ca ions and Synch oniza ion using an In eg a ed P og ammable Chao ic Ci cui Fig. 13. Expe imen al Lissajous igu es, s a e wa e o ms, and powe spec um o he a iable o , .x1Icon 21.47 µA=Icon 32.56 µA= Double Sc oll Chao ic A ac o P ojec ion x1 - x2P ojec ion x1 - x3P ojec ion x2 - x3 Wa e o m x1Wa e o m x2Wa e o m x3 Spec um x1 In e na ional Jou nal o Bi u ca ion and Chaos 19 Fig. 14. Expe imen al Lissajous igu es, s a e wa e o ms, and powe spec um o he a iable o , .x1Icon 21.62 µA=Icon 32.56 µA= 4-4 Pe iodic Window P ojec ion x1 - x2P ojec ion x1 - x3P ojec ion x2 - x3 Wa e o m x1Wa e o m x2Wa e o m x3 Spec um x1 20 Bi u ca ions and Synch oniza ion using an In eg a ed P og ammable Chao ic Ci cui Fig. 15. Expe imen al Lissajous igu es, s a e wa e o ms, and powe spec um o he a iable o , .x1Icon 21.60 µA=Icon 32.58 µA= Double Sc oll Chao ic A ac o P ojec ion x1 - x2P ojec ion x1 - x3P ojec ion x2 - x3 Wa e o m x1Wa e o m x2Wa e o m x3 Spec um x1 In e na ional Jou nal o Bi u ca ion and Chaos 21 Fig. 16. Expe imen al Lissajous igu es, s a e wa e o ms, and powe spec um o he a iable o , .x1Icon 21.65 µA=Icon 32.61 µA= 5-5 Pe iodic Window P ojec ion x1 - x2P ojec ion x1 - x3P ojec ion x2 - x3 Wa e o m x1Wa e o m x2Wa e o m x3 Spec um x1 22 Bi u ca ions and Synch oniza ion using an In eg a ed P og ammable Chao ic Ci cui Fig. 17. Expe imen al Lissajous igu es, s a e wa e o ms, and powe spec um o he a iable o , .x1Icon 21.70 µA=Icon 32.61 µA= Double Sc oll Chao ic A ac o P ojec ion x1 - x2P ojec ion x1 - x3P ojec ion x2 - x3 Wa e o m x1Wa e o m x2Wa e o m x3 Spec um x1 In e na ional Jou nal o Bi u ca ion and Chaos 23 Fig. 18. Expe imen al Lissajous igu es, s a e wa e o ms, and powe spec um o he a iable o , .x1Icon 21.72 µA=Icon 32.63 µA= 6-6 Pe iodic Window P ojec ion x1 - x2P ojec ion x1 - x3P ojec ion x2 - x3 Wa e o m x1Wa e o m x2Wa e o m x3 Spec um x1 24 Bi u ca ions and Synch oniza ion using an In eg a ed P og ammable Chao ic Ci cui Fig. 19. Expe imen al Lissajous igu es, s a e wa e o ms, and powe spec um o he a iable o , .x1Icon 21.79 µA=Icon 32.66 µA= Double Sc oll Chao ic A ac o P ojec ion x1 - x2P ojec ion x1 - x3P ojec ion x2 - x3 Wa e o m x1Wa e o m x2Wa e o m x3 Spec um x1 In e na ional Jou nal o Bi u ca ion and Chaos 25 Fig. 20. Expe imen al Lissajous igu es, s a e wa e o ms, and powe spec um o he a iable o , .x1Icon 21.81 µA=Icon 32.66 µA= 7-7 Pe iodic Window P ojec ion x1 - x2P ojec ion x1 - x3P ojec ion x2 - x3 Wa e o m x1Wa e o m x2Wa e o m x3 Spec um x1 32 Bi u ca ions and Synch oniza ion using an In eg a ed P og ammable Chao ic Ci cui 4.3 In e se Sys em Scheme Fig.27(a) shows he expe imen al se up used o demons a e synch oniza ion by he in e se sys em app oach be ween wo o he manu ac u ed chips. A ol age signal is linea ly con e ed o a cu en and injec ed in he e minal o he i s chip. The ol age gene a ed by his p o o ype is hen ansmi ed o a ecei ing sys em which consis s o a cu en de ec o , a ol age ampli ie and a chao ic oscilla o ma ched wi h ha o he ansmi e . In he ecei e , he signal d i es he cu en de ec o which is a de ice wi h one