# Analysis of Memreactance with Fractional Kinetics

Authors: Rawid Banchuin
Journal: Mathematical Problems in Engineering
 (2020)

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## Abstract

In this work, the analysis of the memreactance, i.e., meminductor and memcapacitor, with fractional-order kinetics has been proposed. The meminductances, memcapacitances, and related parameters due to both DC and periodic input waveforms have been derived. The behavioral analysis has been thoroughly performed with the aid of numerical simulation. The effects of fractional-order kinetics have been explored where both linear and nonlinear dopant drift scenarios have been considered. Moreover, the emulation of memreactance with fractional-order kinetics by using the memristor and the effect of the fractional-order kinetics on the memreactance-based circuits have also been mentioned along with the extension of our results to the fractional-order memreactance.


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## Body

## 1. Introduction

Apart from the basic circuit elements, the circuit elements with memory, i.e., memristor, meminductor, and memcapacitor, have been found by Leon Chua and his colleagues [1, 2]. The kinetics of the memristor has been further generalized in the fractional-order domain as proposed in previous works [3–7] by using the concept of fractional calculus. By such a concept, the fractional derivative which is capable of including the effect of the past state, i.e., memory effect, of any system of the interested unlike the conventional derivative, has been used in the mathematical analysis. There exist many fractional derivatives, e.g., Riemann–Liouville, Liouville–Caputo [8], Caputo–Fabrizio [9], and Atangana–Baleanu [10]. Some of them, e.g., Riemann–Liouville and Liouville–Caputo derivatives, relied on the simple power-law kernel, whereas the others, e.g., Caputo–Fabrizio and Atangana–Baleanu fractional derivatives, employ more complicated kernel functions, e.g., exponential and Mittag–Leffler functions. As a result, different fractional derivatives describe the effect of the past state, i.e., memory effect, of an arbitrary system in different manners. The applications of the fractional calculus concept and fractional derivatives can be found in many research areas, e.g., biomedical engineering [11, 12], control system [13–15], electrical/electronic engineering [16–30], and plasma physics [31, 32].Motivated by the generalization of memristor kinetics, we generalize the kinetics of the meminductor and memcapacitor which are commonly referred to as memreactance [33–35] and have been adopted in many applications, e.g., electronic oscillator [36] and synaptic circuit [37, 38], in the fractional-order domain by applying the fractional calculus concept to the state equation of the memreactance and perform the modelling of such memreactance. By using the obtained results, the meminductances, memcapacitances, and related parameters due to both DC and periodic input waveforms have been derived. Moreover, the behavioral analysis has been thoroughly performed with the aid of numerical simulation with MATHEMATICA. The effects of fractional-order kinetics have been explored where both linear and nonlinear dopant drift scenarios have been considered. The emulation of memreactance with fractional-order kinetics by using the memristor and the effect of the fractional-order kinetics on the memreactance-based circuits have also been mentioned. In addition, the extension of our results to the fractional-order memreactance which employs the interpolate characteristics between the memristor and the memreactance has also been presented.In the following section, the overview of both the meminductor and the memcapacitor will be briefly given followed by the proposed fractional domain generalization and mathematical model in Section3. The resulting meminductances, memcapacitances, and related parameters due to various input waveforms and the behavioral analysis will be given in Section 4 where the DC waveform will be firstly treated followed by the periodic ones. The emulation by using the memristor will be discussed in Section 5. Moreover, the effect of the fractional-order kinetics on the memreactance-based circuits will be studied in Section 6, and the extension of our results to the fractional-order memreactance will be shown in Section 7. Finally, the conclusion will be drawn in Section 8.


## 2. The Overview of Meminductor and Memcapacitor

Meminductor and memcapacitor are nonlinear electrical circuit elements. They can be simply thought of as the inductor and capacitor with memory. The meminductor relates the time integral of flux (ρ(t)) and instantaneous charge (q(t)) through the following constitutive relation [5]:(1)LMt=dρtdqt,where LM(t) denotes the memductance. Since we assume the electromechanical model of the meminductor and the current controlled operation [39] in this work, LM(t) can be given in terms of its minimum and maximum values denoted by Lmin and Lmax and the state variable (xL(t)) as(2)LMt=Lmin+xLtLmax−Lmin,where xL(t) can be given in terms of the meminductor’s current (i(t)) and mobility factor (kL) by (3) if the linear dopant drift model has been assumed. Note also that fxLt stands for the window function which has been used for modelling the boundary effect of the device.(3)dxLtdt=kLitfxLt.On the contrary, the memcapacitance links the instantaneous flux (φ(t)) to the time integral of charge (σ(t)) by using the following relationship [5]:(4)DMt=dφtdσt,where DM(t) denotes the inverse memcapacitance or memelestance. In this research, the principle of memcapacitor physical operation proposed in [40] which is charge controlled has been assumed. As a result, DM(t) can be in terms of the minimum and maximum of DM(t), i.e., Dmin and Dmax, and the memcapacitor’s state variable (xC(t)) as(5)DMt=Dmin+xCtDmax−Dmin,where xC(t) can be given in terms of q(t) and the memcapacitor’s mobility factor (kC) based on the linear dopant drift model by (6). Similarly to fxLt, fxCt denotes the window function of the memcapacitor.(6)dxCtdt=kCqtfxCt.


