Fracmemristor chaotic oscillator with multistable and antimonotonicity properties

Fracmemristor chaotic oscillator with multistable and antimonotonicity properties Haikong Lu a , Jiri Petrzela b, ⇑ , Tomas Gotthans b , Karthikeyan Rajagopal c , Sajad Jafari d , Iqtadar Hussain e a School of Electronic Engineering, Changzhou College of Information Technology, 213164, China b Department of Radio Electronics, Brno University of Technology, 616 00 Brno, Czech Republic c Nonlinear Systems and Applications, Faculty of Electrical and Electronics Engineering, Ton Duc Thang University, Ho Chi Minh City, Viet Nam d Department of Biomedical Engineering, Amirkabir University of Technology, 424 Hafez Ave., Tehran 15875-4413, Iran e Department of Mathematics, Statistics and Physics, Qatar University, Doha 2713, Qatar graphical abstract article info Article history: Received 4 April 2020 Revised 29 May 2020 Accepted 30 May 2020 Available online 17 June 2020 Keywords: Memristor Fracmemristor Chaotic oscillators Multistability Antimonotonicity abstract Memristor is a non-linear circuit element in which voltage-current relationship is determined by the previous values of the voltage and current, generally the history of the circuit. The nonlinearity in this component can be considered as a fractional-order form, which yields a fractional memristor (fracmemristor). In this paper, a fractional-order memristor in a chaotic oscillator is applied, while the other electronic elements are of integer order. The fractional-order range is determined in a way that the circuit has chaotic solutions. Also, the statistical and dynamical features of this circuit are analyzed. Tools like Lyapunov exponents and bifurcation diagram show the existence of multistability and antimonotonicity, two less common properties in chaotic circuits. Ó2020 The Authors. Published by Elsevier B.V. on behalf of Cairo University. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). https://doi.org/10.1016/j.jare.2020.05.025 2090-1232/Ó2020 The Authors. Published by Elsevier B.V. on behalf of Cairo University. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Peer review under responsibility of Cairo University. ⇑ Corresponding author. E-mail addresses: [email protected] (J. Petrzela), [email protected] (T. Gotthans), [email protected] (S. Jafari), [email protected] (I. Hussain). Journal of Advanced Research 25 (2020) 137–145 Contents lists available at ScienceDirect Journal of Advanced Research journal homepage: www.elsevier.com/locate/jare Introduction A memristor is a non-linear circuit circuit element, which is based on nonlinear voltage-current relation. The electrical resistance of this element is related to its previous current, so it has been named memristor (memory resistor) [1]. Circuits and systems containing memristors have been successfully used in image and text encryption, simulating biological systems, electronic and neural networks [2]. Continuous symmetrical, continuous nonsymmetrical, switching and fractional models of memristor with its emulators and realizations are discussed in [3]. Chaotic circuits and systems are interesting topics in nonlinear dynamics [4]. Various chaotic systems have been proposed in recent years [5,6]. Memristive systems show complex dynamical behaviors, like chaos [7], multistability [8], and hidden attractors. Designing and analyzing memristive systems and circuits with particular properties have been considered in different oscillator e.g., Wien-bridge oscillator [9], diode bridge-based oscillator [10] and neuron models [11]. Fractional-order differential equations are in the group of nonlinear and complex systems [12–14]. These systems have shown different complex properties such as hyperchaos [15], selfproducing attractors, and strange maps [16], which enabled them to be used in modeling of biological phenomena, electrical components, controllers, and filters [17]. Multistability and antimonotonicity are two features that have been reported in fractionalorder systems [18]. The predictor–corrector