Hybrid Analog Computer for Modeling Nonlinear Dynamical Systems: The Complete Cookbook

Ci a ion: Rujzl, M.; Polak, L.; Pe zela, J. Hyb id Analog Compu e o Modeling Nonlinea Dynamical Sys ems: The Comple e Cookbook. Senso s 2023,23, 3599. h ps:// doi.o g/10.3390/s23073599 Academic Edi o : Giuseppe Fe i Recei ed: 27 Feb ua y 2023 Re ised: 22 Ma ch 2023 Accep ed: 27 Ma ch 2023 Published: 30 Ma ch 2023 Copy igh : © 2023 by he au ho s. Licensee MDPI, Basel, Swi ze land. This a icle is an open access a icle dis ibu ed unde he e ms and condi ions o he C ea i e Commons A ibu ion (CC BY) license (h ps:// c ea i ecommons.o g/licenses/by/ 4.0/). senso s A icle Hyb id Analog Compu e o Modeling Nonlinea Dynamical Sys ems: The Comple e Cookbook Mi osla Rujzl , Ladisla Polak * and Ji i Pe zela Depa men o Radio Elec onics, Facul y o Elec ical Enginee ing and Communica ion, B no Uni e si y o Technology, Technicka 3082/12, 616 00 B no, Czech Republic; [email p o ec ed] (M.R.); pe zela@ u b .cz (J.P.) *Co espondence: [email p o ec ed] Abs ac : This pape desc ibes a design p ocess o a uni e sal de elopmen ki based on an analog compu e concep ha can model he dynamics o an a bi a ily complex dynamical sys em up o he ou h o de . The cons uc ed de elopmen ki con ains digi al blocks and associa ed analog- o-digi al and digi al- o-analog con e e s (ADCs and DAC), such ha mul iple-segmen ed piecewise-linea inpu –ou pu cha ac e is ics can be used o he syn hesis o he p esc ibed ma hema ical model. Polynomial inpu –ou pu cu es can be implemen ed easily by ou -quad an analog mul iplie s. The p oposed ki was e i ied h ough se e al expe imen al scena ios, s a ing wi h simple sinusoidal oscilla o s and ending wi h gene a o s o con inuous- ime obus chao ic a ac o s. The desc ip ion o each indi idual pa o he de elopmen ki is accompanied by links o echnical documen a ion, allowing skilled eade s in he cons uc ion o elec onic sys ems o eplica e he p oposed unc ional example. Fo his pu pose, he elec ical scheme o he hyb id analog compu e and all impo an sou ce codes a e a ailable online. Keywo ds: analog compu e ; ci cui syn hesis; chao ic sys em; s ange a ac o ; ans e unc ion 1. In oduc ion The cons uc ion o chao ic oscilla o s has been a a o i e opic o analog design engi- nee s and en husias s o many decades. The unique p ope ies o con inuous- ime chao ic signals s ongly mo i a e esea che s o de elop gene a o s o chaos ha a e obus wi h espec o a ia ions in undamen al ci cui pa ame e s and non-ideal in insic p ope ies o he used ac i e elemen s. The chao ic oscilla o s discussed in his pape can be unde s ood as lumped elec onic sys ems whe e ou pu ol ages o cu en s ep esen a s eady s a e ha is ex emely sensi i e o iny changes in ini ial condi ions. These sys ems exhibi a con inuous b oad-band equency spec um and p oduce dense s ange a ac o s wi h a p esc ibed geome ical shape. These ω -limi se s need o be long- e m s able and epea able, a oiding misin e p e a ion wi h chao ic ansien s. Based on he comple e knowledge o he ma hema ical model, which in ol es bo h o dina y di e en ial equa ions and nume ical alues o in e nal pa ame e s a e known, many sys ema ic app oaches can esul in he ci cui y implemen a ion o a chao ic os- cilla o con aining comme cially a ailable componen s only. This pape ocuses on one s aigh o wa d me hod ha has been known o decades: he analog compu e concep . The au ho s’ main aim is o p o ide a de ailed cookbook o cons uc ing a maximally uni e sal de elopmen ki based on he in eg a o block schema ic o a gene al class o ou h-o de dynamical sys ems. This uni e sali y includes a bi a ily shaped polynomial ou pu -inpu unc ions up o he ou h o de , a nonlinea ans e unc ion wi h digi ally con olled slopes o he piecewise-linea segmen s, as well as he numbe and posi ions o b eakpoin s, a iable complexi y o ini ial s a e equa ions (including he possibili y o model cyclically symme ical ec o ields and ac ional-o de nonlinea ci cui s), and he choice o ini ial condi ions (bo h posi i e and nega i e alues) ha can be imposed in o Senso s 2023,23, 3599. h ps://doi.o g/10.3390/s23073599 h ps://www.mdpi.com/jou nal/senso s Senso s 2023,23, 3599 2 o 17 he ci cui . In e es ed eade s will be di ec ed o in e ne s o age si es whe e all echnical documen a ion can be ound, including layou s o indi idual p in ed ci cui boa ds, lis s o passi e and ac i e componen s, design ules, e c. This pape is o ganized as ollows: The ollowing sec ion guides eade s h ough a ca e ully selec ed collec ion o pape s published on he opic o modeling de e minis ic nonlinea dynamics ia elec onic ci cui s, wi h pa icula emphasis on he analog compu e concep . The hi d sec ion discusses he undamen al composi ion o he de elopmen ki , including desc ip ions o he analog and digi al pa s, as well as he in e aces be ween indi idual unc ional blocks. The ou h sec ion deals wi h he cons uc ion o he p in ed ci cui boa d, p o iding ables wi h componen lis s and discussions o design equi e- men s. The i h sec ion is ull o demons a ions, beginning wi h ma hema ical models and ending wi h cap u ed oscilloscope sc eensho s. This sec ion compa es heo y and p ac ice, showing excellen ag eemen be ween he wo. Finally, a ew concluding ema ks a e p esen ed, which mainly ocus on addi ional possible imp o emen s o he p oposed unc ional example. 