## 3. The Generalization and Mathematical Modelling of Memreactance with Fractional-Order Kinetics

After a careful consideration of (2), (3), (5), and (6), it has been found that the memreactance can be mathematically defined by using the following equation:(7)yt=ymin+xtymax−ymin,where(8)dxtdt=kJaitfxt,where, Ja[ ] stands for the integral operator of arbitrary order a and f(x(t)) stands for the window function. It should be mentioned here that a = 0, k = kL, x(t) = xL(t), y(t) =  LMt, ymin =  Lmin, and ymax =  Lmax if the memreactance under consideration is a meminductor. For the memcapacitor on the contrary, a = 1, k = kC, x(t) = xC(t), y(t) = DM(t), ymin = Dmin, and ymax = Dmax.For the memreactance with fractional-order kinetics, we replace d/dt in (8) by the fractional-order derivative of arbitrary real order α, i.e., Dα[ ]. Therefore, we obtain(9)Dαxt=kfxtzt,where zt=Jait.If the linear dopant drift has been assumed, the rectangular window function [2], which is linear, will be adopted. Such window function can be given by(10)fxt=1,0≤xt≤1,0,xt<0∧xt>1.Therefore, (9) becomes(11)Dαxt=kzt,0≤xt≤1,0,xt<0∧xt>1.By takingDa[ ] for both sides of this equation and keeping the above definition of z(t) in mind, we have(12)Dα+axt=kit,0≤xt≤1,0,xt<0∧xt>1.Sincea can be either 0 or 1, we finally obtain(13)Dβxt=kit,0≤xt≤1,0,xt<0∧xt>1,where β = α for the meminductor. On the contrary, β = α + 1 if the memcapacitor had been considered. At this point, it can be seen that the memreactance with fractional-order kinetics can also be defined by using (7) but with (13) as the state equation instead of (8). By using the Riemann–Liouville fractional-order integral [8], we have(14)xt=0,xt<0,x0+kΓβ∫0tt−τβ−1iτdτ,0<xt<1,1,xt≥1,where the integral term in (14) is the Riemann–Liouville fractional-order integral. Note also that x(0) and Г( ) denote the initial value of x(t) and the gamma function [41], respectively.By using (7) and (14) and keeping in mind that the initial value of y(t), i.e., y(0), can be given as(15)y0=ymin+x0ymax−ymin,we obtain(16)yt=ymin,yt≤ymin,y0+kymax−yminΓβ∫0tt−τβ−1iτdτ,ymin<yt<ymax,ymax,yt≥ymax,which can be rewritten in a more compact manner in terms of the nested min[ ] and max[ ] similarly to the previously proposed model of the memristor with fractional-order kinetics [7] as follows:(17)yt=minmaxy0+kymax−yminΓβ∫0tt−τβ−1iτdτ,ymin,ymax.On the contrary, if the nonlinear dopant drift model has been adopted, the analysis will become more complicated. Here, we adopt Jocklecar’s window function [5] which can be given by (18) where p≥1 and p∈I. Note that the aforementioned rectangular window function which merely models the boundary effect in a discrete manner is approximately equivalent to Jocklecar’s window function when p approaches ∞ [5].(18)fxt=1−2xt−12p.By using (18), (13) becomes(19)Dαxt=k1−2xt−12pzt.AsDαxt=Δdαxt/dtα, (19) can be rearranged by following [3] as(20)1−2xt−12p−1dαxt=kztdtα,which is equivalent to(21)1−2xt−12p−1xα−1tdαxtdxtαdxt=kzttα−1dt.Since we definedαxt/dxtα as the αth order Riemann–Liouville fractional derivative [8] of x(t) with respect to x(t), we have(22)1−2xt−12p−1Γ2−αdxt=kzttα−1dt,which is equal to(23)∫x0xt1−2u−12p−1du=kΓ2−α∫0tτα−1zτdτ.If we assumev=2u−1, we obtain(24)∫x0xt1−v2p−1dv=2kΓ2−α∫0tτα−1zτdτ.Since(25)∫1−v2p−1dv=∑r=0∞v2pr+12pr+1,as given in [5], the relationship between x(t) and z(t) can be obtained as(26)∑r=0∞xt2pr+1−x02pr+12pr+1=2kΓ2−α∫0tτα−1zτdτ.As it can be, respectively, seen from (7) that xt=yt−ymin/ymax−ymin, at x0=y0−ymin/ymax−ymin, the relationship between y(t) and z(t) can be finally obtained as follows:(27)∑r=0∞yt−ymin/ymax−ymin2pr+1−y0−ymin/ymax−ymin2pr+12pr+1=2kΓ2−α∫0tτα−1zτdτ.Whenp=1 which means that the dopant drift is highly nonlinear [5] as the degree of nonlinearity is inversely proportional to p and p≥1, (26) becomes(28)∑r=0∞x2r+1t2r+1−∑r=0∞x2r+102r+1=2kΓ2−α∫0tτα−1zτdτ,which is equivalent to(29)tanh−1xt−tanh−1x0=2kΓ2−α∫0tτα−1zτdτ.Therefore, we have the following expression ofx(t):(30)xt=tanhtanh−1x0+2kΓ2−α∫0tτα−1zτdτ.By combining (7) and (30) and keeping the above expression of x(0) in mind, the following expression of y(t) can be obtained:(31)yt=ymin+tanhtanh−1y0−yminymax−ymin+2kΓ2−α∫0tτα−1zτdτymax−ymin.However, it is very hard to obtain the exact expression ofy(t) like (31) when p>1 because it is hard to derive the exact expression of x(t) like (30). As an example for illustration, we let p=2. Thus by using (26), we have(32)∑r=0∞x4r+1t4r+1−∑r=0∞x4r+104r+1=2kΓ2−α∫0tτα−1zτdτ,which is equivalent to(33)tan−12xt−1−tanh−12xt−1=4kΓ2−α∫0tτα−1zτdτ+tan−12x0−1−tanh−12x0−1.Obviously, it can be seen thatx(t) is very hard to be analytically obtained. More complexity can be expected if larger values of p have been assumed.