inpu - and wo ou pu -po s. One o he ou pu e minals ac s as a ol age bu e om he inpu po , and i is connec ed o he e minal o he second chao ic oscilla o p o o ype. The o he e minal p o ides a ol age p opo ional o he cu en low- ing h ough he i s ou pu po , and i is connec ed o a p og ammable ol age ampli ie . This ampli ie , in u n, con ols he ampli ude o he ol age gene a ed by he cu en de ec- o and ob ains he eco e ed signal . In p ac ice, he cu en de ec o and he ol age ampli ie can be unded in a single block o med by an opamp and an ins umen a ion ampli ie . Fig.27(b) illus a es he pe o mance o he se up. The pic u e on he le shows he inpu signal (a sine wa e o 10kHz and ) and he eco e ed signal . As can be seen a nea ly pe ec synch oniza ion is achie ed. On he o he hand, he pic u e on Fig. 26. Synch oniza ion pe o mance o he -d i e sys em.x2 (a) (b) x13 bu , x23 bu , x12 bu , x11 bu , x 21 bu , x 12 bu , s () x11 Φ () x11 bu , = Φ () x21 () s ( ) 350mVpp– () In e na ional Jou nal o Bi u ca ion and Chaos 33 he igh o Fig.27(b) shows he wa e o m o he chao ic modula ed ansmi ed signal, which clea ly keeps no esemblance wi h he injec ed one. Fig.28 shows he powe spec a o he signals in Fig.27(b)-(c). No e ha he signal o noise a io o he eco e ed signal (Fig.28(c)) is g ea e han +55dB wi h less han -0.2dB loss o he inpu signal powe (Fig.28(a)) †4. Also no e ha he spec um o he ansmi ed 4. Fo inpu equencies a ound 15kHz, he signal- o-noise a io ises up o +60dB. con 4 con 3 con 2 in ou con 1 X al Osc. Amp.De . Amp. De . Di e en ial Ampli ie Signal Condi ioning o con 4 con 3 con 2 in ou con 1 X al Osc. Amp.De . Amp. De . Di e en ial Ampli ie Signal Condi ioning o Fig. 27. (a) Simpli ied expe imen al se up o he in e se sys em app oach; (b) Measu ed pe o mance. ime s () () (a) (b) VIC Amp ime s( ) Cu en De ec o ( ) Φ( ) Φ () Co e Chao ic Oscilla o Re e ence Uni VSS VDD Chip 2 VSS VDD Chip 1 Co e Chao ic Oscilla o Re e ence Uni 50 µs/di 5 ms/di x12 x13 x12,bu x13,bu x22 x23 x21,bu x22,bu x23,bu 34 Bi u ca ions and Synch oniza ion using an In eg a ed P og ammable Chao ic Ci cui signal does no p esen a peak a he inpu equency, hus con i ming ha is com- ple ely hidden on he chao ic wa e o m . A lowe one equencies, masking p ope y s ill holds, bu he signal- o-noise a io o he eco e ed signal no ably wo sens. In ac , o inpu equencies below 1kHz, i has been ound ha he signal- o-noise a io d ops down o +40dB, while e aining simila losses a he ecei e . The pe o mance o he in e se sys em se up in Fig.27(a) has been also s a is ically cha ac e ized in ime domain by compa ing he inpu signal wi h he eco e ed signal . We ha e assumed ha consis s o a single one and ha e a ied i s ampli ude and equency. By keeping ack o he eco e ed signal , we can iden i y which a e he be - e condi ions o signal ansmission. Fig.29 shows he o se , a iance and maximal de ia- ion o he eco e ed signal wi h espec o he inpu signal. Special men ion dese es he e olu ion o he a iance wi h he one ampli ude, shown in Fig.29(b). Obse e ha o low one ampli udes (below 350mV), he a iance main ains small (less han ) o inpu equencies be ween 1 and 25kHz. As he ampli ude aises om his alue, he a iance ab up ly inc eases, specially a he bounds o he inpu equency ange. This means ha o ampli udes la ge han abou 350mV, synch oniza ion is los . We ha e iden i ied wo main causes o desynch oniza ion: • The ecei e is unable o keep ack o he ansmi ed signal. • The ansmi e becomes locked a a s able limi cycle ega dless o . The i s cause undamen ally appea s a high inpu equencies, while he second occu s o low inpu equencies. Fo ampli udes lowe han 350mV, he sys em may exhibi spo adic losses o synch oniza ion as indica ed by he maximal de ia ion be ween he inpu and eco e ed signals, shown in Fig.29(c). Howe e , a e a sho ansien , synch oniza ion is again es o ed. s () Φ () Fig. 28. Powe spec a o he (a) inpu signal; (b) ansmi ed signal; and (c) eco e ed signal. (b) (a) (c) s () () s () () 1.5mV2 Φ () In e na ional Jou nal o Bi u ca ion and Chaos 35 We ha e also expe imen ally e alua ed he co ela ion index be ween he inpu and Fig. 29. Time-domain pe o mance o he chao ic modula ion synch oniza ion scheme using wo in eg a ed p o o ypes. 0.0 200.0 400.0 600.0 800.0 1000.0 Tone Ampli ude (mV) -75.0 -50.0 -25.0 0.0 25.0 Vol age (mV) O se Vol age 1 kHz 5 kHz 10 kHz 15 kHz 25 kHz 0.0 200.0 400.0 600.0 800.0 1000.0 Tone Ampli ude (mV) 0.0 3.0 6.0 9.0 12.0 Va iance 0.0 200.0 400.0 600.0 800.0 1000.0 Tone Ampli ude (mV) 0.0 50.0 100.0 150.0 200.0 Vol age (mV) Maximal De ia ion (a) (b) (c) 1 kHz 5 kHz 10 kHz 15 kHz 25 kHz 1 kHz 5 kHz 10 kHz 15 kHz 25 kHz Vol age Squa e (mV2) 36 Bi u ca ions and Synch oniza ion using an In eg a ed P og ammable Chao ic Ci cui he eco e ed signals. This is illus a ed in Fig.30. Obse e ha , o one ampli udes abo e 150mV, co ela ion index is always la ge han 0.9 ega dless o he inpu equency. Tak- ing his in o accoun as well as he p e ious esul s on he a iance, we conclude ha he ampli ude o he inpu signal mus be comp ised be ween 150mV and 350mV, o inpu e- quencies be ween 1 and 25kHz, in o de o gua an ee synch oniza ion. Taking in o accoun he ange o equencies used o and he noise-like appea - ance o he ansmi ed signal , he synch oniza ion scheme in Fig.27(a) could be eadily exploi ed o audio signal enc yp ion. To e alua e he secu i y o he ansmission, we ha e measu ed he co ela ion index be ween he inpu and he ansmi ed signal, assuming again ha consis s o a single one. The esul s a e shown in Fig.31. No e ha he index is close o ze o o e e y inpu equency, excep ing a 1kHz. In his las case, since he ansmi e e ol es in o a s able limi cycle o inpu ampli udes abo e 350mV, he co ela ion index ends o inc ease. 5. Chip Func ion and Block Diag am This sec ion con ains he unc ional desc ip ion and ci cui ealiza ion o he co e chao ic oscilla o . Fo hose eade s wi h sca ce knowledge o in eg a ed ci cui design, some un- damen al concep s will be gi en a he on -end o his desc ip ion. Fig.32 illus a es a sys ema ic p ocedu e o he monoli hic ealiza ion o a bi a y Fig. 30. Co ela ion index be ween he inpu and he eco e ed signals. 