2. B ie O e iew o he Sys ema ic Design o Lumped Chao ic Oscilla o s One o he ea lies lumped chao ic sys ems was named a e i s in en o , P o . L. O. Chua, and was i s ealized in he ea ly 1980s. This sys em is s ill conside ed one o he simples ci cui y ealiza ions in e ms o o al componen coun . I consis s o a hi d-o de passi e ladde immi ance and a i e-segmen odd-symme ic piecewise linea (PWL) esis o [1–3] . A e i s disco e y, Chua’s oscilla o unde wen many modi ica ions mo i a ed by di e en equi emen s, such as emo ing he eal induc o in pa allel esonan ci cui s [ 4 ] o enabling high- equency ope a ions [ 5 ]. Despi e i s age-long bi h da e, his ci cui emains a a o i e educa ional example o demons a ing he bi h and dea h o chaos. I is also a sou ce o chao ic signals wi h a ious le els o en opy o p ac ical applica ions, including modula ion and secu e communica ion echniques. I is a ool used o he g aphical explana ion o chaos e olu ion and is a subjec o ad anced nume ical analysis in gene al [6]. The pa allel connec ion o a highe -o de wo- e minal immi ance and a nonlinea esis o is used in o he implemen a ions o lumped chao ic sys ems. Fo example, e e - ence [ 7 ] shows ha a passi e ladde s uc u e composed o esis o s and capaci o s can p oduce chao ic wa e o ms i e mina ed by a h ee-segmen PWL ac i e esis o wi h speci ic slopes. The main d awbacks o his concep , whe e he linea pa o he ec o ield is ealized by a passi e RC wo- e minal ne wo k, a e he lack o uni e sali y and anspa ency. The i s disad an age a ises om he passi i y o he wo- e minal immi - ance. Fo gi en nume ical alues o pa ame e s o he ma hema ical model (eigen alues associa ed wi h indi idual segmen s o he used PWL unc ion), he wo- e minal immi - ance can con ain one o se e al basic nega i e elemen s [ 8 ] o second-o de accumula ion elemen s [ 9 , 10 ]. The second disad an age is ela ed o he ela ionships be ween he pa- ame e s o he o iginal ma hema ical model and he alues o ci cui elemen s because hese can be oo complica ed o coun by hea . A gene aliza ion o he app oach discussed abo e owa d a bi a y-o de dimensional dynamical sys ems is p o ided in [ 11 ]. Bo h o he d awbacks men ioned abo e can be emo ed by adop ing a design me hod based on he in eg a o block schema ics o he dynamical sys em. I is wo h no ing ha only de e minis ic sys ems will be conside ed in he upcoming sec ions o his pape . An analog compu e can be hough o as an elec onic ci cui ha di ec ly sol es a se o i s -o de o dina y di e en ial equa ions. Di e en ia ion is ypically achie ed using a lossless in eg a o on he igh -hand side o he di e en ial equa ions; ˙ x= (x) a e ypically implemen ed in analog compu e s using di e en ial ampli ie s, in e ing ’summa o s,’ and wo po s wi h he p esc ibed inpu –ou pu cha ac e is ics. In mos cases, indi idual s a e a iables a e ol ages, which a e easily accessible and measu able a he ou pu s o in eg a o s [ 12 – 14 ]. Al hough analog compu e s end o ha e a highe numbe o ac i e elemen s, hese p ope ies can some imes be educed by using linea Senso s 2023,23, 3599 3 o 17 ans o ma ions o he coo dina es. In cu en -mode analog compu e s, simple ci cui nodes ep esen non-weigh ed summa ion blocks o cu en s, making hem po en ially simple han hei ol age-mode coun e pa s [ 15 ]. O cou se, pu ely o p ac ical easons, he comme cial a ailabili y o ac i e elemen s is he decisi e ac o o he ope a ional egime o an analog compu e . I simula ion esul s a e su icien o e i y ci cui design, a wide ange o hypo he ical ac i e elemen s can be conside ed, as discussed in [ 16 ]. The in e nal s uc u e o a e sa ile analog building block ab ica ed using 350 nm CMOS echnology and op imized o he syn hesis o complex nonlinea dynamical sys ems is desc ibed in [ 17 ]. The au ho s no e ha ans-conduc ance mode ac i e elemen s a e pa icula ly bene icial o he syn hesis o chao ic and hype chao ic elec onic sys ems. Le ’s s a wi h he simples examples o in eg a o -based chao ic oscilla o s. Re e - ence [ 18 ] demons a es se e al ci cui ealiza ions o he so-called je k unc ions. In hese cases, he second s a e a iable is he de i a i e o he i s one and he hi d s a e a iable is he ime de i a i e o he second one. Thus, he s a e a iables can be in e p e ed as posi- ion, eloci y, and accele a ion, making he ma hema ical model a membe o New onian mechanics. S a ing wi h ma hema ical models in he o m o a single hi d-o de o dina y di e en ial equa ion wi h an absolu e alue o signum nonlinea i y, he inal au onomous ci cui con ains a cascade connec ion o h ee ideal in e ing in eg a o s supplemen ed by semiconduc o diodes o compa a o s. Bo h simula ion and expe imen al esul s a e p o ided, showing excellen ag eemen be ween heo y and p ac ice. The je k dynamics wi h he absolu e alue nonlinea unc ion ep esen s he o iginal ma hema ical model o a chao ic oscilla o desc ibed in [ 