## 4. Meminductance, Memcapacitance, and Behavioral Analysis

By using our mathematical model, meminductance and memcapacitance of the fractional-order kinetic memreactance excited by various waveforms can be determined and the behavioral analysis can be analyzed where the DC waveform will be firstly considered followed by the AC ones as will be presented in the following sections.


### 4.1. DC Waveform

Mathematically, the DC waveform can be defined asi(t) = IDCu(t), where IDC and u(t) denote the magnitude of the waveform and the unit step function. If we assume the linear dopant drift which yields a linear proportional relationship between the rate of expansion/contraction of the doped region and the applied current (charge) of the meminductor (memcapacitor), the resulting LMt and DM(t) can be obtained by using (17) and by keeping in mind that β = α, k = kL, y(t) = LMt, ymin = Lmin, and ymax = Lmin for the meminductor and β = α + 1, k = kC, y(t) = DM(t), ymin = Dmin, and ymax = Dmax for the memcapacitor as follows:(34)LMt=minmaxLM0+kLLmax−LminIDCtαΓα+1,Lmin,Lmax,(35)DMt=minmaxDM0+kCDmax−DminIDCtα+1Γα+2,Dmin,Dmax.After obtainingLMt and DM(t), LM(t) and memcapacitance (CM(t)) can be, respectively, obtained as the square of LMt and reciprocal of DM(t). By using Lmin = 1 mH, Lmax = 20 mH, L(0) = 5 mH, Dmin = 0.1 MF−1, Dmax = 0.1 GF−1, and D(0) = 0.01 GF−1, LM(t)’s and CM(t)’s with various α’s excited by the DC waveform can be numerically simulated, as depicted in Figures 1–4 where IDC = 1 A has been assumed in Figures 1 and 3 and IDC = −1 A has been assumed in Figures 2 and 4. Moreover, LM(t)’s and CM(t)’s simulated based on the SPICE models of the meminductor and memcapacitor [39, 40] with the rectangular window function have also been included in these figures where the strong agreements between our LM(t)’s and CM(t)’s with α = 1 and their SPICE model-based benchmarks which imply the accuracy of our simulation results can be observed. These figures also show that LM(t) is an increasing and decreasing function of t when IDC > 0 and IDC < 0, respectively, and vice versa for CM(t). However, both LM(t) and CM(t) are saturated at either Lmax or Lmin and Cmax = 1/Dmin or Cmin = 1/Dmax which are the maximum and minimum values of CM(t), after certain saturation times, i.e., tsat,L and tsat,C, for the meminductor and memcapacitor, respectively. It has also been shown that tsat,L and tsat,C are directly proportional to α.Figure 1
L
M(t) due to the DC waveform with IDC > 0: α = 0.75 (red), α = 1 (green), α = 1.25 (blue), and SPICE model (black dots).Figure 2
C
M(t) due to the DC waveform with IDC > 0: α = 0.75 (red), α = 1 (green), α = 1.25 (blue), and SPICE model (black dots).Figure 3
L
M(t) due to the DC waveform with IDC < 0: α = 0.75 (red), α = 1 (green), α = 1.25 (blue), and SPICE model (black dots).Figure 4
C
M(t) due to the DC waveform with IDC < 0: α = 0.75 (red), α = 1 (green), α = 1.25 (blue), and SPICE model (black dots).By using the aforesaid observation, (34) and (35), we have(36)LM0+kLLmax−LminIDCtsat,LαΓα+1=Lmax,IDC>0,Lmin,IDC<0,(37)DM0+kCDmax−DminIDCtsat,Cα+1Γα+2=Dmax,IDC>0,Dmin,IDC<0.Therefore,tsat,L and tsat,C can be immediately given by(38)tsat,L=Γα+1LM0−LmaxkLLmin−LmaxIDC1/α,IDC>0,Γα+1LM0−LminkLLmin−LmaxIDC1/α,IDC<0,(39)tsat,C=Γα+2DM0−DmaxkCDmin−DmaxIDC1/α+1,IDC>0,Γα+2DM0−DminkCDmin−DmaxIDC1/α+1,IDC<0,which shows that they are proportional to the size of difference between the initial and saturated memreactance values which can be either maxima or minima. So, both tsat,L and tsat,C reach their maximum values given, respectively, by tsat,L,M and tsat,C,M, if and only if LM(0) and DM(0) reach their possible peak values given by either Lmin and Dmin when IDC > 0 or Lmax and Dmax when IDC < 0. Thus, tsat,L,M and tsat,C,M can be found as(40)tsat,L,M=Γα+1kLIDC1/α,(41)tsat,C,MAX=Γα+2kCIDC1/α+1.Now, let us assume the nonlinear dopant drift which in turn leads to a nonlinear relationship between such a rate of expansion/contraction of the doped region and the current (charge) of the meminductor (memcapacitor). By using (27), the following LMt and DM(t) with arbitrary p can be obtained:(42)∑r=0∞LMt−Lmin/Lmax−Lmin2pr+1−LM0−Lmin/Lmax−Lmin2pr+12pr+1=2kLIDCΓ2−αtαα,(43)∑r=0∞DMt−Dmin/Dmax−Dmin2pr+1−DM0−Dmin/Dmax−Dmin2pr+12pr+1=2kCΓ2−αIDCtα+1α+1.If we letp=1, the resulting LMt and DM(t) can be obtained by using (31) as follows:(44)LMt=Lmin+tanhtanh−1LM0−LminLmax−Lmin+2kLIDCΓ2−αtααLmax−Lmin,(45)DMt=Dmin+tanhtanh−1DM0−DminDmax−Dmin+2kCΓ2−αIDCtα+1α+1Dmax−Dmin.As previously done for the conventional memcapacitor with integer kinetic transport [5], we assume that the memreactance with the nonlinear dopant drift is saturated at either x(t) = xon, i.e., y(t) = ysat,on, or x(t) = xoff, i.e., y(t) = ysat,off. By using (44) and keeping in mind that xon = xon,L, xoff = xoff,L, ysat,on = Lsat,on, and ysat,off = Lsat,off for the meminductor, the following nonlinear dopant drift model-based saturation time of the meminductor can be obtained:(46)Tsat,L=α2kLIDCΓ2−αtanh−1Lsat,off−LminLmax−Lmin−tanh−1LM0−LminLmax−Lmin1/α,IDC>0,α2kLIDCΓ2−αtanh−1Lsat,on−LminLmax−Lmin−tanh−1LM0−LminLmax−Lmin1/α,IDC<0.On the contrary, the nonlinear dopant drift model-based saturation time of the memcapacitor can be derived by using (45) and keeping in mind that xon = xon,C, xoff = xoff,C, ysat,on = Dsat,on, and ysat,off = Dsat,off as(47)Tsat,C=α+12kCΓ2−αIDCtanh−1Dsat,off−DminDmax−Dmin−tanh−1DM0−DminDmax−Dmin1/α+1,IDC>0,α+12kCΓ2−αIDCtanh−1Dsat,on−DminDmax−Dmin−tanh−1DM0−DminDmax−Dmin1/α+1,IDC<0.Since the maximum saturation time occurred when the memreactance is saturated at eithery(t) = yon given y(0) = yoff or y(t) = yoff given y(0) = yon, the maximal values of Tsat,L and Tsat,C can be, respectively, given by(48)Tsat,L,M=α2kLIDCΓ2−αtanh−1Lsat,off−LminLmax−Lmin−tanh−1Lsat,on−LminLmax−Lmin1/α,IDC>0,α2kLIDCΓ2−αtanh−1Lsat,on−LminLmax−Lmin−tanh−1Lsat,off−LminLmax−Lmin1/α,IDC<0,(49)Tsat,C,M=α+12kCΓ2−αIDCtanh−1Dsat,off−DminDmax−Dmin−tanh−1Dsat,on−DminDmax−Dmin1/α+1,IDC>0,α+12kCΓ2−αIDCtanh−1Dsat,on−DminDmax−Dmin−tanh−1Dsat,off−DminDmax−Dmin1/α+1,IDC<0,which can be simplified as(50)Tsat,L,M=α2kLIDCΓ2−αtanh−1xL,off−tanh−1xL,on1/α,Tsat,C,M=α+12kCΓ2−αIDCtanh−1xC,off−tanh−1xC,on1/α+1.At this point, we simulatetsat,L,M, tsat,C,M, Tsat,L,M, and Tsat,C,M with respect to IDC by assuming that xon,C = xon,L = 0.01 and xoff,C = xoff,L = 0.99 where other parameters similar to those of the simulation of LM(t) and CM(t) have been adopted. The results are depicted in Figures 5 and 6 which show that these saturation times are directly proportional to α but inversely proportional to the magnitude of IDC. It can also be seen that the memreactance with the nonlinear dopant drift except the memcapacitor with α > 1 takes longer time to reach saturation.Figure 5
t
sat,L,M (line) and Tsat,L,M (dashed): α = 0.75 (red), α = 1 (green), and α = 1.25 (blue).Figure 6
t
sat,C,M (line) and Tsat,C,M (dashed): α = 0.75 (red), α = 1 (green), and α = 1.25 (blue).Before we proceed to the subsequent section, it should be mentioned here that(51)Tsat,M,Ltsat,M,L=tanh−1xL,off−tanh−1xL,on2ΓαΓ2−α1/α,Tsat,M,Ctsat,M,C=tanh−1xC,off−tanh−1xC,on2Γ2−αΓα+11/α+1.


