Fracmemristor chaotic oscillator with multistable and antimonotonicity properties

F acmem is o chao ic oscilla o wi h mul is able and an imono onici y p ope ies Haikong Lu a , Ji i Pe zela b, ⇑ , Tomas Go hans b , Ka hikeyan Rajagopal c , Sajad Ja a i d , Iq ada Hussain e a School o Elec onic Enginee ing, Changzhou College o In o ma ion Technology, 213164, China b Depa men o Radio Elec onics, B no Uni e si y o Technology, 616 00 B no, Czech Republic c Nonlinea Sys ems and Applica ions, Facul y o Elec ical and Elec onics Enginee ing, Ton Duc Thang Uni e si y, Ho Chi Minh Ci y, Vie Nam d Depa men o Biomedical Enginee ing, Ami kabi Uni e si y o Technology, 424 Ha ez A e., Teh an 15875-4413, I an e Depa men o Ma hema ics, S a is ics and Physics, Qa a Uni e si y, Doha 2713, Qa a g aphical abs ac a icle in o A icle his o y: Recei ed 4 Ap il 2020 Re ised 29 May 2020 Accep ed 30 May 2020 A ailable online 17 June 2020 Keywo ds: Mem is o F acmem is o Chao ic oscilla o s Mul is abili y An imono onici y abs ac Mem is o is a non-linea ci cui elemen in which ol age-cu en ela ionship is de e mined by he p e- ious alues o he ol age and cu en , gene ally he his o y o he ci cui . The nonlinea i y in his com- ponen can be conside ed as a ac ional-o de o m, which yields a ac ional mem is o ( acmem is o ). In his pape , a ac ional-o de mem is o in a chao ic oscilla o is applied, while he o he elec onic ele- men s a e o in ege o de . The ac ional-o de ange is de e mined in a way ha he ci cui has chao ic solu ions. Also, he s a is ical and dynamical ea u es o his ci cui a e analyzed. Tools like Lyapuno exponen s and bi u ca ion diag am show he exis ence o mul is abili y and an imono onici y, wo less common p ope ies in chao ic ci cui s. Ó2020 The Au ho s. Published by Else ie B.V. on behal o Cai o Uni e si y. This is an open access a icle unde he CC BY license (h p://c ea i ecommons.o g/licenses/by/4.0/). h ps://doi.o g/10.1016/j.ja e.2020.05.025 2090-1232/Ó2020 The Au ho s. Published by Else ie B.V. on behal o Cai o Uni e si y. This is an open access a icle unde he CC BY license (h p://c ea i ecommons.o g/licenses/by/4.0/). Pee e iew unde esponsibili y o Cai o Uni e si y. ⇑ Co esponding au ho . E-mail add esses: [email p o ec ed] (J. Pe zela), [email p o ec ed] (T. Go hans), [email p o ec ed] (S. Ja a i), [email p o ec ed] (I. Hussain). Jou nal o Ad anced Resea ch 25 (2020) 137–145 Con en s lis s a ailable a ScienceDi ec Jou nal o Ad anced Resea ch jou nal homepage: www.else ie .com/loca e/ja e In oduc ion A mem is o is a non-linea ci cui ci cui elemen , which is based on nonlinea ol age-cu en ela ion. The elec ical esis- ance o his elemen is ela ed o i s p e ious cu en , so i has been named mem is o (memo y esis o ) [1]. Ci cui s and sys ems con- aining mem is o s ha e been success ully used in image and ex enc yp ion, simula ing biological sys ems, elec onic and neu al ne wo ks [2]. Con inuous symme ical, con inuous nonsymme i- cal, swi ching and ac ional models o mem is o wi h i s emula- o s and ealiza ions a e discussed in [3]. Chao ic ci cui s and sys ems a e in e es ing opics in