19 ]. Phase po ai s and bi u ca ion diag ams calcula ed o a change in a single esis o a e p o ided, al hough only he simula ion esul s associa ed wi h he inal chao ic oscilla o a e p esen ed. The same design app oach associa ed wi h he linea pa o he ec o ield, bu wi h mo e ci cui ealiza ions o a ious nonlinea ans e unc ions, is o e ed o eade s in [ 20 ]. In [ 21 ], he au ho s desc ibe he algeb aically simples je k unc ion capable o exhibi ing obus chaos. The inal oscilla o con ains ou s anda d ope a ional ampli ie s, which a e a ailable as a single in eg a ed ci cui , and one diode. The au ho s also p o ide a pho o o an analog compu e econ igu able ki , al hough i is much less uni e sal han he de elopmen ki p esen ed in his pape . In [ 22 ], he au ho s subs i u e wo ac i e in eg a o s wi h passi e RC o RLC low-pass il e s. Such ci cui s can be conside ed a b idge be ween he wo ypes o ealiza ions men ioned abo e. A simple jump unc ion is implemen ed by an ope a ional ampli ie -based compa a o as he only nonlinea i y. The pu e concep o an analog compu e can be ound in many pape s dealing wi h he design o chao ic sys ems. Fo example, a s ep-by-s ep desc ip ion o a chao ic oscilla o based on a gi en se o o dina y di e en ial equa ions is p o ided in [ 23 ]. The e, he au ho s we e able o make a signi ican disco e y: a obus chao ic solu ion associa ed wi h a sys em ha ing one s able equilib ium. Nine algeb aic e ms a e p ac ically implemen ed using wo analog mul iplie s and six ol age eedback ope a ional ampli ie s. In he wo k [ 24 ], he au ho s p esen new chao ic dynamical sys ems wi h su aces o equilib ia and hei demons a ion ci cui y ealiza ions. As s a ed in he pape , eal measu emen s can be conside ed an al e na i e o nume ical simula ions, which a e always subjec o ounding and app oxima ion e o s and equi e a long ime. A simila ci cui concep , only wi h s anda d ope a ional ampli ie s and ou -quad an mul iplie s, is u ilized o expe imen al e i ica ion o sys ems wi h only quad a ic and cubic e ms in he [ 25 ]. In he design o a chao ic dynamical sys em wi h mul iple chao ic a ac o s p esen ed in [ 26 ], ou AD633 analog mul iplie s and he same numbe o ope a ional ampli ie s a e he ac i e elemen s employed. I is demons a ed ha each s ange a ac o e ealed du ing nume ical analysis can also be obse ed in he expe imen al measu emen o he p ac ical ci cui . Thus, imposing a se o ini ial condi ions in o he p ope nodes o he chao ic oscilla o is possible and seems o be an easy ask. The di icul , bu e en ually success ul ci cui y ealiza ion o a chao ic sys em wi h ci cula and squa e equilib ium is he subjec o e e ence [ 27 ]. The basin o a ac ion leading o he desi ed chao ic a ac o Senso s 2023,23, 3599 4 o 17 is a he na ow and loca ed a away om he o igin o he s a e space. Ne e heless, he au ho s we e able o e i y he p esc ibed chao ic a ac o o e a long ime (because o he chosen sho ime cons an o he u ilized in eg a o s) in his case as well. O cou se, chao ic dynamical sys ems ha ing o he geome ical shapes o degene a ed equilib ia can also be modeled using he analog compu e app oach. Fo example, in [ 28 ], he au ho s p esen a chao ic sys em wi h a ounded squa e equilib ium, and bo h nume ical analysis and ue expe imen al measu emen show e y good ag eemen . The ci cui ealiza ion comp ises only an in e ing summing in eg a o and analog mul iplie s AD633, wi hou di e en ial and summing ampli ie s. In [ 29 ], he au ho s p esen wel e new chao ic dynamical sys ems whe e a single in e nal pa ame e can be used o s udy he in e ac ions be ween he gene a ed chao ic a ac o and he mul iple-dimensional equilib ium s uc u e, such as planes, cu ed planes, ings, and sphe es. Fo expe imen s, wo chao ic sys ems a e chosen as examples and ealized as elec onic ci cui s using cheap and o - he-shel ope a ional ampli ie s LM741. The equi ed quad a ic e ms a e implemen ed using analog mul iplie s connec ed as squa e s, wi h nine and se en such ac i e elemen s needed o he i s and second oscilla o s, espec i ely. Chao ic ci cui s designed using he analog compu e app oach, especially hose employing ope a ional ansconduc ance ampli ie s only, can se e as a pa e n o ull on-chip implemen a ions using CMOS echnology [ 30 ]. Ad an ageously, hese s uc u es con ain no esis o s, i.e., he inal oscilla o s occupy e y small chip a eas. In cases whe e he nonlinea i y has a e y complica ed shape, i s ealiza ion using disc e e analog componen s can be di icul , and e en impossible. Fo example, in [ 31 ], he au ho s ocus on he ci cui design o a sys em ha gene a es a mul i-wing bu e ly a ac o . The co esponding ma hema ical model con ains one scala nonlinea unc ion in he o m o a sum o signum jumps wi h di e en b eakpoin s. Since each jump unc ion mus be implemen ed using a compa a o , a e y complex inal ci cui can be expec ed. This assump ion is p o en co ec since he co esponding oscilla o con ains mo e han i y ope a ional ampli ie s. The sine unc ion can be app oxima ed by a ini e powe se ies con aining only e en powe s o he inpu a iable. Simila ly, he cosine unc ion app oxima ion