### 4.2. Periodic Waveforms

For the sinusoidal waveform with arbitrary phase (θ), i.e., i(t) = Imsin(ωt + θ), where Im and ω, respectively, denote its peak value and angular frequency, LMt with the linear dopant drift can be given by using (17) with β = α, k = kL, y(t) = LMt, ymin = Lmin, and ymax = Lmin as(52)LMt=minmaxLM0−kLLmin−LmaxImtαΓα+1sinθF121;α2+12,α2+1;−14ωt2+ωtcosθα+1F121;α2+1,α2+32;−14ωt2,Lmin,Lmax,where F12;,; denotes a generalized hypergeometric function with p=1 and q=2 [42].At the steady state,LMt is reduced to the following equation which can be given by using our model and the asymptotic approximation of the sinusoidal function’s fractional-order integration [43]:(53)LMt=minmaxLM0−kLLmin−LmaxImωαcosθ+1−α2π−cosωt+θ+1−α2π,Lmin,Lmax.By using (53) with θ = π/2 rad, Lmin = 1 mH, Lmax = 20 mH, and L(0) = 5 mH, we can simulate the ρ(t)-q(t) and φ(t)-i(t) characteristics of the fractional-order kinetic meminductor with various α’s, as depicted in Figures 7–10 where Im = 10 mA has been assumed in Figures 7 and 9. On the contrary, Im = −10 mA has been adopted in Figures 8 and 10. Moreover, the vertical axes of these figures have been scaled up by 100 for visibility. From Figures 7 and 8, it can be seen that the one-to-one correspondence between ρ(t) and q(t) ceased to be existed when α ≠ 1 which means that the constitutive relation of the meminductor is ambiguous when its kinetic transport is of fractional order. Therefore, the unambiguous constitutive relation which is one of the basic fingerprints of the meminductor [39] cannot be preserved by the fractional-order domain generalization.Figure 7
ρ(t)-q(t) of the meminductor under the periodic waveform with Im > 0: α = 0.75 (red), α = 1 (green), and α = 1.25 (blue).Figure 8
ρ(t)-q(t) of the meminductor under the periodic waveform with Im < 0: α = 0.75 (red), α = 1 (green), and α = 1.25 (blue).Figure 9
φ(t)-i(t) of the meminductor under the periodic waveform with Im > 0: α = 0.75 (red), α = 1 (green), α = 1.25 (blue), and SPICE model (black dots).Figure 10
φ(t)-i(t) of the meminductor under the periodic waveform with Im < 0: α = 0.75 (red), α = 1 (green), α = 1.25 (blue), and SPICE model (black dots).However, despite the asymmetricities, the pinched hysteresis loop in theφ(t)-i(t) characteristic which is also a basic fingerprint of the meminductor can be preserved by the fractional-order domain generalization as can be seen from Figures 9 and 10 where the strong agreements between the φ(t)-i(t) Lissajous curve of the device with α = 1 and the SPICE meminductor model counterpart with rectangular window function that verify our results can be observed. Moreover, it can be observed that the lobe area of the pinched hysteresis loop which refers to the memory effect and linearity is inversely proportional to α. Therefore, more memory effect and less linearity can be obtained by using the meminductor with fractional-order kinetics of lower α, and vice versa.When the kinetic transport is fractional, the meminductor becomes unbalanced unlessα = 1 as it has nonzero average power consumption caused by nonzero stored energy after an integer number of periods. In order to illustrate this issue, the areas of the upper lobe and lower lobe of the φ(t)-i(t) Lissajous curve, i.e., AUL and ALL, must be firstly calculated. For doing so, we use the following equations:(54)AUL=∫−θ/ωπ−θ/ωφtdit,(55)ALL=∫π−θ/ω2π−θ/ωφtdit.Sinceφ(t) = LM(t)i(t), where i(t) = Imsin(ωt + θ), AUL and ALL can be obtained by using (53)–(55) as follows:(56)AUL=−kIm3π12ω2αsinαπ2Lmax−Lmin2×16ωαLM0+kImLmax−Lmin3πcosαπ2−16sinαπ2−θ,(57)ALL=kIm3π12ω2αsinαπ2Lmax−Lmin2×16ωαLM0+kImLmax−Lmin16sinαπ2−θ−3πcosαπ2.Therefore,AUL + ALL can be immediately found as(58)AUL+ALL=−k2Im4π4ω2αsinαπLmax−Lmin2.SinceAUL and ALL are, respectively, referred to as the energy intake and energy dissipated during the positive and negative half cycles, AUL + ALL refers to the total stored energy after a period and its multiple is referred to as the aforesaid net stored energy after an integer number of periods. AUL + ALL ≠ 0, and so do its multiple and such net stored energy; thus, nonzero average power consumption is obtained and the aforementioned unbalance occurred. However, it can be seen from (51) that AUL + ALL = 0 and so do its multiple, the net stored energy, and the average power consumption if we let α = 1. Therefore, the balance can be achieved with this specific value of α. It can also be seen from (56)–(58) that the total stored energy after a period, net stored energy after an integer number of periods, and average power consumption are independent of θ despite the energy intake and energy dissipated during the positive and negative half cycles are dependent.Despite employing fractional-order kinetic transport, the meminductor still has identical time instants at whichφ(t) and i(t) cross zero levels; that is, the fractional-order domain generalization preserves this basic fingerprint of the meminductor. For illustration, we simulate φ(t) of the fractional-order meminductor with various α’s, as depicted in Figures 11 and 12 where i(t) has also been included and φ(t)’s have been multiplied by 100 for visibility.Figure 11
φ(t)-i(t) of the meminductor under the periodic waveform with Im > 0: α = 0.75 (red), α = 1 (green), α = 1.25 (blue), and i(t) (black).Figure 12
φ(t)-i(t) of the meminductor under the periodic waveform with Im < 0: α = 0.75 (red), α = 1 (green), α = 1.25 (blue), and i(t) (black).From Figures11 and 12 where θ = π/2 rad, Lmin = 1 mH, Lmax = 20 mH, and L(0) = 5 mH have also been assumed, it can be seen that time instants at which both φ(t) and i(t) cross zero levels are identical. Without regarding any specific device, we commonly denote these time instants by tx0. Since i(t) = Imsin(ωt + θ), tx0 can be obtained by(59)tx0=nπ−θω,where n = {1, 2, 3, …}.Now, the effects of the nonlinear dopant drift will be studied. By using (27), the resulting LMt with arbitrary p can be found as(60)∑r=0∞LMt−Lmin/Lmax−Lmin2pr+1−LM0−Lmin/Lmax−Lmin2pr+12pr+1=jkLImΓ2−αω−αexpjθΓα,−jωt−exp−jθΓα,jωt,where Г( ) denotes the incomplete gamma function [44].On the contrary,LMt with p=1 can be obtained by using (31) as follows:(61)LMt=Lmin+tanhtanh−1LM0−LminLmax−Lmin+jkLImΓ2−αω−αexpjθΓα,−jωt−exp−jθΓα,jωtLmax−Lmin.If we letα approach 0, ∫0tτα−1zτdτ approaches ∫0tτ−1zτdτ. Thus by also, respectively, applying (27) and (31), we have(62)∑r=0∞LMt−Lmin/Lmax−Lmin2pr+1−LM0−Lmin/Lmax−Lmin2pr+12pr+1=2kLImΓ2−αsinθCiωt+cosθSiωt,LMt=Lmin+tanhtanh−1LM0−LminLmax−Lmin+2kLImΓ2−αsinθCiωt+cosθSiωtLmax−Lmin,where Si( ) and Ci( ) stand for sine and cosine integral functions [45].Ifα approaches 1 on the contrary, ∫0tτα−1zτdτ approaches ∫0tzτdτ. As a result, we have(63)∑r=0∞LMt−Lmin/Lmax−Lmin2pr+1−LM0−Lmin/Lmax−Lmin2pr+12pr+1=−2kLImΓ2−αcosωτ+θω,LMt=Lmin+tanhtanh−1LM0−LminLmax−Lmin−2kLImΓ2−αcosωτ+θωLmax−Lmin.With these results, we can simulateLM(t) with the effect of the nonlinear dopant drift as dashed curves in Figures 13 and 14 where those based on the linear dopant drift have also been included as normal curves for comparison. We also simulate the φ(t)-i(t) characteristic, as shown in Figures 15 and 16 where the dashed curves have been adopted for being distinguished from the linear dopant drift-based φ(t)-i(t) depicted in Figures 9 and 10. Here, we choose p=1 where θ = π/2 rad, Lmin = 1 mH, Lmax = 20 mH, and L(0) = 5 mH have also been assumed. From these figures where the strong agreements between our results with the nonlinear dopant drift and those based on the SPICE model with Jocklecar’s window function and p=1 can be observed, and Figures 9 and 10, it can be seen that the nonlinear dopant drift reduces the effects of α on the variation in LM(t) and variation in nonlinearity of the device; that is, the device is more robust to the effects of fractional kinetic transport. We have also found that the meminductor becomes more nonlinear with increasing α and with a more symmetric pinched hysteresis loop if the nonlinear dopant drift has been assumed for ceteris paribus.Figure 13
L
M(t) of the meminductor with linear (normal curve) and nonlinear (dashed curve) dopant drifts under the periodic waveform with Im > 0: α = 0.75 (red), α = 1 (green), α = 1.25 (blue), and SPICE model (black).Figure 14