0.0 200.0 400.0 600.0 800.0 1000.0 Tone Ampli ude (mV) 0.50 0.75 1.00 Co ela ion Index 1 kHz 5 kHz 10 kHz 15 kHz 25 kHz s () Φ () s () In e na ional Jou nal o Bi u ca ion and Chaos 37 nonlinea dynamical sys ems. This p ocedu e s ongly elies upon p ope hie a chical p ob- lem decomposi ion as shown in Fig.32, which pa icula izes o he well-known double- sc oll a ac o . The i s s ep in he me hodology is o iden i y he se o equa ions desc ib- ing he dynamics. This co esponds o he beha io al le el a he op o he hie a chy. The ob ained desc ip ion maps down o he block le el, which de ines a ne wo k syn hesis a chi- ec u e o he p oblem. A he block le el, he di e en ope a o s, o unc ional building blocks, equi ed o physical ealiza ion, as well as hei in e connec ion, a e clea ly iden i- ied. Each o hese blocks mus be subsequen ly mapped down o a collec ion o in e con- nec ed ci cui elemen s, hus de ining a ci cui le el. Two di e en suble els can be iden i ied; one con aining only idealized elemen s ( o ins ance VCCS’s), and ano he whe e hese idealized elemen s a e ealized using a ailable ci cui p imi i es o he echnol- ogy. Fig.32 illus a es bo h suble els. Obse e ha he ci cui le el in e s choosing he physical na u e o he a iables which suppo in o ma ion low (usually ol ages, cu en s o bo h). Bo om le el in he VLSI design hie a chy de ine he layou phase, whe e ci cui p imi i es a e codi ied in o geome ical objec s equi ed o p ocessing and ab ica ion. In his pape , we will be mainly in e es ed in he wo i s s eps o he hie a chy, i.e., in he beha io al and block le el design aspec s o he chao ic oscilla o . Technical de ails a he ci cui and layou le els will be published elsewhe e. Fig. 31. Co ela ion index be ween he inpu and he ansmi ed signals. 0.0 200.0 400.0 600.0 800.0 1000.0 Tone Ampli ude (mV) -0.25 0.00 0.25 0.50 Co ela ion Index 1 kHz 5 kHz 10 kHz 15 kHz 25 kHz 38 Bi u ca ions and Synch oniza ion using an In eg a ed P og ammable Chao ic Ci cui Fig. 32. Syn hesis ou e owa ds monoli hic nonlinea ci cui s. τxdx d ⁄()αyx– x()–[]= τydy d ⁄()xy–z+= τzdz d ⁄()βy–=      x() bx ab– 2 ------------xE+xE––{}+= 1/τy −α/τx α/τx −1/τy ∑∫ (.) 1/τy −β/τz xyz −α/τx zyx τττ + _ + _ + _ βy αw w +_ zw Vi1Vi2 IQ Polysilicon n+ di usion Beha io al Le el ∑∫ ∑∫ Block Le el Ci cui Le el Physical Le el In e na ional Jou nal o Bi u ca ion and Chaos 39 5.1 Beha io al Le el Desc ip ion The ma hema ical model o he designed chao ic oscilla o is a canonical sys em (which will be de ined below) o he amily o con inuous, odd-symme ic, h ee- egion piecewise- linea (PWL) ec o ields in . Membe s o his amily, deno ed he ea e by , a e gene ally ep esen ed by he ollowing hi d o de con inuous- ime nonlinea s a e equa ion [Chua e al., 1986], (10) which can be mapped on o he analog compu e concep shown in Fig.33. In he abo e equa ion, ep esen s he ime-in eg a ion cons an ; is he s a e-space ec o ; is a eal in e ible squa e ma ix de ining he linea pa o he sys em; and a e eal 3-dimensional ec o s; and he nonlinea map is a eal- alued con inuous PWL unc ion gi en by (11) whe e is a eal scale ac o , wi h no in luence on he quali a i e dynamic beha io o he sys em. The unc ion hus de ined, di ides in o an inne egion con aining he o igin, and wo ou e egions and , in such a way ha , . Acco d- ing o Eq. (11), he wo pa allel bounda y planes sepa a ing om he ou e egions and , a e gi en espec i ely by, ℜ3L3 τ d dx () Fx ()[]Ax () B D†x ()[]+== τx () x1 () x2 () x3 (),,[] † = Aaij []= Bbi []=Ddi []= () Fig. 33. Block diag am o he membe s o he amily .L3 ρ (ρ) Bp Bp – Σ1 τ∫ A B (•) x x .