nea he o igin con ains odd powe s o he inpu a iable. The u iliza ion o he cosine unc ion in a non-au onomous chao ic oscilla o (d i en by a single- one sinusoidal signal) based on he analog compu e concep is sugges ed in [ 32 ], al hough eal measu emen esul s a e no p o ided. Bo h igonome ic unc ions men ioned abo e can be used in he so-called laby in h chaos gene a o [ 33 ]. Howe e , low sys em dissipa ion leads o a la ge s a e space olume occupied by he desi ed a ac o . Since achie ing a wide dynamical ange o ac i e de ices can be p oblema ic, a digi al app oach is sugges ed in [ 34 ]. Because he ou pu o digi al blocks is no con inuously alued smoo h unc ion, he esul ing ma hema ical model is piecewise cons an a he han PWL. Ne e heless, bo h simula ion and expe imen al esul s emain in good ag eemen , p o ing his concep o digi aliza ion o nonlinea eedback unc ion. An ex ension o he in eg a o block schema ic o a bi a y-dimensional ma hema ical models is no a p oblem, making i a good choice o hype chao ic sys ems [ 35 ]. The analog compu e idea can be applied no only o make a ci cui equi alen o a gi en ma hema ical model bu also o accu a e dynamical modeling o exis ing lumped elec onic ne wo ks o any kind [ 36 , 37 ], including non-elec onic sys ems. Howe e , since each elec onic sys em con ains many ac i e elemen s, hei pa asi ic p ope ies can se iously a ec he global dynamics. Thus, o emo e unwan ed dis u bances a highe equencies, nume ical alues o pa asi ic p ope ies should be kep as negligible as possible by p ope choice o he ime cons an . Fo u he eading, e e ence [38] is ecommended. Besides ully analog and mixed analog-digi al ealiza ions o chao ic dynamical sys- ems, he e a e o he possibili ies o he expe imen al e i ica ion o he long- e m s uc- u al s abili y o a chao ic ci cui ’s egime. A Field P og ammable Analog A ay (FPAA) de elopmen ki usually o e s a g aphical use in e ace and limi ed-size window whe e p e-de ined building blocks such as ’summa o s’, il e s, and in eg a o s can be easily Senso s 2023,23, 3599 5 o 17 placed and in e connec ed. In [ 39 ], bo h analog compu e -based and FPAA-based eal- iza ions o ac ional-o de chao ic sys ems a e p oposed. This ac ionali y means ha one o se e al di e en ial equa ions a e o non-in ege o de , usually be ween ze o and one. Fo una ely, such ma hema ical models can be implemen ed by an analog compu e , using a con en ional in e ing in eg a o whe e he capaci o is eplaced wi h a ac al capaci o (a wo- e minal de ice ha beha es simila ly o a hyb id be ween a esis o and capaci o ) app oxima ed in he equency domain. Re e ence [ 40 ] shows ano he op ion o documen ing he exis ence o chaos as a solu ion o some dynamical sys em. A ield p og ammable ga e a ay (FPGA) pla o m can be conside ed a b idge be ween ully ana- log ealiza ion and pu ely ma hema ical analysis. In his wo k, he au ho s add ess he in e es ing p oblem o he co-exis ence o equilib ium poin s and s ange a ac o s. They disco e a dynamical sys em ha gene a es ei he a sel -exci ed o hidden chao ic a ac o depending on he nume ical alue o an in e nal pa ame e . E en his e y special ea u e does no p e en he design enginee om successi e ealiza ion. O cou se, in e es ing FPGA-based ealiza ions can be ound in o he publica ions, such as [41–45]. I should be poin ed ou ha all o he pape s men ioned abo e con ain much mo e aluable pa s han ci cui ealiza ions o chao ic sys ems, namely deep nume ical in es- iga ions, including bi u ca ion diag ams, he spec um o Lyapuno exponen s, basins o a ac ion o indi idual a ac o s, e c. Mo eo e , he p o ided lis o pape s dealing wi h he syn hesis o chao ic oscilla o s is by no means comple e. Fo a u he s udy o he design p inciples leading o chaos gene a o s, we sugges e e ing o he comp ehensi e e iew wo k [ 46 ]. In addi ion o he nume ous pape s ci ed he ein, a as numbe o aluable pape s can also be ound h ough an in e ne sea ch. Mem-elemen s belong o p omising ci cui componen s ha con ain bo h nonlinea i y and in e nal ine ia. Combined wi h p ope addi ional ci cui y, hese a e good candida es o he gene a ion o obus chaos. Mo eo e , mem-elemen s can be included in s anda d analog compu e s and b oaden hei applica ion po en ial, as sugges ed in [47]. Many esea che s s ill ocus hei a en ion on he p oblem o inding new chao ic dynamical sys ems wi h speci ic p ope ies o ma hema ical desc ip ions o some eal physical phenomena. Since ci cui y ealiza ion and expe imen al e i ica ion a e s anda d componen s o such scien i ic pape s [ 48 ], he numbe o no el chao ic ci cui s will likely inc ease in he nea u u e. Thus, he impo ance o a e sa ile analog de elopmen pla o m is gua an eed. 3. Cons uc ion o a Hyb id Compu e As b ie ly men ioned ea lie , analog compu e echniques can be used o model and simula e nonlinea dynamical sys ems in he con inuous ime domain. Howe e , pu ely analog compu e s canno p ocess chao ic signals using digi al-based algo i hms. In his pape , a hyb id compu e is in oduced o analyze sys ems exhibi ing chao ic solu ions in bo h he con inuous and disc e e domains. The hyb id compu