nonlinea dynamics [4]. Va ious chao ic sys ems ha e been p oposed in ecen yea s [5,6]. Mem is- i e sys ems show complex dynamical beha io s, like chaos [7], mul is abili y [8], and hidden a ac o s. Designing and analyzing mem is i e sys ems and ci cui s wi h pa icula p ope ies ha e been conside ed in di e en oscilla o e.g., Wien-b idge oscilla o [9], diode b idge-based oscilla o [10] and neu on models [11]. F ac ional-o de di e en ial equa ions a e in he g oup o non- linea and complex sys ems [12–14]. These sys ems ha e shown di e en complex p ope ies such as hype chaos [15], sel - p oducing a ac o s, and s ange maps [16], which enabled hem o be used in modeling o biological phenomena, elec ical compo- nen s, con olle s, and il e s [17]. Mul is abili y and an imono- onici y a e wo ea u es ha ha e been epo ed in ac ional- o de sys ems [18]. The p edic o –co ec o me hod o he Adams-Bash o h-Moul on (ABM) algo i hm can be used o dis- c e ize ac ional-o de equa ions, especially when sys ems a e highly sensi i e. Se e al s udies ha e been done ecen ly o de elop and ealize he ac ional-o de elemen . F ac ional pa ame e s o hese ele- men s p o ide lexibili y and deg ees o eedom in compu a ional modeling [19], con ol enginee ing [20,21], and il e designing [22]. Al hough he ac ional-o de o m o he h ee con en ional elemen s has been explo ed well, s udying his o m o mem is o s ill is a new opic. S ep, DC, sinusoidal, and non-sinusoidal pe i- odic esponses o he ac ional-o de mem is o ha e been ana- lyzed in [23,24]. Some esea ches show ha sa u a ion ime o his elemen changes when ac ional o de and ol age change [23,24]. Also, conside ing ac ional o de makes a cha ge- con olled mem is o ha e wo hys e esis loop in i s V-I plane [25].To compa e he e ec o using ac ional mem is o , e e ence [26] shows ha a wide ange o equency is gene a ed using he mem is o wi h ac ional-o de elemen s, a he han in ege ones. Also, conside ing ac ional-o de mem is i e Chua’s ci cui makes i a non-smoo h sys em which shows di e en bi u ca ions such as angen o g azing ones [27]. As ac ional-sys ems a e in he g oup o complex sys ems, hey need ele an analyzing ools. To analyze he s a is ical p ope ies o he sys ems, equilib ia, eigen alues, and s abili y should be checked. In hese sys ems, he s abili y depends on he alue o he o de in addi ion o he eigen alues. Also, o analyze he dynamical p ope ies o he sys ems, Lyapuno exponen s (LEs) shows he di e gence o he adjacen ini ial condi ions. Wol ’s algo i hm [28] is a well-known algo i hm ha nume ically es i- ma es he LEs o he sys em. In ha case, he posi i i y o he la - ges Lyapuno exponen (LLE) o he sys em shows he chao ici y o he sys em. The bi u ca ion diag am o he sys ems is ano he ool o analyze he a ac o s o he sys ems as he con olling pa ame e (s) changes. Using bi u ca ion diag am, one can explo e he mul is abili y and an imono onici y o he sys em. We comple ely in oduce he acmem is o and Twin-T oscilla- o ma hema ical model and ci cui in Sec ion 2. The s a is ical and dynamical p ope ies o he p oposed ac ional-o de model a e analyzed in Sec ion 3. We also