ρ D D†x ()[] 1 2 ---D†x () Bp +D†x () Bp ––{}= Bp () ℜ3D0 D+1 D1– Fx() Fx–()–= D0D+1 D1– 40 Bi u ca ions and Synch oniza ion using an In eg a ed P og ammable Chao ic Ci cui (12) I is wo h no ing ha he quali a i e beha io o any membe o he amily is solely de e mined by he h ee eigen alues associa ed o he inne egion o he ec o ield , and he h ee eigen alues associa ed o he ou e egions [Chua e al., 1986]. By canonical sys ems o we mean hose ec o ields in such ha , wi h only 7 nonze o pa ame e s, a e able o syn hesize almos e e y p esc ibed se o eigen alue pa - e ns, and hence, o ep oduce almos e e y possible quali a i e dynamics in †5 [Chua & Lin, 1990; Chua, 1993]. A well-known example o canonical sys em in is he Chua´s oscilla o which is endowed wi h a ich epe oi e o nonlinea dynamical phenomena, including all kinds o bi u ca ions and ou es o chaos (pe iod-doubling, in e mi ency and o us b eakdown). Ac ually he numbe o s ange a ac o s which can be gene a ed wi h Chua´s oscilla o o m a zoo wi h mo e han 30 di e en exempla s (see [Chua e al., 1993] o a nice collec ion o colo pla es co esponding o all hese a ac o s). F om an in eg a ed design pe spec i e, canonical sys ems dese es special a en ion: Since sys em pa ame e s mus be mapped in o physical de ices, hose models wi h a mini- mum numbe o nonze o pa ame e s will be a p io i he mos ad an ageous in e ms o sys- em complexi y and a ea consump ion. In ou design, we ha e aken ad an age o he opological conjugacy p ope y o canonical sys ems in , no o ep oduce as much as possible dynamic beha io s, bu o iden i y which o hese sys ems is he bes sui ed o he monoli hic implemen a ion o a pa icula chao ic a ac o . Acco dingly, he beha io al le el desc ip ion o ou p o o ype ha e been ob ained a e applying he ollowing algo i hm: • Calcula e he eigen alues associa ed wi h he sys em candida e in whose a ac- o is o be ep oduced by canonical sys ems, up o opological conjugacy. • Iden i y he pa ame e alues which mus ake e e y canonical sys em in so ha co esponding eigen alues coincide wi h hose ob ained in he p e ious s ep. • Selec ha canonical sys em o hose p e iously iden i ied which sa is ies as close as possible a se o op imiza ion c i e ia de i ed om mic oelec onic expe ience. 