e is composed o classical analog elemen s, supplemen ed by ADCs, digi al po en iome e s, and con ol ci cui s. Figu e 1shows he block diag am o he p oposed hyb id compu e ; Figu e 2shows i s p ac ical ealiza ion. The indi idual componen s o he p oposed concep a e implemen ed as sepa a e ci cui s and hen connec ed by ex e nal wi es acco ding o he gi en scheme. The hyb id compu e is highly e sa ile and can be used no only o he simula ion o nonlinea dynamic sys ems. The powe supply ol age is symme ical: ± 15 V o he analog pa and 5 V o he digi al pa . Some blocks a e designed o be pu ely analog, while o he s ha e analog unc ions complemen ed by digi al pa s, allowing o he se ing o indi idual blocks o he sensing o analog quan i ies o digi al signal p ocessing. Fu he in o ma ion, including sou ce iles equi ed o ebuild he hyb id compu e , can be ound a h ps://www.gi hub.com/m ujzl/Hyb id_compu e _ 1 (accessed on 27 Feb ua y 2023) In he ollowing sec ions, he indi idual componen s o he p oposed hyb id compu e a e desc ibed in de ail. Senso s 2023,23, 3599 6 o 17 ++ MCU ATMEGA 64 Keypad + display IC ADC PWL ∫ ∫ ∫ ∫ ∑ ∑ ∑ ∑ + + + + + + -1 -1 -1 -1 A A A A SRC 1 SRC 2 SRC 3 Po en iome e s Digi al po en iome e s Ampli ie s Inpu s Summa o s In eg a o s Ou pu s Mul iplie s In e o s Cons an sou ces PC Figu e 1. Block diag am o he p oposed hyb id compu e . Figu e 2. P in ed ci cui boa d (PCB) o he p oposed hyb id compu e . 3.1. In eg a o s and ’Summa o s’ In eg a o s a e he undamen al analog blocks o sol ing sys ems desc ibed by o di- na y di e en ial equa ions. The in oduced hyb id compu e consis s o ou in eg a o blocks implemen ed using comme cially a ailable monoli hic op-amp AD844. When he compensa ion e minal is u ilized, AD844 ac s as a second-gene a ion posi i e cu en con eyo (CCII+) wi h a decoupling ol age ollowe . The ad an age o using AD844 is he ease o implemen ing a swi chable sign o he cons an o in eg a ion, allowing o bo h posi i e and nega i e in eg a o s o be ealized. Ano he ad an age o using his elemen is Senso s 2023,23, 3599 7 o 17 he g ounded capaci o , which acili a es he ealiza ion o ini ial condi ion se ings wi hou he p esence o la ge leakage cu en s. The change in sign is ensu ed by a le e swi ch. Nex , a simple o a y swi ch can be used o change he ime cons an o he in eg a o . I is possible o choose om ou ime cons an s (1 µ s, 100 µ s, 10 ms , 1s) and one ex e nally connec able cons an . Such a solu ion allows o he use o an a bi a y alue o he ime cons an o o connec a ac al elemen o an in eg a ion membe , inc easing he e sa ili y o he compu e . The whole in eg a o can be desc ibed as ollows: uou ( ) = ±1 RC Zuin( )d +uIC(0), (1) whe e R and C deno e he esis o s and capaci o s o ming he selec ed ime cons an , and uIC is he ol age ac oss he capaci o s a ime = 0 s, i.e., he ini ial condi ion o he sys em, which can be se by he IC block (see below). The inpu o indi idual in eg a o s a e usually ’summa o s’; he numbe in he p e- sen ed hyb id compu e is he same as he numbe o in eg a o s, i.e., ou summing elemen s. Howe e , he ou pu s o hese ’summa o s’ can be disconnec ed om he inpu s o he in eg a o s. This solu ion is sui able i he simula ed sys em needs o be sol ed using he me hod o dec easing he o de o de i a i e (explained in he sec ion below). The summa ion blocks a e implemen ed by a s anda d ci cui using he LT1209 ope a ional ampli ie , which adds he applied ol ages wi h opposi e signs. This opposi e sign can be compensa ed by swi ching in eg a o s. Consequen ly, he ’summa o s’ can be desc ibed as: uou ( ) = − N ∑ n=1 un( )(2) whe e un( ) is he indi idual inpu ol age. The ’summa o s’ a e designed o eigh inpu s; he e o e, N=8. 3.2. Mul iplie s and Coun ing-Based Po en iome e s The designed analog compu e should be used p ima ily o simula ions o non- linea dynamic sys ems wi h chao ic o hype chao ic beha io . In gene al, wo ypes o nonlinea i ies a e used in he modeling o hese sys ems, namely piecewise linea (PWL) unc ions and polynomial nonlinea i y. To c ea e polynomial nonlinea i ies, he analog compu e has eigh (analog) mul iplie s ha ope a e in all ou quad an s. Fo his pu pose, a well-known in eg a ed ci cui AD633 wi h a supply ol age in a ange o ± 15 ˜ V is employed. Thanks o a simple di iding cons an , i allows o easy implemen a ion o he mul iplica ion o wo ol ages. The o mula o he ou pu ol age is: uou ( ) = [x1( )−x2( )][(y1( )−y2( )] K+z( )(3) whe e x1( ) , x2( ) , y1( ) , y2( ) a e he inpu ol ages used o c ea e he inal p oduc , z( ) is he inpu ol age ha can be added o he inal p oduc , and K is he so-called in e - nal di ision cons an ( K= 10 o he ci cui AD633). This cons an mus be aken in o accoun when p og amming he analog compu e and p epa ing he p og amming scheme. Howe e , due o i s sui able decimal alue, he elimina ion o his componen is easie . I is no necessa y o use analog mul iplie s only o he ealiza ion o nonlinea i ies. I one o he inpu ol ages is a cons an , i is possible o implemen a simple mul iplica ion by a cons an (e.g., a sys em pa ame e ) in his way. The coun ing-based po en iome e s a e connec