method of the Adams-Bashforth-Moulton (ABM) algorithm can be used to discretize fractional-order equations, especially when systems are highly sensitive. Several studies have been done recently to develop and realize the fractional-order element. Fractional parameters of these elements provide flexibility and degrees of freedom in computational modeling [19], control engineering [20,21], and filter designing [22]. Although the fractional-order form of the three conventional elements has been explored well, studying this form of memristor still is a new topic. Step, DC, sinusoidal, and non-sinusoidal periodic responses of the fractional-order memristor have been analyzed in [23,24]. Some researches show that saturation time of this element changes when fractional order and voltage change [23,24]. Also, considering fractional order makes a chargecontrolled memristor have two hysteresis loop in its V-I plane [25].To compare the effect of using fractional memristor, reference [26] shows that a wider range of frequency is generated using the memristor with fractional-order elements, rather than integer ones. Also, considering fractional-order memristive Chua’s circuit makes it a non-smooth system which shows different bifurcations such as tangent or grazing ones [27]. As fractional-systems are in the group of complex systems, they need relevant analyzing tools. To analyze the statistical properties of the systems, equilibria, eigenvalues, and stability should be checked. In these systems, the stability depends on the value of the order in addition to the eigenvalues. Also, to analyze the dynamical properties of the systems, Lyapunov exponents (LEs) shows the divergence of the adjacent initial conditions. Wolf’s algorithm [28] is a well-known algorithm that numerically estimates the LEs of the system. In that case, the positivity of the largest Lyapunov exponent (LLE) of the system shows the chaoticity of the system. The bifurcation diagram of the systems is another tool to analyze the attractors of the systems as the controlling parameter(s) changes. Using bifurcation diagram, one can explore the multistability and antimonotonicity of the system. We completely introduce the fracmemristor and Twin-T oscillator mathematical model and circuit in Section 2. The statistical and dynamical properties of the proposed fractional-order model are analyzed in Section 3. We also explain the stability of the equilibriums, the Lyapunov exponents, bifurcation diagram, multistability, and antimonotonicity of the proposed model in that section. Finally, the conclusion of this work is presented in Section 4. Fracmemristor Twin-T oscillator (FTT) The fractional-order form of the memristor is given by [24], R m ¼R qþ1 in  C qþ2ðÞ C qðÞ gR on R off  Z t 0 t s ðÞ qþ1 ts ðÞd s 2 43 5 1 qþ1 ð1Þ in which R m ,R on ,R off and Rin denote the moment, minimum, maximum, and initial value resistances of the memristor, respectively. Also, g and q are the memristor constant and the fractional-order which varies in the range of 0;1 ðÞ . It should be noted that the memristor in (1) becomes integer-order, when q¼1. The oscillator, which is considered in this paper, is Twin-T memristor oscillator [29]. Unlike most of the fractional-order systems which consider all the elements as fractional ones, we just study the effect of the fractional-order memristor in integerorder Twin-T oscillator. In [29], the authors proposed a memristor emulator which contains an op-amp based integer-order integrator. We replace the integer-order integrator with the fractionalorder one discussed in [30].Fig. 1 shows the fracmemristor emulator, and Fig. 2 shows the Twin-T oscillator with this fracmemristor. In Fig. 1, the value of the resistors is R D =A –1 Rwhere A 1 ¼ 1þq 1q and qrepresents the fractional order of the system [30]. The voltage-current relationship of the memristor emulator with fractional-order integrator will be i¼MV /  V¼ VgV gV 2 / ðÞ R / ¼ 1 R / 1g 2 V 2 /  V d a V / dt a ¼ V / R D C /  V RC / ð2Þ where M(V / ) is a continuous