L
M(t) of the meminductor with linear (normal curve) and nonlinear (dashed curve) dopant drifts under the periodic waveform with Im < 0: α = 0.75 (red), α = 1 (green), α = 1.25 (blue), and SPICE model (black).Figure 15
φ(t)-i(t) of the meminductor with the nonlinear dopant drift under the periodic waveform with Im > 0: α = 0.75 (red), α = 1 (green), α = 1.25 (blue), and SPICE model (black dots).Figure 16
φ(t)-i(t) of the meminductor with the nonlinear dopant drift under the periodic waveform with Im < 0: α = 0.75 (red), α = 1 (green), α = 1.25 (blue), and SPICE model (black dots).At this point, the memcapacitor with fractional-order kinetics will be considered where cosinusoidal with arbitraryθ, i.e., i(t) = Imcos(ωt + θ), will be assumed. Assuming the linear dopant drift, DM(t) can be given by using (17) with β = α + 1, k = kC, y(t) = DM(t), ymin = Dmin, and ymax = Dmax as(64)DMt=DM0−kCDmin−DmaxImtα+1Γα+2sinθF121;α+12+12,α+12+1;−14π2−ωt−2θ2+π/2−ωt−2θcosθα+2F121;α+12+1,α+12+32;−14π2−ωt−2θ2.Similarly toLMt, DM(t) is reduced to a simplified version at the steady state given by(65)DMt=DM0−kCDmin−DmaxImωα+1cosθ−α2π−cos1−α2π−ωt+θ.By using (65) with θ = 0 rad, Dmin = 0.1 MF−1, Dmax = 0.1 GF−1, and D(0) = 0.01 GF−1, we can simulate the φ(t)-σ(t) and vt-q(t) characteristics with various α’s, as shown in Figures 17–20 where Im = 0.1 μA has been assumed in Figures 17 and 19. On the contrary, Im = −0.1 μA has been adopted in Figures 18 and 20. Moreover, the vertical axes of these figures have been scaled down by 107 for visibility. Similarly to the meminductor, it has been found that the constitutive relation of the memcapacitor with fractional-order kinetics becomes ambiguous unless α ≠ 1 as the one-to-one correspondence between φ(t) and σ(t) ceased to be existed, as shown in Figures 17 and 18. Thus, it can be stated that the fractional-order domain generalization cannot preserve the unambiguous constitutive relation which is one of the basic fingerprints of the memcapacitor [40]. However, the 2nd basic fingerprint, i.e., the pinched hysteresis loop in the vt-q(t) characteristic, can be preserved despite the asymmetricity when α ≠ 1, as shown in Figures 19 and 20 where the strong agreements between the vt-q(t) characteristic with α = 1 and the SPICE memcapacitor model-based counterpart with rectangular window function can be observed. Similarly to the meminductor, it can be seen that the lobe area of the pinched hysteresis loop of the memcapacitor with fractional-order kinetics is also inversely proportional to α. Therefore, more memory effect and less linearity can also be obtained by using the memcapacitor with lower α, and vice versa.Figure 17
φ(t)-σ(t) of the memcapacitor under the periodic waveform with Im > 0: α = 0.75 (red), α = 1 (green), and α = 1.25 (blue).Figure 18
φ(t)-σ(t) of the memcapacitor under the periodic waveform with Im < 0: α = 0.75 (red), α = 1 (green), and α = 1.25 (blue).Figure 19
v
t-q(t) of the memcapacitor under the periodic waveform with Im > 0: α = 0.75 (red), α = 1 (green), α = 1.25 (blue), and SPICE model (black dots).Figure 20
v
t-q(t) of the memcapacitor under the periodic waveform with Im < 0: α = 0.75 (red), α = 1 (green), α = 1.25 (blue), and SPICE model (black dots).Unlike the meminductor, we have found that the fractional-order kinetic memcapacitor is balanced. For illustration, the areas of the upper lobe and lower lobe of thevt-q(t) pinched hysteresis loop, i.e., AUC and ALC, must be calculated by using the following equations:(66)AUC=∫−θ/ωπ−θ/ωvtdqt,(67)ALC=∫π−θ/ω2π−θ/ωvtdqt.Sincevt = DM(t)q(t) and qt=∫0tiτdτ, where i(t) = Imcos(ωt + θ), AUC and ALC can be obtained by using (65)–(67) as follows:(68)AUC=−23kCIm3ωα+3sinαπ2Dmax−Dmin,ALC=23kCIm3ωα+3sinαπ2Dmax−Dmin,which show that AUC + ALC = 0. Since AUC and ALC, respectively, refer to the energy intake and energy dissipated during the positive and negative half cycles, AUC + ALC and its multiple can be interpreted in a similar manner to AUL + ALL and its multiple. So, AUC + ALC = 0 means that zero average power consumption is obtained, and therefore, the memcapacitor with fractional-order kinetics is balanced. Moreover, both AUC and ALC are independent of θ, and so do the energy intake and energy dissipated during the positive and negative half cycles unlike those of the fractional-order kinetic meminductor.Finally, it can be seen that the fractional-order domain generalization preserves the 3rd basic fingerprint of the memcapacitor, i.e., identical time instants at which vt and q(t) cross zero levels. As illustrations, we simulate vt of the fractional-order kinetic memcapacitor with various α’s, as depicted in Figures 21 and 22 where q(t) has also been included, vt’s have been divided by 107 for visibility, and θ = 0 rad, Dmin = 0.1 MF−1, Dmax = 0.1 GF−1, and D(0) = 0.01 GF−1 have been assumed. Obviously, these figures show that vt and q(t) employ identical tx0. Since i(t) = Imcos(ωt + θ), qt=Im/ωsinωt+θ as qt=∫0tiτdτ. Therefore, tx0 can be obtained as given by (59).Figure 21
v
t of the memcapacitor under the periodic waveform with Im > 0: α = 0.75 (red), α = 1 (green), α = 1.25 (blue), and q(t) (black).Figure 22
v
t of the memcapacitor under the periodic waveform with Im < 0: α = 0.75 (red), α = 1 (green), α = 1.25 (blue), and q(t) (black).Finally, the effects of the nonlinear dopant drift will be analyzed. By using (27) and (31), the resulting DMt with arbitrary p and p=1 can be, respectively, found as(69)∑r=0∞DMt−Dmin/Dmax−Dmin2pr+1−DM0−Dmin/Dmax−Dmin2pr+12pr+1=jkCΓ2−αImω1+αexpjθΓα,−jωt−exp−jθΓα,jωt,DMt=Dmin+tanhtanh−1DM0−DminDmax−Dmin+jkCΓ2−αImω1+αexpjθΓα,−jωt−exp−jθΓα,jωtDmax−Dmin.Ifα approaches 0, we have(70)∑r=0∞DMt−Dmin/Dmax−Dmin2pr+1−DM0−Dmin/Dmax−Dmin2pr+12pr+1=2kCΓ2−αImωciωtsinθ+siωtcosθ,DMt=Dmin+tanhtanh−1DM0−DminDmax−Dmin+2kCΓ2−αImωciωtsinθ+siωtcosθDmax−Dmin,as ∫0tτα−1zτdτ approaches ∫0tτ−1zτdτ.But ifα approaches 1 instead, we have(71)∑r=0∞DMt−Dmin/Dmax−Dmin2pr+1−DM0−Dmin/Dmax−Dmin2pr+12pr+1=−2kCΓ2−αImcosωt+θω2,DMt=Dmin+tanhtanh−1DM0−DminDmax−Dmin−2kCΓ2−αImcosωt+θω2Dmax−Dmin.This is because∫0tτα−1zτdτ approaches ∫0tzτdτ.By keeping in mind thatCM(t) = 1/DM(t) and applying the above equations, we can simulate CM(t) affected by the nonlinear dopant drift as dashed curved in Figures 23 and 24 where those based on the linear dopant drift have been included as normal curves for comparison. We also simulate the vt-q(t) characteristic, as shown in Figures 25 and 26 where the dashed curves have been used for being distinguished from the linear dopant drift-based vt-q(t) depicted in Figures 19 and 20. Here, we choose p=1 where θ = 0 rad, Dmin = 0.1 MF−1, Dmax = 0.1 GF−1, and D(0) = 0.01 GF−1 have also been assumed for ceteris paribus.Figure 23
C
M(t) of the memcapacitor with linear (normal curve) and nonlinear (dashed curve) dopant drifts under the periodic waveform with Im > 0: α = 0.75 (red), α = 1 (green), α = 1.25 (blue), and SPICE model (black dots).Figure 24
C
M(t) of the memcapacitor with linear (normal curve) and nonlinear (dashed curve) dopant drifts under the periodic waveform with Im < 0: α = 0.75 (red), α = 1 (green), α = 1.25 (blue), and SPICE model (black dots).Figure 25
v
t-q(t) of the memcapacitor with the nonlinear dopant drift under the periodic waveform with Im > 0: α = 0.75 (red), α = 1 (green), α = 1.25 (blue), and SPICE model (black dots).Figure 26
v
t-q(t) of the memcapacitor with the nonlinear dopant drift under the periodic waveform with Im > 0: α = 0.75 (red), α = 1 (green), α = 1.25 (blue), and SPICE model (black dots).From Figures23–26, where strong agreements between our results with the nonlinear dopant drift and those based on the SPICE model with Jocklecar’s window function and p=1 can be observed, and Figures 19 and 20, it can be seen that the nonlinear dopant drift reduces the effects of α on the variation in CM(t) and variation in nonlinearity of the device. Therefore, the memcapacitor with the nonlinear dopant drift is more robust to the effects of fractional kinetic transport. Similarly to the meminductor, the memcapacitor with the nonlinear dopant drift also becomes more nonlinear with increasing α and employs a more symmetric pinched hysteresis loop than that with the linear dopant drift.