ed in he o m o ol age di ide s, which in ol es a s anda d connec ion me hod used in analog compu e s. This opology allows o ol age mul iplica ion by a cons an in he ange o 0 o 1. The sol ed asks a e ypically no malized o he size o he machine uni , which e e s o he ol age alue ha he solu ion in he s a e space should no exceed in any dimension. Du ing no maliza ion, Senso s 2023,23, 3599 8 o 17 he alues o he coe icien s o he indi idual e ms o he equa ions a e changed, esul ing in alues ypically in he ange o 0 o 1. In he designed analog compu e , he size o he machine uni is se o 10 V. In addi ion o s anda d analog po en iome e s, ol age di ide s a e also implemen ed using wo digi al po en iome e s, which can be con olled bo h wi h in e nal con ol elemen s and wi h a compu e applica ion. 3.3. In e e s, Ampli ie s, Cons an Sou ces Some basic ci cui blocks ha allow o he simula ion o dynamic sys ems in se e al ways a e included in he analog pa o he p esen ed hyb id compu e . These blocks p ima ily consis o ou in e ing ope a ional ampli ie s wi h gain one, which ealize an o dina y ol age in e e o he connec ed a iable. In addi ion, he e a e ou non-in e ing ampli ie s a ailable, including a po en iome e in he eedback loop. Such a con igu a ion o non-in e ing ope a ional ampli ie s allows o he ealiza ion o mul iplica ion ope a ions by a cons an (which can be se in a ange o 1 o 5) gi en by he alue o he po en iome e . The use o hese blocks makes i possible o no malize he simula ed sys em, e en o a a iable ha does no con ain he highes alue o he pa ame e . Howe e , his op ion equi es mo e expe ience in he de i a ion o p og amming schemes. Cons an ol age sou ces a e he inal analog-based componen s o he p oposed concep . They allow o he se ing o he ou pu ol age in he ange o he analog componen ’s supply ol age, which is symme ically ± 15 V. In his design, cons an ol age sou ces a e u ilized o implemen ing pa ame e s in he analog mul iplie s (see Sec ion 3.2). Since he mul iplie s implemen o mula (3), he ou pu ol age o he cons an ol age sou ce can be used as one o he ol ages, enabling a wide ange o sys em pa ame e alues o be ealized. A he same ime, he di iding cons an s o he mul iplie s can be exploi ed o inc eased e sa ili y. 3.4. Digi ally Con olled Componen s The p esen ed hyb id compu e , in addi ion o he p e iously men ioned digi al po en iome e s ealizing he so-called coun ing po en iome e s, con ains o he digi al componen s, such as an IC block, PWL block, and ADCs o moni o ing he ou pu s o in eg a o s (s a e a iables). All digi al componen s a e con olled by he main p ocesso ATMEGA64 o he hyb id compu e , and can be achie ed in wo ways. By de aul , he digi al pa s can be con olled om an in eg a ed keyboa d, while in o ma ion abou he cu en s a e o each block is displayed on an in eg a ed OLED display. Ano he op ion is o p o ide con ol by commands ia a pe sonal compu e . In o de o simpli y and imp o e he wo k wi h he command-based sys em, an app op ia e desk op applica ion (g aphical use in e ace), see Figu e 3was c ea ed in he in e ac i e de elopmen en i on- men App Designe (h ps://www.ma hwo ks.com/p oduc s/ma lab/app-designe .h ml) (accessed on 27 Feb ua y 2023) which is a ailable in he MATLAB p og am en i onmen . As men ioned in Sec ion 3.1, he in eg a o blocks ha e he abili y o disconnec he capaci o s and p e-cha ge hem o a speci ic ol age alue, allowing he ini ial condi ions o he sys em o be se . A digi ally con olled ini ial condi ion (IC) block is a ailable in he hyb id compu e . I is implemen ed using a digi al po en iome e connec ed as a ol age di ide (in he ange o he supply ol age) wi h a decoupling op-amp. In his wo k, he MAX5437 digi al po en iome e om Maxim In eg a ed is used, which allows i s alue o be changed in 128 s eps. Gi en a ol age ange o 30 V, his co esponds o a change o app oxima ely 0.2 V pe s ep. The IC block is con olled by he main p ocesso ia he se ial pe iphe al in e ace (SPI). Senso s 2023,23, 3599 9 o 17 Figu e 3. Desk op applica ion o con olling he hyb id compu e . A PWL block is implemen ed o c ea e nonlinea i ies using a piecewise-linea unc ion. Fo his pu pose, he comme cially a ailable AD844 ci cui is connec ed as a ol age ampli ie . By changing he pola i y o he ampli ie , which is ealized by a swi ch, he quad an s in which he PWL unc ion is loca ed a e also changed. The esis o connec ed o he compensa ion node o he AD844 ci cui is implemen ed as a digi al po en iome e , allowing he gain alue o be changed in s eps. A he inpu o he ampli ie , he AD7322 analog- o-digi al con e e is used; wi h i s maximum bipola ange o ± 10 ˜ V, i co e s he ange gi en by he machine uni . The whole PWL block is con olled by a sepa a e ATMEGA8 mic op ocesso uni (MCU). The MCU con ains a lookup able (modi iable by a compu e applica ion) ha includes he ol age sec ions and hei co esponding gain alues. I he p ocesso de ec s a ansi ion be ween he ol age sec ions, de e mined by he lookup able ADC, i changes he esis ance o he digi al po en iome e o se he gain o he desi ed alue. Con olling his block ia he in eg a ed con ol uni s