linear impedance function related to the voltage of the memristor V / and equals MV /  ¼ 1 R / 1g 2 V 2 /  . Using KVL in Fig. 2, we can derive the 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 where MðwÞ¼ a þbw 2 ,x=V a ,y=V b ,z=V c and w=V / . In this article, we used the Predict Evaluate Correct Evaluate (PECE) method of ABM, which its convergence and accuracy are discussed in [31]. To use the PECE method, we first consider a fractional-order dynamical system as D q x¼ft;xðÞ;0tTð4Þ where x k 0ðÞ¼x k 0 for k2[0, n–1]. This equation is analogous to the Volterra integral equation as xtðÞ¼X n1 k¼0 x k 0 t k k!þ1 C qðÞ Z t 0 f s ;xðÞ t s ðÞ 1q d s ð5Þ which can be discretized as x h t nþ1 ðÞ¼ X n1 k¼0 x ðkÞ 0 t kþ1 n k!þh q C qþ2ðÞ ft nþ1 ;x p h t nþ1 ðÞ  þh q C qþ2ðÞ Xa j;nþ1 ft j ;x h t j  ð6Þ wherein (6),h¼ T N and t n ¼nh as h2[0, N]. Also, we have 138 H. Lu et al. / Journal of Advanced Research 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 t nþ1 ðÞ¼ P n1 k¼0 x ðkÞ 0 t kþ1 n k! þ h q C 2ðÞ P n j¼0 b j;nþ1 ft j x h t j  b j;nþ1 ¼ h q q njþ1ðÞ q njðÞ q  ð7Þ The estimated error is e¼Max x t i ðÞx h ðt i Þjj¼0ðh p Þwhile j¼0;1;;Nand p¼Minð2;1þqÞ. Using the above, the fourth state of the FTT discrete form 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 where l=1. To solve the equation, the fourth-order Runge-Kutta method is used for the first three states, and PECE is used for the fractionalorder state in (3). Eq. (3) can be discretized 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 where K ð1Þ x nðÞ¼hf x xnðÞ;ynðÞ;znðÞ;wðnÞ½ K ð2Þ x nðÞ¼hf 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 nðÞ¼hf 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 nðÞ¼hf 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 Similarly, the Runge-Kutta coefficients for the other two states (y,z) can be calculated as (12). For the parameter values of 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, the 2D phase portraits of the FTT system are shown in Fig. 3. Analysis of the FTT oscillator Equilibrium points, corresponding eigenvalues, stability, LEs, and bifurcation diagram of the FTT are examined to the system in this section. Fig. 1. Memristor emulator with the fractional-order integrator. Fig. 2. Twin-T oscillator with fracmemristor (F M ). H. Lu et al. / Journal of Advanced Research 25 (2020) 137–145 139 Statistical analysis of the system The FTT system shows three fixed points as below E 1 ¼½0;0;0;0;E 2 ¼0;a 10 a 9 ffiffiffiffiffiffiffi  a b r;0;ffiffiffiffiffiffiffi  a b r  ; E 3 ¼0;a 10 a 9 ffiffiffiffiffiffiffi  a b r;0;ffiffiffiffiffiffiffi  a b r  ð13Þ The Jacobian matrix of the FTT system 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 equation detðdiagðk M q1 ;k M q2 ;k M q3 ;k M q4 ÞJ E i Þ¼0 yields the generalized characteristic polynomial of the FTT system. In this equation, q 1 ¼q 2 ¼q 3 ¼1, q 4 ¼0:99 and Mis the least common multiple (LCM) of q i for i¼1;;4. The characteristic equations at E 1 ;E 2 and E 3 are given by (15),(16) and (17) respectively. 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 Corollary 1. The fixed points should be unstable to the FTT system exhibit chaotic dynamics. So the essential condition is any kof the equilibrium points should satisfy the following inequality q>2 p arctan Im kðÞjj Re kðÞ  ð18Þ The eigenvalues of the FTT at the equilibrium E 1 when a¼3are k 1,2 = 0.5000 ± 0.8660i and k 3 = –2, which to satisfy (18), we have q> 0.97. Corollary 2. A chaotic attractor exists in the FTT if the corresponding equilibrium points show instability. So the essential condition is that the roots of the characteristic equations (15),(16) and (17) should satisfy the following inequality p 2Mmin i arg k i ðÞfg0ð19Þ It can be concluded from [32] that the system is unstable as not all the roots of the equations (15),(16) and (17) satisfy the condition (19). Hence, we can conclude the existence of chaotic oscillations like its integer-order system discussed in [29] when q> 0.97. Lyapunov exponents Wolfs algorithm is used to derive