## 4.1. DC Waveform

Mathematically, the DC waveform can be defined asi(t) = IDCu(t), where IDC and u(t) denote the magnitude of the waveform and the unit step function. If we assume the linear dopant drift which yields a linear proportional relationship between the rate of expansion/contraction of the doped region and the applied current (charge) of the meminductor (memcapacitor), the resulting LMt and DM(t) can be obtained by using (17) and by keeping in mind that β = α, k = kL, y(t) = LMt, ymin = Lmin, and ymax = Lmin for the meminductor and β = α + 1, k = kC, y(t) = DM(t), ymin = Dmin, and ymax = Dmax for the memcapacitor as follows:(34)LMt=minmaxLM0+kLLmax−LminIDCtαΓα+1,Lmin,Lmax,(35)DMt=minmaxDM0+kCDmax−DminIDCtα+1Γα+2,Dmin,Dmax.After obtainingLMt and DM(t), LM(t) and memcapacitance (CM(t)) can be, respectively, obtained as the square of LMt and reciprocal of DM(t). By using Lmin = 1 mH, Lmax = 20 mH, L(0) = 5 mH, Dmin = 0.1 MF−1, Dmax = 0.1 GF−1, and D(0) = 0.01 GF−1, LM(t)’s and CM(t)’s with various α’s excited by the DC waveform can be numerically simulated, as depicted in Figures 1–4 where IDC = 1 A has been assumed in Figures 1 and 3 and IDC = −1 A has been assumed in Figures 2 and 4. Moreover, LM(t)’s and CM(t)’s simulated based on the SPICE models of the meminductor and memcapacitor [39, 40] with the rectangular window function have also been included in these figures where the strong agreements between our LM(t)’s and CM(t)’s with α = 1 and their SPICE model-based benchmarks which imply the accuracy of our simulation results can be observed. These figures also show that LM(t) is an increasing and decreasing function of t when IDC > 0 and IDC < 0, respectively, and vice versa for CM(t). However, both LM(t) and CM(t) are saturated at either Lmax or Lmin and Cmax = 1/Dmin or Cmin = 1/Dmax which are the maximum and minimum values of CM(t), after certain saturation times, i.e., tsat,L and tsat,C, for the meminductor and memcapacitor, respectively. It has also been shown that tsat,L and tsat,C are directly proportional to α.Figure 1
L
M(t) due to the DC waveform with IDC > 0: α = 0.75 (red), α = 1 (green), α = 1.25 (blue), and SPICE model (black dots).Figure 2
C
M(t) due to the DC waveform with IDC > 0: α = 0.75 (red), α = 1 (green), α = 1.25 (blue), and SPICE model (black dots).Figure 3
L
M(t) due to the DC waveform with IDC < 0: α = 0.75 (red), α = 1 (green), α = 1.25 (blue), and SPICE model (black dots).Figure 4
C
M(t) due to the DC waveform with IDC < 0: α = 0.75 (red), α = 1 (green), α = 1.25 (blue), and SPICE model (black dots).By using the aforesaid observation, (34) and (35), we have(36)LM0+kLLmax−LminIDCtsat,LαΓα+1=Lmax,IDC>0,Lmin,IDC<0,(37)DM0+kCDmax−DminIDCtsat,Cα+1Γα+2=Dmax,IDC>0,Dmin,IDC<0.Therefore,tsat,L and tsat,C can be immediately given by(38)tsat,L=Γα+1LM0−LmaxkLLmin−LmaxIDC1/α,IDC>0,Γα+1LM0−LminkLLmin−LmaxIDC1/α,IDC<0,(39)tsat,C=Γα+2DM0−DmaxkCDmin−DmaxIDC1/α+1,IDC>0,Γα+2DM0−DminkCDmin−DmaxIDC1/α+1,IDC<0,which shows that they are proportional to the size of difference between the initial and saturated memreactance values which can be either maxima or minima. So, both tsat,L and tsat,C reach their maximum values given, respectively, by tsat,L,M and tsat,C,M, if and only if LM(0) and DM(0) reach their possible peak values given by either Lmin and Dmin when IDC > 0 or Lmax and Dmax when IDC < 0. Thus, tsat,L,M and tsat,C,M can be found as(40)tsat,L,M=Γα+1kLIDC1/α,(41)tsat,C,MAX=Γα+2kCIDC1/α+1.Now, let us assume the nonlinear dopant drift which in turn leads to a nonlinear relationship between such a rate of expansion/contraction of the doped region and the current (charge) of the meminductor (memcapacitor). By using (27), the following LMt and DM(t) with arbitrary p can be obtained:(42)∑r=0∞LMt−Lmin/Lmax−Lmin2pr+1−LM0−Lmin/Lmax−Lmin2pr+12pr+1=2kLIDCΓ2−αtαα,(43)∑r=0∞DMt−Dmin/Dmax−Dmin2pr+1−DM0−Dmin/Dmax−Dmin2pr+12pr+1=2kCΓ2−αIDCtα+1α+1.If we letp=1, the resulting LMt and DM(t) can be obtained by using (31) as follows:(44)LMt=Lmin+tanhtanh−1LM0−LminLmax−Lmin+2kLIDCΓ2−αtααLmax−Lmin,(45)DMt=Dmin+tanhtanh−1DM0−DminDmax−Dmin+2kCΓ2−αIDCtα+1α+1Dmax−Dmin.As previously done for the conventional memcapacitor with integer kinetic transport [5], we assume that the memreactance with the nonlinear dopant drift is saturated at either x(t) = xon, i.e., y(t) = ysat,on, or x(t) = xoff, i.e., y(t) = ysat,off. By using (44) and keeping in mind that xon = xon,L, xoff = xoff,L, ysat,on = Lsat,on, and ysat,off = Lsat,off for the meminductor, the following nonlinear dopant drift model-based saturation time of the meminductor can be obtained:(46)Tsat,L=α2kLIDCΓ2−αtanh−1Lsat,off−LminLmax−Lmin−tanh−1LM0−LminLmax−Lmin1/α,IDC>0,α2kLIDCΓ2−αtanh−1Lsat,on−LminLmax−Lmin−tanh−1LM0−LminLmax−Lmin1/α,IDC<0.On the contrary, the nonlinear dopant drift model-based saturation time of the memcapacitor can be derived by using (45) and keeping in mind that xon = xon,C, xoff = xoff,C, ysat,on = Dsat,on, and ysat,off = Dsat,off as(47)Tsat,C=α+12kCΓ2−αIDCtanh−1Dsat,off−DminDmax−Dmin−tanh−1DM0−DminDmax−Dmin1/α+1,IDC>0,α+12kCΓ2−αIDCtanh−1Dsat,on−DminDmax−Dmin−tanh−1DM0−DminDmax−Dmin1/α+1,IDC<0.Since the maximum saturation time occurred when the memreactance is saturated at eithery(t) = yon given y(0) = yoff or y(t) = yoff given y(0) = yon, the maximal values of Tsat,L and Tsat,C can be, respectively, given by(48)Tsat,L,M=α2kLIDCΓ2−αtanh−1Lsat,off−LminLmax−Lmin−tanh−1Lsat,on−LminLmax−Lmin1/α,IDC>0,α2kLIDCΓ2−αtanh−1Lsat,on−LminLmax−Lmin−tanh−1Lsat,off−LminLmax−Lmin1/α,IDC<0,(49)Tsat,C,M=α+12kCΓ2−αIDCtanh−1Dsat,off−DminDmax−Dmin−tanh−1Dsat,on−DminDmax−Dmin1/α+1,IDC>0,α+12kCΓ2−αIDCtanh−1Dsat,on−DminDmax−Dmin−tanh−1Dsat,off−DminDmax−Dmin1/α+1,IDC<0,which can be simplified as(50)Tsat,L,M=α2kLIDCΓ2−αtanh−1xL,off−tanh−1xL,on1/α,Tsat,C,M=α+12kCΓ2−αIDCtanh−1xC,off−tanh−1xC,on1/α+1.At this point, we simulatetsat,L,M, tsat,C,M, Tsat,L,M, and Tsat,C,M with respect to IDC by assuming that xon,C = xon,L = 0.01 and xoff,C = xoff,L = 0.99 where other parameters similar to those of the simulation of LM(t) and CM(t) have been adopted. The results are depicted in Figures 5 and 6 which show that these saturation times are directly proportional to α but inversely proportional to the magnitude of IDC. It can also be seen that the memreactance with the nonlinear dopant drift except the memcapacitor with α > 1 takes longer time to reach saturation.Figure 5
t
sat,L,M (line) and Tsat,L,M (dashed): α = 0.75 (red), α = 1 (green), and α = 1.25 (blue).Figure 6
t
sat,C,M (line) and Tsat,C,M (dashed): α = 0.75 (red), α = 1 (green), and α = 1.25 (blue).Before we proceed to the subsequent section, it should be mentioned here that(51)Tsat,M,Ltsat,M,L=tanh−1xL,off−tanh−1xL,on2ΓαΓ2−α1/α,Tsat,M,Ctsat,M,C=tanh−1xC,off−tanh−1xC,on2Γ2−αΓα+11/α+1.