would be complica ed, so i is he only block con olled ia he c ea ed desk op applica ion. Measu ed esul s a e o en equi ed o be in digi al o m. Fo his eason, wo AD7322 dual-channel ADCs a e employed a he ou pu s o he in eg a o s (s a e a iables, e- spec i ely), which allow (as in he PWL block) ope a ing in he 10 ˜ V bipola ange o he machine uni . The ADCs a e di ec ly connec ed o he ATMEGA64 MCU ia he SPI bus. The eco ding unc ion used o collec digi al esul s has wo modes. The i s mode allows di ec con ol ia he in eg a ed keyboa d, enabling he con ol p ocesso o ead da a om he ADCs wi h he maximum da a a e and send he da a di ec ly o he se ial line o he compu e . This mode sends a aw da a s eam wi hou any egula ion, sending he da a ega dless o whe he hey a e ecei ed o no . In he second mode, connec ion ia he desk op applica ion is assumed. The desk op applica ion, when he eco ding mode op ion is selec ed, allows us o se he sampling a e o he ADC om 1Hz o 50 kHz. A he beginning o he eco ding, he eco ded da a a e di ec ly displayed in g aphs ha show all a ailable a iable planes (XY, XZ, XW, YZ, YW, ZW). Nex , he da a (as eco ded a ac o s) can be sa ed as images ( eco ded a ac o s) o as da a iles (in o ma s such as CSV, TXT, e c.). Thus, he eco ded da a can be analyzed o line by o he ools o algo- i hms ha a e gene ally used o s udying chao ic sys ems (e.g., Lyapuno exponen s o bi u ca ion analyses). 4. Simula ions wi h Hyb id Compu e s Va ious ypes o calcula ions and simula ions can be pe o med on analog and hyb id compu e s. In addi ion o he al eady men ioned s udy o linea and nonlinea dynamical sys ems, hese de ices can be used o sol e ans e unc ions, sys ems o algeb aic equa- ions, op imiza ion p oblems, and ind complex oo s o polynomials [ 49 ]. Fo his pu pose, he ollowing i e poin s mus be pe o med o a success ul simula ion: 1. Ob aining a ma hema ical desc ip ion o he p oblem o be sol ed; Senso s 2023,23, 3599 16 o 17 13. Sambas, A.; Vaidyana han, S.; Tlelo-Cuau le, E.; Zhang, S.; Guillen-Fe nandez, O.; Hidaya , Y.; Gunda a, G. A no el chao ic sys em wi h wo ci cles o equilib ium poin s: Mul is abili y, elec onic ci cui and FPGA ealiza ion. Elec onics 2019 ,8, 1211. [C ossRe ] 14. Pe zela, J.; H ubos, Z.; Go hans, T. Modeling de e minis ic chaos using elec onic ci cui s. Radioenginee ing 2011,20, 438–444. 15. Pe zela, J.; Go hans, T.; Guzan, M. Cu en -mode ne wo k s uc u es dedica ed o simula ion o dynamical sys ems wi h plane con inuum o equilib ium. J. Ci cui s Sys . Compu . 2018,27, 1830004. [C ossRe ] 16. Biolek, D.; Senani, R.; Biolko a, V.; Kolka, Z. Ac i e elemen s o analog signal p ocessing: Classi ica ion, e iew, and new p oposals. Radioenginee ing 2008,17, 15–32. 17. Pe zela, J.; So ne , R. New nonlinea ac i e elemen dedica ed o modeling chao ic dynamics wi h complex polynomial ec o ields. En opy 2019,21, 871. [C ossRe ] 18. Sp o , J.C. Simple chao ic sys ems and ci cui s. Am. J. Phys. 2000,68, 758–763. [C ossRe ] 19. Kie s, K.; Klein, T.; Kolb, J.; P ice, S.; Sp o , J.C. Chaos in a nonlinea analog compu e . In . J. Bi u c. Chaos 2004 ,14, 2867–2873. [C ossRe ] 20. Sp o , J.C. A new class o chao ic ci cui . Phys. Le . A 2000,266, 19–23. [C ossRe ] 21. Sp o , J.C. A new chao ic je k ci cui . IEEE T ans. Ci cui s Sys . II Exp ess B ie s 2011,58, 240–243. [C ossRe ] 22. Pipe , J.R.; Sp o , J.C. Simple au onomous chao ic ci cui s. IEEE T ans. Ci cui s Sys . II Exp ess B ie s 2010 ,57, 730–734. [C ossRe ] 23. Kapi aniak, T.; Mohammadi, S.A.; Mekhile , S.; Alsaadi, F.E.; Haya , T.; Pham, V.T. A new chao ic sys em wi h s able equilib ium: En opy analysis, pa ame e es ima ion, and ci cui design. En opy 2018,20, 670. [C ossRe ] 24. Ja a i, S.; Sp o , J.; Pham, V.T.; Volos, C.; Li, C. Simple chao ic 3D lows wi h su aces o equilib ia. Nonlinea Dyn. 2016 ,86, 1349–1358. [C ossRe ] 25. Alma oud, O.A.; Tamba, V.K.; G assi, G.; Pham, V.T. An oscilla o wi hou linea e ms: In ini e equilib ia, chaos, ealiza ion, and applica ion. Ma hema ics 2021,9, 3315. [C ossRe ] 26. Lai, Q.; Akgul, A.; Li, C.; Xu, G.; Ça u¸so˘glu, Ü. A new chao ic sys em wi h mul iple a ac o s: Dynamic analysis, ci cui ealiza ion and S-Box design. En opy 2017,20, 12. [C ossRe ] 27. Go hans, T.; Sp o , J.C.; Pe zela, J. Simple chao ic low wi h ci cle and squa e equilib ium. In . J. Bi u c. Chaos 2016 ,26, 1650137. [C ossRe ] 28. Pham, V.T.; Ja a i, S.; Volos, C.; Giakoumis, A.; Vaidyana han, S.; Kapi aniak, T. A chao ic sys em wi h equilib ia loca ed on he ounded squa e loop and i s ci cui implemen a ion. IEEE T ans. Ci cui s Sys . II Exp ess B ie s 2016,63, 878–882. [C ossRe ] 29. Li, C.; Peng, Y.; Tao, Z.; Sp o , J.C.; Ja a i, S. Coexis ing in ini e equilib ia and chaos. In . J. Bi u c. Chaos 2021 ,31, 2130014. [C ossRe ] 30. Valencia-Ponce, M.A.; González-Zapa a, A.M.; de la F aga, L.G.; Sanchez-Lopez, C.; Tlelo-Cuau le, E. In eg a ed Ci cui Design o F ac ional-O de Chao ic Sys ems Op imized by Me aheu is ics. Elec onics 2023,12, 413. [C ossRe ] 31. Ma, J.; Wang, L.; Duan, S.; Xu, Y. A mul i-wing bu e ly chao ic sys em and i s implemen a ion. In . J. Ci cui Theo y Appl. 2017 , 45, 1873–1884. [C ossRe ] 32. Pham, V.T.; Ali, D.S.; Al-Saidi, N.M.; Rajagopal, K.; Alsaadi, F.E.; Ja a i, S. A no el mega-s able chao ic ci cui . Radioenginee ing 2020,29, 140–146. [C ossRe ] 33. Sp o , J.C.; Chlou e akis, K.E. Laby in h chaos. In . J. Bi u c. Chaos 2007,17, 2097–2108. [C ossRe ] 34. Go hans, T.; Pe zela, J. Expe imen al s udy o he sampled laby in h chaos. Radioenginee ing 2011,20, 873–879. 