the Lyapunov exponents of the FTT system and check the chaoticity of the system for different values of the parameters. Also, the fractional-order predictor–corrector solver fde12 is used instead of the ordinary differential equation (ODE) solvers [33]. The Lyapunov exponents of the FTT Fig. 3. The phase portraits of the FTT system 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 et al. / Journal of Advanced Research 25 (2020) 137–145 system for different values of the fractional order qare shown in Fig. 4. Bifurcation diagram To investigate the impact of the parameters on the FTT oscillator, we derived the bifurcation plots where we plotted the local maxima of the state variables versus the control parameter. We have considered a 1 as the bifurcation parameter and the local maxima of xin Fig. 5a. The FTT takes a period-doubling route to the chaos, which is similarly supported by the Lyapunov exponents shown in Fig. 5b. The fractional order for the bifurcation plot is taken as q¼0:99;and the other parameters are considered as used in Fig. 3. Also, to show the effect of the parameters a 4 and a 1 , the 2D Fig. 4. Lyapunov exponents of the FTT system as qincreases. This fig. shows that the system exhibits different responses. Fig. 5. a) The bifurcation plot of the FTT versus the parametera 1 and b) the corresponding LEs. H. Lu et al. / Journal of Advanced Research 25 (2020) 137–145 141 bifurcation diagram of the system is plotted in Fig. 6. This figure shows the different ranges of the parameters which yield stable equilibrium, strange attractor, and unbounded responses. Multistability To study the multistability, the forward (parameter increases) and backward (parameter decreases) bifurcations are considered. The initial condition for each parameter is the final value of the trajectory in the previous parameter. In Fig. 7, parameter a 4 is the bifurcation parameter, and the local maxima of the state variable yare plotted when the fractional order equals q¼0:99:Fig. 7a shows the bifurcation of the FTT system while the forward and backward shown in blue and red, respectively. Fig. 7b shows the corresponding LEs. We could see the coexistence of chaotic attractors for 0:6694 a 4 0:7092, period-8 limit cycles for 0:6568 a 4 0:6664 and period-4 limit cycles for 0:6105 a 4 0:6567:The various coexisting attractors for different values of the parameter a 4 are shown in Fig. 8. We use the same forward and backward continuation to check the multistability and coexisting attractors for the fractional order q. Also, the other parameters are considered as used for Fig. 3. We could identify the coexistence of period-2 limit cycles for 0:98 q0:9867, period 4 limit cycles for 0:9868  q0:9883;and chaotic attractors for 0:9887 q0:9948 as seen in Fig. 9.Fig. 10 shows the various coexisting limit cycles and chaotic attractors for different values of the fractional orderq. To better analyze the coexisting attractors of the system, the Basin of attraction of the system is considered in the x-z plane when y(0) = 0 and w(0) = 0. In Fig. 11, cyan and magenta color show unbounded and chaotic responses of the system, respectively. Antimonotonicity Antimonotonicity, a complex behavior in nonlinear systems, means the occurrence of period-doubling and inverse perioddoubling. In the bifurcation diagram of these systems, the periodic Fig. 6. 2D bifurcation diagram for a 1 and a 4 when the fractional-order equals 0.99. Fig. 7. a) The bifurcation plot of the FTT versus a 4 which forward and backward are shown in blue and red dots, respectively. b) The corresponding LEs are also plotted. 