## 4.2. Periodic Waveforms

For the sinusoidal waveform with arbitrary phase (θ), i.e., i(t) = Imsin(ωt + θ), where Im and ω, respectively, denote its peak value and angular frequency, LMt with the linear dopant drift can be given by using (17) with β = α, k = kL, y(t) = LMt, ymin = Lmin, and ymax = Lmin as(52)LMt=minmaxLM0−kLLmin−LmaxImtαΓα+1sinθF121;α2+12,α2+1;−14ωt2+ωtcosθα+1F121;α2+1,α2+32;−14ωt2,Lmin,Lmax,where F12;,; denotes a generalized hypergeometric function with p=1 and q=2 [42].At the steady state,LMt is reduced to the following equation which can be given by using our model and the asymptotic approximation of the sinusoidal function’s fractional-order integration [43]:(53)LMt=minmaxLM0−kLLmin−LmaxImωαcosθ+1−α2π−cosωt+θ+1−α2π,Lmin,Lmax.By using (53) with θ = π/2 rad, Lmin = 1 mH, Lmax = 20 mH, and L(0) = 5 mH, we can simulate the ρ(t)-q(t) and φ(t)-i(t) characteristics of the fractional-order kinetic meminductor with various α’s, as depicted in Figures 7–10 where Im = 10 mA has been assumed in Figures 7 and 9. On the contrary, Im = −10 mA has been adopted in Figures 8 and 10. Moreover, the vertical axes of these figures have been scaled up by 100 for visibility. From Figures 7 and 8, it can be seen that the one-to-one correspondence between ρ(t) and q(t) ceased to be existed when α ≠ 1 which means that the constitutive relation of the meminductor is ambiguous when its kinetic transport is of fractional order. Therefore, the unambiguous constitutive relation which is one of the basic fingerprints of the meminductor [39] cannot be preserved by the fractional-order domain generalization.Figure 7
ρ(t)-q(t) of the meminductor under the periodic waveform with Im > 0: α = 0.75 (red), α = 1 (green), and α = 1.25 (blue).Figure 8
ρ(t)-q(t) of the meminductor under the periodic waveform with Im < 0: α = 0.75 (red), α = 1 (green), and α = 1.25 (blue).Figure 9
φ(t)-i(t) of the meminductor under the periodic waveform with Im > 0: α = 0.75 (red), α = 1 (green), α = 1.25 (blue), and SPICE model (black dots).Figure 10
φ(t)-i(t) of the meminductor under the periodic waveform with Im < 0: α = 0.75 (red), α = 1 (green), α = 1.25 (blue), and SPICE model (black dots).However, despite the asymmetricities, the pinched hysteresis loop in theφ(t)-i(t) characteristic which is also a basic fingerprint of the meminductor can be preserved by the fractional-order domain generalization as can be seen from Figures 9 and 10 where the strong agreements between the φ(t)-i(t) Lissajous curve of the device with α = 1 and the SPICE meminductor model counterpart with rectangular window function that verify our results can be observed. Moreover, it can be observed that the lobe area of the pinched hysteresis loop which refers to the memory effect and linearity is inversely proportional to α. Therefore, more memory effect and less linearity can be obtained by using the meminductor with fractional-order kinetics of lower α, and vice versa.When the kinetic transport is fractional, the meminductor becomes unbalanced unlessα = 1 as it has nonzero average power consumption caused by nonzero stored energy after an integer number of periods. In order to illustrate this issue, the areas of the upper lobe and lower lobe of the φ(t)-i(t) Lissajous curve, i.e., AUL and ALL, must be firstly calculated. For doing so, we use the following equations:(54)AUL=∫−θ/ωπ−θ/ωφtdit,(55)ALL=∫π−θ/ω2π−θ/ωφtdit.Sinceφ(t) = LM(t)i(t), where i(t) = Imsin(ωt + θ), AUL and ALL can be obtained by using (53)–(55) as follows:(56)AUL=−kIm3π12ω2αsinαπ2Lmax−Lmin2×16ωαLM0+kImLmax−Lmin3πcosαπ2−16sinαπ2−θ,(57)ALL=kIm3π12ω2αsinαπ2Lmax−Lmin2×16ωαLM0+kImLmax−Lmin16sinαπ2−θ−3πcosαπ2.Therefore,AUL + ALL can be immediately found as(58)AUL+ALL=−k2Im4π4ω2αsinαπLmax−Lmin2.SinceAUL and ALL are, respectively, referred to as the energy intake and energy dissipated during the positive and negative half cycles, AUL + ALL refers to the total stored energy after a period and its multiple is referred to as the aforesaid net stored energy after an integer number of periods. AUL + ALL ≠ 0, and so do its multiple and such net stored energy; thus, nonzero average power consumption is obtained and the aforementioned unbalance occurred. However, it can be seen from (51) that AUL + ALL = 0 and so do its multiple, the net stored energy, and the average power consumption if we let α = 1. Therefore, the balance can be achieved with this specific value of α. It can also be seen from (56)–(58) that the total stored energy after a period, net stored energy after an integer number of periods, and average power consumption are independent of θ despite the energy intake and energy dissipated during the positive and negative half cycles are dependent.Despite employing fractional-order kinetic transport, the meminductor still has identical time instants at whichφ(t) and i(t) cross zero levels; that is, the fractional-order domain generalization preserves this basic fingerprint of the meminductor. For illustration, we simulate φ(t) of the fractional-order meminductor with various α’s, as depicted in Figures 11 and 12 where i(t) has also been included and φ(t)’s have been multiplied by 100 for visibility.Figure 11
φ(t)-i(t) of the meminductor under the periodic waveform with Im > 0: α = 0.75 (red), α = 1 (green), α = 1.25 (blue), and i(t) (black).Figure 12
φ(t)-i(t) of the meminductor under the periodic waveform with Im < 0: α = 0.75 (red), α = 1 (green), α = 1.25 (blue), and i(t) (black).From Figures11 and 12 where θ = π/2 rad, Lmin = 1 mH, Lmax = 20 mH, and L(0) = 5 mH have also been assumed, it can be seen that time instants at which both φ(t) and i(t) cross zero levels are identical. Without regarding any specific device, we commonly denote these time instants by tx0. Since i(t) = Imsin(ωt + θ), tx0 can be obtained by(59)tx0=nπ−θω,where n = {1, 2, 3, …}.Now, the effects of the nonlinear dopant drift will be studied. By using (27), the resulting LMt with arbitrary p can be found as(60)∑r=0∞LMt−Lmin/Lmax−Lmin2pr+1−LM0−Lmin/Lmax−Lmin2pr+12pr+1=jkLImΓ2−αω−αexpjθΓα,−jωt−exp−jθΓα,jωt,where Г( ) denotes the incomplete gamma function [44].On the contrary,LMt with p=1 can be obtained by using (31) as follows:(61)LMt=Lmin+tanhtanh−1LM0−LminLmax−Lmin+jkLImΓ2−αω−αexpjθΓα,−jωt−exp−jθΓα,jωtLmax−Lmin.If we letα approach 0, ∫0tτα−1zτdτ approaches ∫0tτ−1zτdτ. Thus by also, respectively, applying (27) and (31), we have(62)∑r=0∞LMt−Lmin/Lmax−Lmin2pr+1−LM0−Lmin/Lmax−Lmin2pr+12pr+1=2kLImΓ2−αsinθCiωt+cosθSiωt,LMt=Lmin+tanhtanh−1LM0−LminLmax−Lmin+2kLImΓ2−αsinθCiωt+cosθSiωtLmax−Lmin,where Si( ) and Ci( ) stand for sine and cosine integral functions [45].Ifα approaches 1 on the contrary, ∫0tτα−1zτdτ approaches ∫0tzτdτ. As a result, we have(63)∑r=0∞LMt−Lmin/Lmax−Lmin2pr+1−LM0−Lmin/Lmax−Lmin2pr+12pr+1=−2kLImΓ2−αcosωτ+θω,LMt=Lmin+tanhtanh−1LM0−LminLmax−Lmin−2kLImΓ2−αcosωτ+θωLmax−Lmin.With these results, we can simulateLM(t) with the effect of the nonlinear dopant drift as dashed curves in Figures 13 and 14 where those based on the linear dopant drift have also been included as normal curves for comparison. We also simulate the φ(t)-i(t) characteristic, as shown in Figures 15 and 16 where the dashed curves have been adopted for being distinguished from the linear dopant drift-based φ(t)-i(t) depicted in Figures 9 and 10. Here, we choose p=1 where θ = π/2 rad, Lmin = 1 mH, Lmax = 20 mH, and L(0) = 5 mH have also been assumed. From these figures where the strong agreements between our results with the nonlinear dopant drift and those based on the SPICE model with Jocklecar’s window function and p=1 can be observed, and Figures 9 and 10, it can be seen that the nonlinear dopant drift reduces the effects of α on the variation in LM(t) and variation in nonlinearity of the device; that is, the device is more robust to the effects of fractional kinetic transport. We have also found that the meminductor becomes more nonlinear with increasing α and with a more symmetric pinched hysteresis loop if the nonlinear dopant drift has been assumed for ceteris paribus.Figure 13