explain he s abili y o he equilib- iums, he Lyapuno exponen s, bi u ca ion diag am, mul is abil- i y, and an imono onici y o he p oposed model in ha sec ion. Finally, he conclusion o his wo k is p esen ed in Sec ion 4. F acmem is o Twin-T oscilla o (FTT) The ac ional-o de o m o he mem is o is gi en by [24], R m ¼R qþ1 in  C qþ2ðÞ C qðÞ gR on R o  Z 0  s ðÞ qþ1 s ðÞd s 2 43 5 1 qþ1 ð1Þ in which R m ,R on ,R o and Rin deno e he momen , minimum, maximum, and ini ial alue esis ances o he mem is o , espec- i ely. Also, g and q a e he mem is o cons an and he ac ional-o de which a ies in he ange o 0;1 ðÞ . I should be no ed ha he mem is o in (1) becomes in ege -o de , when q¼1. The oscilla o , which is conside ed in his pape , is Twin-T mem is o oscilla o [29]. Unlike mos o he ac ional-o de sys- ems which conside all he elemen s as ac ional ones, we jus s udy he e ec o he ac ional-o de mem is o in in ege - o de Twin-T oscilla o . In [29], he au ho s p oposed a mem is o emula o which con ains an op-amp based in ege -o de in eg a- o . We eplace he in ege -o de in eg a o wi h he ac ional- o de one discussed in [30].Fig. 1 shows he acmem is o emula- o , and Fig. 2 shows he Twin-T oscilla o wi h his acmem is o . In Fig. 1, he alue o he esis o s is R D =A –1 Rwhe e A 1 ¼ 1þq 1q and q ep esen s he ac ional o de o he sys em [30]. The ol age-cu en ela ionship o he mem is o emula o wi h ac ional-o de in eg a o will be i¼MV /  V¼ VgV gV 2 / ðÞ R / ¼ 1 R / 1g 2 V 2 /  V d a V / d a ¼ V / R D C /  V RC / ð2Þ whe e M(V / ) is a con inuous linea impedance unc ion ela ed o he ol age o he mem is o V / and equals MV /  ¼ 1 R / 1g 2 V 2 /  . Using KVL in Fig. 2, we can de i e he dimensionless model [29] as _ x¼a 1 MwðÞyþa 2 zþa 3 x; _ y¼a 4 MwðÞyþa 5 zþa 6 x;_ z¼a 7 xþa 8 z; D q w¼a 9 yþa 10 w ð3Þ whe e MðwÞ¼ a þbw 2 ,x=V a ,y=V b ,z=V c and w=V / . In his a icle, we used he P edic E alua e Co ec E alua e (PECE) me hod o ABM, which i s con e gence and accu acy a e discussed in [31]. To use he PECE me hod, we i s conside a ac ional-o de dynamical sys em as D q x¼ ;xðÞ;0 Tð4Þ whe e x k 0ðÞ¼x k 0 o k2[0, n–1]. This equa ion is analogous o he Vol e a in eg al equa ion as x ðÞ¼X n1 k¼0 x k 0 k k!þ1 C qðÞ Z 0 s ;xðÞ  s ðÞ 1q d s ð5Þ which can be disc e ized as x h nþ1 ðÞ¼ X n1 k¼0 x ðkÞ 0 kþ1 n k!þh q C qþ2ðÞ nþ1 ;x p h nþ1 ðÞ  þh q C qþ2ðÞ Xa j;nþ1 j ;x h j  ð6Þ whe ein (6),h¼ T N and n ¼nh as h2[0, N]. Also, we ha e 138 H. Lu e al. / Jou nal o Ad anced Resea ch 25 (2020) 137–145 a j;nþ1 ¼ n qþ1 nqðÞðnþ1Þ qþ1 ;j¼0 2njþ1ðÞ qþ1 ;1jn 1;j¼nþ1 8 > < > : x p h nþ1 ðÞ¼ P n1 k¼0 x ðkÞ 0 kþ1 n k! þ h q C 2ðÞ P n j¼0 b j;nþ1 j x h j  b j;nþ1 ¼ h q q njþ1ðÞ q njðÞ q  ð7Þ The es ima ed e o is e¼Max x i ðÞx h ð i Þjj¼0ðh p Þwhile j¼0;1;;Nand p¼Minð2;1þqÞ. Using he abo e, he ou h s a e o he FTT disc e e o m is w nþ1 ¼ w 0 þ h q C qþ2ðÞ a 9 y p nþ1 þa 10 w p nþ1  þ h q C qþ2ðÞ P n j¼0 g j;nþ1 a 9 y j þa 10 w j  hi 8 < : 9 = ; ð8Þ as w p nþ1 ¼w 0 þ1 C qþ2ðÞ X n j¼0 x j;nþ1 a 9 y j þa 10 w j  ð9Þ and g l;j;nþ1 ¼ n qþ1 nqðÞðnþ1Þ qþ1 ;j¼0 njþ2ðÞ qþ1 þnjðÞ qþ1 2njþ1ðÞ qþ1 ;1jn 1;j¼nþ1 8 > < > : x l;j;nþ1 ¼ h q q