142 H. Lu et al. / Journal of Advanced Research 25 (2020) 137–145 attractors double as parameter increases and instantly joining periodic attractors form smaller ones, so emerging antimonotonicity. To examine antimonotonicity, the bifurcation of the FTT oscillator system is considered as a 4 increases while the fractional-order q¼0:99 and parameter a 1 has some different fixed values (Fig. 12). Conclusion To investigate memory-dependent systems and consider history in the electronic circuit, we can use the memristor element. In this article, we showed that using fractional-order memristor in an integer-order oscillator circuit enables the system to show complex behaviors. For example, we concluded and showed that in some range of the fractional order, q>0:97, the system can show chaotic responses. Multistability, the existence of two or more attractors for a fixed value of the parameter, and antimonotonicity, the existence of period-doubling route to chaos and inverse of it, are the properties that this system shows in different value of the parameters. Precise ranges of the parameters are derived using the bifurcation diagram or its corresponding LyaFig. 9. The bifurcation plot of the FTT versus qwhen forward and backward continuations are shown in blue and red, respectively, which shows coexisting attractors in this system. Fig. 10. Various coexisting limit cycles and strange attractors when the initial conditions are set to 1;0;0;0½(shown in blue) and 1;0;0;0½(shown in red) for different values of q. Fig. 11. Basin of attraction of the system in the x-z plane when y(0) = 0 and w (0) = 0. In this figure, cyan and magenta color show unbounded and chaotic responses. Fig. 8. Various coexisting limit cycles and chaotic attractors when the initial conditions are 1;0;0;0½(shown in blue) and 1;0;0;0½(shown in red) for different values of a 4 . H. Lu et al. / Journal of Advanced Research 25 (2020) 137–145 143 punov exponents. We also use a 2D bifurcation diagram to show the different attractors of the system as two different controlling parameters change. Compliance with ethics requirements This article does not contain any studies with human or animal subjects Declaration of Competing Interest The authors have declared no conflict of interest Acknowledgement Research described in this paper was supported by Grant Agency of Czech Republic through project number 19-22248S. For research, infrastructure of the SIX Center was used. References [1] Chua L. Memristor-the missing circuit element. IEEE Transactions on circuit theory 1971;18:507–19. [2] Radwan AG, Fouda ME. On the mathematical modeling of memristor, memcapacitor, and meminductor 2015;vol:26. [3] Radwan AG, Fouda ME. In: ‘‘Memristor mathematical models and emulators,” On the Mathematical Modeling of Memristor, Memcapacitor, and Meminductor. Springer; 2015. p. 51–84. [4] Petrzela J, Gotthans T, Guzan M. Current-mode network structures dedicated for simulation of dynamical systems with plane continuum of equilibrium. Journal of Circuits, Systems and Computers 2018;27:1830004. [5] Gotthans T, Sprott JC, Petrzela J. Simple chaotic flow with circle and square equilibrium. Int J Bifurcation Chaos 2016;26:1650137. [6] Zhang Y, Liu Z, Wu H, Chen S, Bao B. Extreme multistability in memristive hyper-jerk system and stability mechanism analysis using dimensionality reduction model. The European Physical Journal Special Topics 2019;228:1995–2009. [7] Chen M, Sun M, Bao H, Hu Y, Bao B. Flux-Charge Analysis of Two-MemristorBased Chua’s Circuit: Dimensionality Decreasing Model for Detecting Extreme Multistability. IEEE Trans Ind Electron 2019. [8] Chen M, Feng Y, Bao H, Bao B, Wu H, Xu Q. Hybrid State Variable Incremental Integral for Reconstructing Extreme Multistability in Memristive Jerk System with Cubic Nonlinearity. Complexity 2019;2019. [9] Bao H, Wang N, Wu H, Song Z, Bao B. Bi-stability in an improved memristorbased third-order Wien-bridge oscillator. IETE Technical Review 2019;36:109–16. [10] Bao B, Wu P, Bao H, Wu H, Zhang X, Chen M. Symmetric periodic bursting behavior and bifurcation mechanism in a third-order memristive diode bridge-based oscillator. Chaos, Solitons Fractals 2018;109:146–53. [11] P. Bertsias, C. Psychalinos, and A. S. Elwakil, ‘‘Fractional-Order Mihalas–Niebur Neuron Model Implementation Using Current-Mirrors,” in 2019 6th International Conference on Control, Decision and Information Technologies (CoDIT), 2019, pp. 872-875. [12] Allagui A, Freeborn TJ, Elwakil