5. P ope ly speaking, canonical sys ems a e said o be opologically conjuga e o he class , whe e is a se o ze o measu e. U+1 xℜ3 ∈D†xBp ={}= U1– xℜ3 ∈D†xBp –={}= L3 µ1µ2and µ3 , F() ν 1ν2and ν3 , L3L3 L3 L3 ˜L3ε0 –= ε0 L3 L3 L3 L3 In e na ional Jou nal o Bi u ca ion and Chaos 41 Le us examine each s ep o he algo i hm. The i s s ep begins wi h he selec ion o he pa icula chao ic a ac o o be syn he- sized. Among he wide numbe o candida es o e ed by he amily , we ha e conside ed he so-called double-sc oll a ac o , shown in Fig.34, which a ises om he well-known Chua´s ci cui [Chua, 1992]. The easons behind his elec ion is h ee old. Fi s , and mos impo an , because he e a e se e al expe imen al e idences using disc e e componen s ha he model allows he obse a ion o chaos synch oniza ion phenomena. Second, because he e is an ex ense heo e ical backg ound conce ning i s dynamic beha io [Madan, 1993], wha supposes an in aluable help du ing he syn hesis oo owa ds an in eg a ed p o o ype. Finally, because i is one o he simples models p oposed so a o he gene a ion o cha- o ic signals, and a p io i, will esul in a easie silicon implemen a ion. I is wo h no ing ha he double-sc oll a ac o has been p e iously syn hesized by mic oelec onic ci cui s (in ully monoli hic o m in [Rod íguez-Vázquez & Delgado-Res i- u o, 1993] and in pa ial monoli hic o m in [C uz & Chua, 1993]). A common ea u e o bo h chips is ha hei beha io al le el desc ip ion we e de i ed di ec ly om Chua´s ci - cui , and hence, no a emp o pe o mance op imiza ion om an IC design iewpoin was done. L3 F ig. 34. The Chua’s double-sc oll chao ic a ac o . 48 Bi u ca ions and Synch oniza ion using an In eg a ed P og ammable Chao ic Ci cui onic Ci cui s,” Pa A: Tu o ials and Re iews, IEEE T ans. on Ci cui s and Sys ems-I 40(10); Pa B: Bi u ca ion and Chaos, IEEE T ans. on Ci cui s and Sys ems-I 40(11); Pa C: Applica ions, IEEE T ans. on Ci cui s and Sys ems-II 40(10). Chua, L. O., Wu, C. W., Huang, A. & Zhong, G.-Q. [1993] “A Uni e sal Ci cui o S udying and Gene a ing Chaos -- Pa I: Rou es o Chaos, and Pa II: S ange A ac o s,” IEEE T ans. on Ci cui s and Sys ems-I 40(10), 732-761. C uz, J. M. & Chua, L. O. [1993] “An IC Chip o Chua’s Ci cui ,” IEEE T ans. on Ci cui s and Sys ems-II 40(10), 614-625. Delgado-Res i u o, M., Rod íguez-Vázquez, A., Espejo, S. & Hue as, J. L. [1992] “A Cha- o ic Swi ched-Capaci o Ci cui o 1/ γ Gene a ion,” IEEE T ans. on Ci cui s and Sys- ems 39(4), 325-328. Delgado-Res i u o, M., Medei o, F. & Rod íguez-Vázquez, A., [1993] “Nonlinea Swi ched- Cu en CMOS IC o Random Signal Gene a ion,” Elec onic Le e s 29(25), 2190-2191. Delgado-Res i u o, M. & Rod íguez-Vázquez, A., [1994] “Swi ched-Cu en Chao ic Neu- ons,” Elec onic Le e s 30(5), 429-430. F eeman, W. J. [1992] “Tu o ial on Neu obiology: F om Single Neu ons o B ain Chaos,” In . J. Bi u ca ion and Chaos 2(3), 451-482. Hasle , M. [1994] “Synch oniza ion P inciples and Applica ions”. P oc. o he 1994 IEEE In . Symp. on Ci cui s and Sys ems (Tu o ials), Chap e 6.2, 314-327. Hein, S. [1993] “Exploi ing Chaos o Supp ess Spu ious Tones in Gene al Double-Loop SD Modula o s,” IEEE T ans. on Ci cui s and Sys ems-II 40(10), 651-659. Ho io, H. & Suyama, K. [1995] “Expe imen al Ve i ica ion o Signal T ansmission Using Synch onized SC Chao ic Neu al Ne wo ks,” IEEE T ans. on Ci cui s and Sys ems-I 42(7), 393-395. Koca e , L. J., Halle, K. S., Ecke , K., Pa li z, U. & Chua, L. O. [1992] “Expe imen al Dem- ons a ion o