njþ1ðÞ q njðÞ q  ;0jn  ð10Þ whe e l=1. To sol e he equa ion, he ou h-o de Runge-Ku a me hod is used o he i s h ee s a es, and PECE is used o he ac ional- o de s a e in (3). Eq. (3) can be disc e ized as xnþ1 ðÞ ¼xn ðÞ þ 1 6 K ð1Þ x n ðÞ þ2K ð2Þ x n ðÞ þ2K ð3Þ x n ðÞ þK ð4Þ x n ðÞ hi ynþ1ðÞ¼ynðÞþ 1 6 K ð1Þ y nðÞþ2K ð2Þ y nðÞþ2K ð3Þ y nðÞþK ð4Þ y nðÞ hi znþ1ðÞ¼znðÞþ 1 6 K ð1Þ z nðÞþ2K ð2Þ z nðÞþ2K ð3Þ z nðÞþK ð4Þ z nðÞ hi wðnþ1Þ¼ wðnÞþ h q C qþ2ðÞ a 9 y p nþ1 þa 10 w p nþ1  þ h q C qþ2ðÞ P n j¼0 g j;nþ1 a 9 y j þa 10 w j  hi 8 > < > : 9 > = > ; ð11Þ whe e K ð1Þ x nðÞ¼h x xnðÞ;ynðÞ;znðÞ;wðnÞ½ K ð2Þ x nðÞ¼h x xnðÞþ K ð1Þ x nðÞ 2 ;ynðÞþ K ð1Þ y nðÞ 2 ;znðÞþ K ð1Þ z nðÞ 2 þ K ð1Þ w nðÞ 2  K ð3Þ x nðÞ¼h x xnðÞþ K ð2Þ x nðÞ 2 ;ynðÞþ K ð2Þ y nðÞ 2 ;znðÞþ K ð2Þ z nðÞ 2 þ K ð2Þ w nðÞ 2  K ð4Þ x nðÞ¼h x xnðÞþ K ð3Þ x nðÞ 2 ;ynðÞþ K ð3Þ y nðÞ 2 ;znðÞþ K ð3Þ z nðÞ 2 þ K ð3Þ w nðÞ 2  ð12Þ Simila ly, he Runge-Ku a coe icien s o he o he wo s a es (y,z) can be calcula ed as (12). Fo he pa ame e alues o a 1 ¼9, a 2 ¼0:77, a 3 ¼0:07, a 4 ¼0:75, a 5 ¼0:42, a 6 ¼0:0382, a 7 ¼3:532, a 8 ¼3:85, a 9 ¼10, a 10 ¼1, a ¼1,b¼0:01 and q¼0:99, he 2D phase po ai s o he FTT sys em a e shown in Fig. 3. Analysis o he FTT oscilla o Equilib ium poin s, co esponding eigen alues, s abili y, LEs, and bi u ca ion diag am o he FTT a e examined o he sys em in his sec ion. Fig. 1. Mem is o emula o wi h he ac ional-o de in eg a o . Fig. 2. Twin-T oscilla o wi h acmem is o (F M ). H. Lu e al. / Jou nal o Ad anced Resea ch 25 (2020) 137–145 139 S a is ical analysis o he sys em The FTT sys em shows h ee ixed poin s as below E 1 ¼½0;0;0;0;E 2 ¼0;a 10 a 9 ffiffiffiffiffiffiffi  a b ;0;ffiffiffiffiffiffiffi  a b  ; E 3 ¼0;a 10 a 9 ffiffiffiffiffiffiffi  a b ;0;ffiffiffiffiffiffiffi  a b  ð13Þ The Jacobian ma ix o he FTT sys em is JðXÞ¼ a 3 a 1 ðbw 2 þ a Þa 2 2a 1 bwy a 6 a 4 ðbw 2 þ a Þa 5 2a 4 bwy a 7 0a 8 0 0a 9 0a 10   ð14Þ The equa ion de ðdiagðk M q1 ;k M q2 ;k M q3 ;k M q4 ÞJ E i Þ¼0 yields he gene alized cha ac e is ic polynomial o he FTT sys em. In his equa ion, q 1 ¼q 2 ¼q 3 ¼1, q 4 ¼0:99 and Mis he leas common mul iple (LCM) o q i o i¼1;;4. The cha ac e is ic equa ions a E 1 ;E 2 and E 3 a e gi en by (15),(16) and (17) espec i ely. k 399 þk 300 þ3:03k 299 þ3:03k 200 0:72866k 199 0:72866k 100 þ10:189725k 99 þ10:189725 ¼0ð15Þ k 399 þk 300 þ3:78k 299 þ5:28k 200 þ2:45014k 199 þ8:80774k 100 20:37945 ¼0ð16Þ k 399 þk 300 þ3:78k 299 þ5:28k 200 þ2:45014k 199 þ8:80774k 100 20:37945 ¼0ð17Þ Co olla y 1. The ixed poin s should be uns able o he FTT sys em exhibi chao ic dynamics. So he essen ial condi ion is any ko he equilib ium poin s should sa is y he ollowing inequali y q>2 p a c an Im kðÞjj Re kðÞ  ð18Þ The eigen alues o he FTT a he equilib ium E 1 when a¼3a e k 1,2 = 0.5000 ± 0.8660i and k 3 = –2, which o sa is y (18), we ha e q> 0.97. Co olla y 2. A chao ic a ac o exis s in he FTT i he co esponding equilib ium poin s show ins abili y. So he essen ial condi ion is ha he oo s o he cha ac e is ic equa ions (15),(16) and (17) should sa is y he ollowing inequali y p 2Mmin i a g k i ðÞ g0ð19Þ I can be concluded om [32] ha