AS, Fouda ME, Maundy BJ, Radwan AG, et al. Review of fractional-order electrical characterization of supercapacitors. J Power Sources 2018;400:457–67. [13] Semary MS, Fouda ME, Hassan HN, Radwan AG. Realization of fractional-order capacitor based on passive symmetric network. J Adv Res 2019;18:147–59. [14] Hamed EM, Fouda ME, Alharbi AG, Radwan AG. Experimental verification of triple lobes generation in fractional memristive circuits. IEEE Access 2018;6:75169–80. [15] Peng D, Sun K, He S, Zhang L, Alamodi AO. Numerical analysis of a simplest fractional-order hyperchaotic system. Theor Appl Mech Lett 2019;9:220–8. [16] Peng Y, Sun K, Peng D, Ai W. Dynamics of a higher dimensional fractionalorder chaotic map. Physica A 2019;525:96–107. [17] Bertsias P, Psychalinos C, Maundy BJ, Elwakil AS, Radwan AG. Partial fraction expansion–based realizations of fractional-order differentiators and integrators using active filters. Int J Circuit Theory Appl 2019;47:513–31. [18] He S, Sun K, Peng Y. Detecting chaos in fractional-order nonlinear systems using the smaller alignment index. Phys Lett A 2019;383:2267–71. [19] Y. Yu, M. Shi, H. Kang, M. Chen, and B. Bao, ‘‘Hidden dynamics in a fractionalorder memristive Hindmarsh–Rose model,” Nonlinear Dynamics, pp. 1-16, 2020. [20] A. T. Mohamed, M. F. Mahmoud, L. A. Said, and A. G. Radwan, ‘‘Design of FOPID Controller for a DC Motor Using Approximation Techniques,” in: 2019 Novel Intelligent and Leading Emerging Sciences Conference (NILES), 2019, pp. 142145. [21] Tolba MF, AboAlNaga BM, Said LA, Madian AH, Radwan AG. Fractional order integrator/differentiator: FPGA implementation and FOPID controller application. AEU-International Journal of Electronics and Communications 2019;98:220–9. [22] Radwan AG, Fouda ME. Optimization of fractional-order RLC filters. Circuits, Systems, and Signal Processing 2013;32:2097–118. [23] Fouda ME, Radwan AG. Fractional-order memristor response under dc and periodic signals. Circuits, Systems, and Signal Processing 2015;34:961–70. [24] Fouda M, Radwan A. On the fractional-order memristor model. Journal of Fractional calculus and applications 2013;4:1–7. [25] Si G, Diao L, Zhu J. Fractional-order charge-controlled memristor: theoretical analysis and simulation. Nonlinear Dyn 2017;87:2625–34. [26] Rashad SH, Hamed EM, Fouda ME, AbdelAty AM, Said LA, Radwan AG. On the analysis of current-controlled fractional-order memristor emulator. In: in 2017 6th International Conference on Modern Circuits and Systems Technologies (MOCAST). p. 1–4. [27] Yu Y, Wang Z. Initial state dependent nonsmooth bifurcations in a fractionalorder memristive circuit. Int J Bifurcation Chaos 2018;28:1850091. Fig. 12. Bifurcation of the FTT oscillator with a 4 for q¼0:99 and different fixed values of a 1 which claims existence of antimonotonicity in this system. 144 H. Lu et al. / Journal of Advanced Research 25 (2020) 137–145 [28] Wolf A, Swift JB, Swinney HL, Vastano JA. Determining Lyapunov exponents from a time series. Physica D 1985;16:285–317. [29] Zhou L, Wang C, Zhang X, Yao W. Various Attractors, Coexisting Attractors and Antimonotonicity in a Simple Fourth-Order Memristive Twin-T Oscillator. Int J Bifurcation Chaos 2018;28:1850050. [30] Muñiz-Montero C, García-Jiménez LV, Sánchez-Gaspariano LA, Sánchez-López C, González-Díaz VR, Tlelo-Cuautle E. New alternatives for analog implementation of fractional-order integrators, differentiators and PID controllers based on integer-order integrators. Nonlinear Dyn 2017;90:241–56. [31] Diethelm K, Freed AD. The FracPECE subroutine for the numerical solution of differential equations of fractional order. Forschung und wissenschaftliches Rechnen 1998;1999:57–71. [32] Deng W, Li C, Lü J. Stability analysis of linear fractional differential system with multiple time delays. Nonlinear Dyn 2007;48:409–16. [33] R. Garrappa, ‘‘Predictor-corrector PECE method for fractional differential equations,” MATLAB Central File Exchange [File ID: 32918], 2011. H. Lu et al. / Journal of Advanced Research 25 (2020) 137–145 145