Secu e Communica ions ia Chao ic Synch oniza ion,” In . J. Bi u ca ion and Chaos 2(4), 709-713. Madan, R. N. (edi o ) [1993] Chua’s Ci cui : A Pa adigm o Chaos (Wo ld Scien i ic, Sin- gapo e). Ma sumo o, G., Aiha a, K., Hanyu, Y., Takahashi, N., Yoshizawa, S. & Nagumo, J. [1987] “Chaos and Phase Locking in No mal Squid Axons,” Phys. Le . A123, 162-166. In e na ional Jou nal o Bi u ca ion and Chaos 49 McGonigal, G. C. & Elmas y, M. I. [1987] “Gene a ion o Noise by Elec onic I e a ion o he Logis ic Map,” IEEE T ans. on Ci cui s and Sys ems 34(8), 981-983. Mu ch, A. R. & Ba es, R. H. T. [1990] “Colo ed Noise Gene a ion h ough De e minis ic Chaos,” IEEE T ans. on Ci cui s and Sys ems 37(5), 608-613. Nozawa, H. [1992] “A Neu al Ne wo k Model as a Globally Coupled Map and Applica ions based on Chaos,” Chaos 2(3), 377-386. Oppenheim, A. V., Wo nell, G. W., Isabelle, S. H. & Cuomo, K. M. [1992] “Signal P ocess- ing in he Con ex o Chao ic Signals,” P oc. IEEE In . Con . on Acous ics, Speech and Signal P ocessing IV, 117-120. Rod íguez-Vázquez, A., Hue as, J. L., Rueda, A., Pé ez-Ve dú, B. & Chua, L. O., [1987] “Chaos om Swi ched-Capaci o Ci cui s: Disc e e Maps”. P oceedings o he IEEE 75(8), 1090-1106. Rod íguez-Vázquez, A., Espejo, S., Hue as, J. L. & Ma in, J. D. [1990] “Analog Building Blocks o Noise and T uly Random Numbe Gene a ion in CMOS VLSI,” P oc. Eu o- pean Con . on Solid-S a e Ci cui s, 225-228. Rod íguez-Vázquez, A., Delgado-Res i u o, M., Espejo, S. & Hue as, J. L. [1991] “Swi ched Capaci o B oadband Noise Gene a o o CMOS VLSI,” Elec onic Le e s 27(21), 1913-1915. Rod íguez-Vázquez, A. & Delgado-Res i u o, M. [1993] “CMOS Design o Chao ic Oscil- la o s Using S a e Va iables: A Monoli hic Chua’s Ci cui ,” IEEE T ans. on Ci cui s and Sys ems-II 40(10), 596-613. Rod íguez-Vázquez, A. & Delgado-Res i u o, M. [1994] “Gene a ion o Chao ic Signals using Cu en -Mode Techniques,” J. o In elligen and Fuzzy Sys ems 2(1), 15-37. Sch eie , R. [1991] “Noise-Shaped Coding,” Ph. D. Thesis, Uni e si y o To on o. 50 Bi u ca ions and Synch oniza ion using an In eg a ed P og ammable Chao ic Ci cui FIGURE CAPTIONS Fig. 1. (a) Chip a chi ec u e; (b) Expe imen al se up showing oscilloscope, chip wi h ou uning esis o s, and he ba e y pack. Fig. 2. Nonlinea i y o he chao ic oscilla o . Fig. 3. Va ia ion o he PWL cha ac e is ics o he nonlinea i y wi h: (a) ; (b) ; (c) he cen al slope, (con ol a iable ); and (d) he ou e slopes, (con- ol a iable ). Fig. 4. Au oma ic Tuning Mechanism. Fig. 5. Expe imen al Lissajous igu es, s a e wa e o ms, and powe spec um o he a iable o , . Fig. 6. Expe imen al Lissajous igu es, s a e wa e o ms, and powe spec um o he a iable o , . Fig. 7. Expe imen al Lissajous igu es, s a e wa e o ms, and powe spec um o he a iable o , . Fig. 8. Expe imen al Lissajous igu es, s a e wa e o ms, and powe spec um o he a iable o , . Fig. 9. Expe imen al Lissajous igu es, s a e wa e o ms, and powe spec um o he a iable o , . Fig. 10. Expe imen al Lissajous igu es, s a e wa e o ms, and powe spec um o he a iable o , . Fig. 11. Expe imen al Lissajous igu es, s