he sys em is uns able as no all he oo s o he equa ions (15),(16) and (17) sa is y he condi- ion (19). Hence, we can conclude he exis ence o chao ic oscilla- ions like i s in ege -o de sys em discussed in [29] when q> 0.97. Lyapuno exponen s Wol s algo i hm is used o de i e he Lyapuno exponen s o he FTT sys em and check he chao ici y o he sys em o di e en alues o he pa ame e s. Also, he ac ional-o de p edic o –co - ec o sol e de12 is used ins ead o he o dina y di e en ial equa ion (ODE) sol e s [33]. The Lyapuno exponen s o he FTT Fig. 3. The phase po ai s o he FTT sys em in (x-y), (y-z), (z-w) and (w-x) plane when a 1 ¼9, a 2 ¼0:77, a 3 ¼0:07, a 4 ¼0:75, a 5 ¼0:42, a 6 ¼0:0382, a 7 ¼3:532, a 8 ¼3:85, a 9 ¼10, a 10 ¼1, a¼1,b¼0:01, and q¼0:99. 140 H. Lu e al. / Jou nal o Ad anced Resea ch 25 (2020) 137–145 sys em o di e en alues o he ac ional o de qa e shown in Fig. 4. Bi u ca ion diag am To in es iga e he impac o he pa ame e s on he FTT oscilla- o , we de i ed he bi u ca ion plo s whe e we plo ed he local maxima o he s a e a iables e sus he con ol pa ame e . We ha e conside ed a 1 as he bi u ca ion pa ame e and he local max- ima o xin Fig. 5a. The FTT akes a pe iod-doubling ou e o he chaos, which is simila ly suppo ed by he Lyapuno exponen s shown in Fig. 5b. The ac ional o de o he bi u ca ion plo is aken as q¼0:99;and he o he pa ame e s a e conside ed as used in Fig. 3. Also, o show he e ec o he pa ame e s a 4 and a 1 , he 2D Fig. 4. Lyapuno exponen s o he FTT sys em as qinc eases. This ig. shows ha he sys em exhibi s di e en esponses. Fig. 5. a) The bi u ca ion plo o he FTT e sus he pa ame e a 1 and b) he co esponding LEs. H. Lu e al. / Jou nal o Ad anced Resea ch 25 (2020) 137–145 141 bi u ca ion diag am o he sys em is plo ed in Fig. 6. This igu e shows he di e en anges o he pa ame e s which yield s able equilib ium, s ange a ac o , and unbounded esponses. Mul is abili y To s udy he mul is abili y, he o wa d (pa ame e inc eases) and backwa d (pa ame e dec eases) bi u ca ions a e conside ed. The ini ial condi ion o each pa ame e is he inal alue o he ajec o y in he p e ious pa ame e . In Fig. 7, pa ame e a 4 is he bi u ca ion pa ame e , and he local maxima o he s a e a i- able ya e plo ed when he ac ional o de equals q¼0:99:Fig. 7a shows he bi u ca ion o he FTT sys em while he o wa d and backwa d shown in blue and ed, espec i ely. Fig. 7b shows he co esponding LEs. We could see he coexis ence o chao ic a ac- o s o 0:6694 a 4 0:7092, pe iod-8 limi cycles o 0:6568 a 4 0:6664 and pe iod-4 limi cycles o 0:6105 a 4 0:6567:The a ious coexis ing a ac o s o di e - en alues o he pa ame e a 4 a e shown in Fig. 8. We use he same o wa d and backwa d con inua ion o check he mul is abili y and coexis ing a ac o s o he ac ional o de q. Also, he o he pa ame e s a e conside ed as used o Fig. 3. We could iden i y he coexis ence o pe iod-2 limi cycles o 0:98 q0:9867, pe iod 4 limi cycles o 0:9868  q0:9883;and chao ic a ac o s o 0:9887 q0:9948 as seen in Fig. 9.Fig. 10 shows he a ious coexis ing limi cycles and chao ic a ac o s o di e en alues o he ac ional o de q. To be e analyze he coexis ing a ac o s o he sys em, he Basin o a ac ion o he sys em is conside ed in he x-z plane when y(0) = 0 and w(0) = 0. In Fig. 11, cyan and magen a colo show unbounded and chao ic esponses o he sys em, espec i ely. An imono onici y An imono onici y, a complex beha io in nonlinea sys ems, means he occu ence o pe iod-doubling and in e se pe iod- doubling. In he bi u ca ion diag am o hese sys ems, he pe iodic Fig. 6. 