35. Liu, L.; Liu, C. Theo e ical analysis and ci cui e i ica ion o ac ional-o de chao ic beha io in a new hype chao ic sys em. Ma h. P obl. Eng. 2014,2014, 682408. [C ossRe ] 36. Pe zela, J. E idence o s ange a ac o s in class C ampli ie wi h single bipola ansis o : Polynomial and piecewise-linea case. En opy 2021,23, 175. [C ossRe ] 37. Pe zela, J. Chao ic and hype chao ic dynamics o a Clapp oscilla o . Ma hema ics 2022,10, 1868. [C ossRe ] 38. Munoz-Pacheco, J.; Tlelo-Cuau le, E.; Toxqui-Toxqui, I.; Sanchez-Lopez, C.; T ejo-Gue a, R. F equency limi a ions in gene a ing mul i-sc oll chao ic a ac o s using CFOAs. In . J. Elec on. 2014,101, 1559–1569. [C ossRe ] 39. Sil a-Juá ez, A.; Tlelo-Cuau le, E.; De La F aga, L.G.; Li, R. FPAA-based implemen a ion o ac ional-o de chao ic oscilla o s using i s -o de ac i e il e blocks. J. Ad . Res. 2020,25, 77–85. [C ossRe ] 40. Ka hikeyan, R.; Aki , A.; Ja a i, S.; Ani ha, K.; Ismail, K. Chao ic chameleon: Dynamic analysis, ci cui implemen a ion, FPGA design and ac ional-o de o m wi h basic analysis. Chaos Soli ons F ac als 2017,103, 476–487. 41. Rajagopal, K.; Ka hikeyan, A.; Du aisamy, P. Hype chao ic chameleon: F ac ional o de FPGA implemen a ion. Complexi y 2017 , 2017, 8979408. [C ossRe ] 42. Rajagopal, K.; Li, C.; Naza imeh , F.; Ka hikeyan, A.; Du aisamy, P.; Ja a i, S. Chao ic dynamics o modi ied Wien b idge oscilla o wi h ac ional o de mem is o . Radioenginee ing 2019,28, 165–174. [C ossRe ] 43. Rajagopal, K.; Ka hikeyan, A.; S ini asan, A. Dynamical analysis and FPGA implemen a ion o a chao ic oscilla o wi h ac ional-o de mem is o componen s. Nonlinea Dyn. 2018,91, 1491–1512. [C ossRe ] 44. Benkouide , K.; Vaidyana han, S.; Sambas, A.; Tlelo-Cuau le, E.; El-La i , A.A.A.; Abd-El-A y, B.; Be mudez-Ma quez, C.F.; Sulaiman, I.M.; Awwal, A.M.; Kumam, P. A New 5-D Mul is able Hype chao ic Sys em Wi h Th ee Posi i e Lyapuno Exponen s: Bi u ca ion Analysis, Ci cui Design, FPGA Realiza ion and Image Enc yp ion. IEEE Access 2022,10, 90111–90132. [C ossRe ] Senso s 2023,23, 3599 17 o 17 45. Lahcene, M.; Nou eddine, C.; Lo enz, P.; Adda, A.P. Secu ing in o ma ion using a p oposed eliable chaos-based s eam ciphe : Wi h eal- ime FPGA-based wi eless connec ion implemen a ion. Nonlinea Dyn. 2023,111, 801–830. [C ossRe ] 46. Pe zela, J. Chaos in Analog Elec onic Ci cui s: Comp ehensi e Re iew, Sol ed P oblems, Open Topics and Small Example. Ma hema ics 2022,10, 4108. [C ossRe ] 47. Ding, S.; Wang, N.; Bao, H.; Chen, B.; Wu, H.; Xu, Q. Mem is o synapse-coupled piecewise-linea simpli ied Hop ield neu al ne wo k: Dynamics analysis and ci cui implemen a ion. Chaos Soli ons F ac als 2023,166, 112899. [C ossRe ] 48. Sp o , J.C. A p oposed s anda d o he publica ion o new chao ic sys ems. In . J. Bi u c. Chaos 2011 ,21, 2391–2394. [C ossRe ] 49. Busca ino, A.; Fo una, L.; F asca, M.; Sciu o, G. A Concise Guide o Chao ic Elec onic Ci cui s; Sp inge : Be lin/Heidelbe g, Ge many, 2014. 50. Sp o , J.C.; Linz, S.J. Algeb aically simple chao ic lows. In . J. Chaos Theo y Appl. 2000,5, 1–20. 51. Ma wan, N.; Romano, M.C.; Thiel, M.; Ku hs, J. Recu ence plo s o he analysis o complex sys ems. Phys. Rep. 2007 ,438, 237–329. [C ossRe ] 52. Sp o , J.C. Some simple chao ic lows. Phys. Re . E 1994,50, R647. [C ossRe ] [PubMed] 53. H ubos, Z.; Go hans, T.; Pe zela, J. No el ci cui implemen a ion o he Nóse-Hoo e he mos a ed dynamic sys em. In P oceedings o he 2011 34 h In e na ional Con e ence on Telecommunica ions and Signal P ocessing (TSP), Budapes , Hunga y, 18–20 Augus 2011; pp. 307–311. 54. Ka imo , T.; Nepomuceno, E.G.; D uzhina, O.; Ka imo , A.; Bu uso , D. Chao ic oscilla o s as induc i e senso s: Theo y and p ac ice. Senso s 2019,19, 4314. [C ossRe ] [PubMed] 55. Ka imo , T.; D uzhina, O.; Va nik, V.; I ano a, E.; Kulagin, M.; Ponoma e a, V.; Vo oshilo a, A.; Rybin, V. Sensi i i y Op imiza ion and Expe imen al S udy o he Long-Range Me al De ec o Based on Chao ic Du ing Oscilla o . Senso s 2022 ,22, 5212. [C ossRe ] [PubMed] 56. Ko ne a, W.; Ga cia-Mo eno, E.; Sena, A.L. Noise ac i a ed dc signal senso based on chao ic Chua ci cui . Commun. Nonlinea Sci. Nume . Simul. 2015,24, 145–152. [C ossRe ] 57. Pospíšil, J.; Kolka, Z.; Ho ská, J.; B zoboha ` y, J. Simples ODE equi alen s o Chua’s equa ions. In . J. Bi u c. Chaos 2000 ,10, 1–23. [C ossRe ] Disclaime /Publishe ’s No e: The s a emen s, opinions and da a con ained in all publica ions a e solely hose o he indi idual au ho (s) and con ibu o (s) and no o MDPI and/o he edi o (s). MDPI and/o he edi o (s) disclaim esponsibili y o any inju y o people o p ope y esul ing om any ideas, me hods, ins uc ions o p oduc s e e ed o in he con en .