a e wa e o ms, and powe spec um o he a iable o , . Fig. 12. Expe imen al Lissajous igu es, s a e wa e o ms, and powe spec um o he a iable o , . Fig. 13. Expe imen al Lissajous igu es, s a e wa e o ms, and powe spec um o he a iable o , . Fig. 14. Expe imen al Lissajous igu es, s a e wa e o ms, and powe spec um o he a iable o , . Fig. 15. Expe imen al Lissajous igu es, s a e wa e o ms, and powe spec um o he Icon 1 Icon 4m0Icon 2m1 Icon 3 x1Icon 21.0 µA=Icon 32.35 µA= x1Icon 21.04 µA=Icon 32.35 µA= x1Icon 21.065 µA=Icon 32.35 µA= x1Icon 21.07 µA=Icon 32.35 µA= x1Icon 21.12 µA=Icon 32.35 µA= x1Icon 21.135 µA=Icon 32.35 µA= x1Icon 21.15 µA=Icon 32.35 µA= x1Icon 21.24 µA=Icon 32.47 µA= x1Icon 21.47 µA=Icon 32.56 µA= x1Icon 21.62 µA=Icon 32.56 µA= In e na ional Jou nal o Bi u ca ion and Chaos 51 a iable o , . Fig. 16. Expe imen al Lissajous igu es, s a e wa e o ms, and powe spec um o he a iable o , . Fig. 17. Expe imen al Lissajous igu es, s a e wa e o ms, and powe spec um o he a iable o , . Fig. 18. Expe imen al Lissajous igu es, s a e wa e o ms, and powe spec um o he a iable o , . Fig. 19. Expe imen al Lissajous igu es, s a e wa e o ms, and powe spec um o he a iable o , . Fig. 20. Expe imen al Lissajous igu es, s a e wa e o ms, and powe spec um o he a iable o , . Fig. 21. Expe imen al Lissajous igu es, s a e wa e o ms, and powe spec um o he a iable o , . Fig. 22. (a) Expe imen al se up o an -linea coupling synch oniza ion scheme; (b)-(c) Co ela ion indexes be ween and , espec i ely. Fig. 23. Synch oniza ion pe o mance o he -linea coupling sys em o . Fig. 24. Measu emen s om an -linea coupling synch oniza ion scheme. (a)-(b) Co ela ion indexes be ween and , espec i ely. (c)-(d) Synch oniza ion pe o mance o . Fig. 25. (a) Mas e -Sla e simpli ied expe imen al se up; (b)-(c) Measu ed pe o - mance. Fig. 26. Synch oniza ion pe o mance o he -d i e sys em. Fig. 27. (a) Simpli ied expe imen al se up o he in e se sys em app oach; (b) Mea- su ed pe o mance. Fig. 28. Powe spec a o he (a) inpu signal; (b) ansmi ed signal; and (c) eco - e ed signal. Fig. 29. Time-domain pe o mance o he chao ic modula ion synch oniza ion scheme using wo in eg a ed p o o ypes. Fig. 30. Co ela ion index be ween he inpu and he eco e ed signals. x1Icon 21.60 µA=Icon 32.58 µA= x1Icon 21.65 µA=Icon 32.61 µA= x1Icon 21.70 µA=Icon 32.61 µA= x1Icon 21.72 µA=Icon 32.63 µA= x1Icon 21.79 µA=Icon 32.66 µA= x1Icon 21.81 µA=Icon 32.66 µA= x1Icon 21.85 µA=Icon 32.66 µA= x1 x 12 bu ,x22 bu , – x 13 bu ,x23 bu , – x1 R1200kΩ= x2 x 11 bu ,x21 bu , – x 13 bu ,x23 bu , – R225kΩ= x2 52 Bi u ca ions and Synch oniza ion using an In eg a ed P og ammable Chao ic Ci cui Fig. 31. Co ela ion index be ween he inpu and he ansmi ed signals. Fig. 32. Syn hesis ou e owa ds monoli hic nonlinea ci cui s. Fig. 33. Block diag am o he membe s o he amily . Fig. 34. The Chua’s double-sc oll chao ic a ac o . Fig. 35. E olu ion o he linea ized sys em eigen alues wi h pa ame e . Fig. 36. (a) Gm − C block diag am o he co e chao ic oscilla o ; (b) Ideal model o he linea ansconduc o s; (c) Ideal model o he PWL blocks. L3 m