2D bi u ca ion diag am o a 1 and a 4 when he ac ional-o de equals 0.99. Fig. 7. a) The bi u ca ion plo o he FTT e sus a 4 which o wa d and backwa d a e shown in blue and ed do s, espec i ely. b) The co esponding LEs a e also plo ed. 142 H. Lu e al. / Jou nal o Ad anced Resea ch 25 (2020) 137–145 a ac o s double as pa ame e inc eases and ins an ly joining pe i- odic a ac o s o m smalle ones, so eme ging an imono onici y. To examine an imono onici y, he bi u ca ion o he FTT oscilla o sys em is conside ed as a 4 inc eases while he ac ional-o de q¼0:99 and pa ame e a 1 has some di e en ixed alues (Fig. 12). Conclusion To in es iga e memo y-dependen sys ems and conside his- o y in he elec onic ci cui , we can use he mem is o elemen . In his a icle, we showed ha using ac ional-o de mem is o in an in ege -o de oscilla o ci cui enables he sys em o show complex beha io s. Fo example, we concluded and showed ha in some ange o he ac ional o de , q>0:97, he sys em can show chao ic esponses. Mul is abili y, he exis ence o wo o mo e a ac o s o a ixed alue o he pa ame e , and an imono- onici y, he exis ence o pe iod-doubling ou e o chaos and in e se o i , a e he p ope ies ha his sys em shows in di e en alue o he pa ame e s. P ecise anges o he pa ame e s a e de i ed using he bi u ca ion diag am o i s co esponding Lya- Fig. 9. The bi u ca ion plo o he FTT e sus qwhen o wa d and backwa d con inua ions a e shown in blue and ed, espec i ely, which shows coexis ing a ac o s in his sys em. Fig. 10. Va ious coexis ing limi cycles and s ange a ac o s when he ini ial condi ions a e se o 1;0;0;0½(shown in blue) and 1;0;0;0½(shown in ed) o di e en alues o q. Fig. 11. Basin o a ac ion o he sys em in he x-z plane when y(0) = 0 and w (0) = 0. In his igu e, cyan and magen a colo show unbounded and chao ic esponses. Fig. 8. Va ious coexis ing limi cycles and chao ic a ac o s when he ini ial condi ions a e 1;0;0;0½(shown in blue) and 1;0;0;0½(shown in ed) o di e en alues o a 4 . H. Lu e al. / Jou nal o Ad anced Resea ch 25 (2020) 137–145 143 puno exponen s. We also use a 2D bi u ca ion diag am o show he di e en a ac o s o he sys em as wo di e en con olling pa ame e s change. Compliance wi h e hics equi emen s This a icle does no con ain any s udies wi h human o animal subjec s Decla a ion o Compe ing In e es The au ho s ha e decla ed no con lic o in e es Acknowledgemen Resea ch desc ibed in his pape was suppo ed by G an Agency o Czech Republic h ough p ojec numbe 19-22248S. Fo esea ch, in as uc u e o he SIX Cen e was used. Re e ences [1] Chua L. Mem is o - he missing ci cui elemen . IEEE T ansac ions on ci cui heo y 1971;18:507–19. 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