L
M(t) of the meminductor with linear (normal curve) and nonlinear (dashed curve) dopant drifts under the periodic waveform with Im > 0: α = 0.75 (red), α = 1 (green), α = 1.25 (blue), and SPICE model (black).Figure 14
L
M(t) of the meminductor with linear (normal curve) and nonlinear (dashed curve) dopant drifts under the periodic waveform with Im < 0: α = 0.75 (red), α = 1 (green), α = 1.25 (blue), and SPICE model (black).Figure 15
φ(t)-i(t) of the meminductor with the nonlinear dopant drift under the periodic waveform with Im > 0: α = 0.75 (red), α = 1 (green), α = 1.25 (blue), and SPICE model (black dots).Figure 16
φ(t)-i(t) of the meminductor with the nonlinear dopant drift under the periodic waveform with Im < 0: α = 0.75 (red), α = 1 (green), α = 1.25 (blue), and SPICE model (black dots).At this point, the memcapacitor with fractional-order kinetics will be considered where cosinusoidal with arbitraryθ, i.e., i(t) = Imcos(ωt + θ), will be assumed. Assuming the linear dopant drift, DM(t) can be given by using (17) with β = α + 1, k = kC, y(t) = DM(t), ymin = Dmin, and ymax = Dmax as(64)DMt=DM0−kCDmin−DmaxImtα+1Γα+2sinθF121;α+12+12,α+12+1;−14π2−ωt−2θ2+π/2−ωt−2θcosθα+2F121;α+12+1,α+12+32;−14π2−ωt−2θ2.Similarly toLMt, DM(t) is reduced to a simplified version at the steady state given by(65)DMt=DM0−kCDmin−DmaxImωα+1cosθ−α2π−cos1−α2π−ωt+θ.By using (65) with θ = 0 rad, Dmin = 0.1 MF−1, Dmax = 0.1 GF−1, and D(0) = 0.01 GF−1, we can simulate the φ(t)-σ(t) and vt-q(t) characteristics with various α’s, as shown in Figures 17–20 where Im = 0.1 μA has been assumed in Figures 17 and 19. On the contrary, Im = −0.1 μA has been adopted in Figures 18 and 20. Moreover, the vertical axes of these figures have been scaled down by 107 for visibility. Similarly to the meminductor, it has been found that the constitutive relation of the memcapacitor with fractional-order kinetics becomes ambiguous unless α ≠ 1 as the one-to-one correspondence between φ(t) and σ(t) ceased to be existed, as shown in Figures 17 and 18. Thus, it can be stated that the fractional-order domain generalization cannot preserve the unambiguous constitutive relation which is one of the basic fingerprints of the memcapacitor [40]. However, the 2nd basic fingerprint, i.e., the pinched hysteresis loop in the vt-q(t) characteristic, can be preserved despite the asymmetricity when α ≠ 1, as shown in Figures 19 and 20 where the strong agreements between the vt-q(t) characteristic with α = 1 and the SPICE memcapacitor model-based counterpart with rectangular window function can be observed. Similarly to the meminductor, it can be seen that the lobe area of the pinched hysteresis loop of the memcapacitor with fractional-order kinetics is also inversely proportional to α. Therefore, more memory effect and less linearity can also be obtained by using the memcapacitor with lower α, and vice versa.Figure 17
φ(t)-σ(t) of the memcapacitor under the periodic waveform with Im > 0: α = 0.75 (red), α = 1 (green), and α = 1.25 (blue).Figure 18
φ(t)-σ(t) of the memcapacitor under the periodic waveform with Im < 0: α = 0.75 (red), α = 1 (green), and α = 1.25 (blue).Figure 19
v
t-q(t) of the memcapacitor under the periodic waveform with Im > 0: α = 0.75 (red), α = 1 (green), α = 1.25 (blue), and SPICE model (black dots).Figure 20
v
t-q(t) of the memcapacitor under the periodic waveform with Im < 0: α = 0.75 (red), α = 1 (green), α = 1.25 (blue), and SPICE model (black dots).Unlike the meminductor, we have found that the fractional-order kinetic memcapacitor is balanced. For illustration, the areas of the upper lobe and lower lobe of thevt-q(t) pinched hysteresis loop, i.e., AUC and ALC, must be calculated by using the following equations:(66)AUC=∫−θ/ωπ−θ/ωvtdqt,(67)ALC=∫π−θ/ω2π−θ/ωvtdqt.Sincevt = DM(t)q(t) and qt=∫0tiτdτ, where i(t) = Imcos(ωt + θ), AUC and ALC can be obtained by using (65)–(67) as follows:(68)AUC=−23kCIm3ωα+3sinαπ2Dmax−Dmin,ALC=23kCIm3ωα+3sinαπ2Dmax−Dmin,which show that AUC + ALC = 0. Since AUC and ALC, respectively, refer to the energy intake and energy dissipated during the positive and negative half cycles, AUC + ALC and its multiple can be interpreted in a similar manner to AUL + ALL and its multiple. So, AUC + ALC = 0 means that zero average power consumption is obtained, and therefore, the memcapacitor with fractional-order kinetics is balanced. Moreover, both AUC and ALC are independent of θ, and so do the energy intake and energy dissipated during the positive and negative half cycles unlike those of the fractional-order kinetic meminductor.Finally, it can be seen that the fractional-order domain generalization preserves the 3rd basic fingerprint of the memcapacitor, i.e., identical time instants at which vt and q(t) cross zero levels. As illustrations, we simulate vt of the fractional-order kinetic memcapacitor with various α’s, as depicted in Figures 21 and 22 where q(t) has also been included, vt’s have been divided by 107 for visibility, and θ = 0 rad, Dmin = 0.1 MF−1, Dmax = 0.1 GF−1, and D(0) = 0.01 GF−1 have been assumed. Obviously, these figures show that vt and q(t) employ identical tx0. Since i(t) = Imcos(ωt + θ), qt=Im/ωsinωt+θ as qt=∫0tiτdτ. Therefore, tx0 can be obtained as given by (59).Figure 21
v
t of the memcapacitor under the periodic waveform with Im > 0: α = 0.75 (red), α = 1 (green), α = 1.25 (blue), and q(t) (black).Figure 22
v
t of the memcapacitor under the periodic waveform with Im < 0: α = 0.75 (red), α = 1 (green), α = 1.25 (blue), and q(t) (black).Finally, the effects of the nonlinear dopant drift will be analyzed. By using (27) and (31), the resulting DMt with arbitrary p and p=1 can be, respectively, found as(69)∑r=0∞DMt−Dmin/Dmax−Dmin2pr+1−DM0−Dmin/Dmax−Dmin2pr+12pr+1=jkCΓ2−αImω1+αexpjθΓα,−jωt−exp−jθΓα,jωt,DMt=Dmin+tanhtanh−1DM0−DminDmax−Dmin+jkCΓ2−αImω1+αexpjθΓα,−jωt−exp−jθΓα,jωtDmax−Dmin.Ifα approaches 0, we have(70)∑r=0∞DMt−Dmin/Dmax−Dmin2pr+1−DM0−Dmin/Dmax−Dmin2pr+12pr+1=2kCΓ2−αImωciωtsinθ+siωtcosθ,DMt=Dmin+tanhtanh−1DM0−DminDmax−Dmin+2kCΓ2−αImωciωtsinθ+siωtcosθDmax−Dmin,as ∫0tτα−1zτdτ approaches ∫0tτ−1zτdτ.But ifα approaches 1 instead, we have(71)∑r=0∞DMt−Dmin/Dmax−Dmin2pr+1−DM0−Dmin/Dmax−Dmin2pr+12pr+1=−2kCΓ2−αImcosωt+θω2,DMt=Dmin+tanhtanh−1DM0−DminDmax−Dmin−2kCΓ2−αImcosωt+θω2Dmax−Dmin.This is because∫0tτα−1zτdτ approaches ∫0tzτdτ.By keeping in mind thatCM(t) = 1/DM(t) and applying the above equations, we can simulate CM(t) affected by the nonlinear dopant drift as dashed curved in Figures 23 and 24 where those based on the linear dopant drift have been included as normal curves for comparison. We also simulate the vt-q(t) characteristic, as shown in Figures 25 and 26 where the dashed curves have been used for being distinguished from the linear dopant drift-based vt-q(t) depicted in Figures 19 and 20. Here, we choose p=1 where θ = 0 rad, Dmin = 0.1 MF−1, Dmax = 0.1 GF−1, and D(0) = 0.01 GF−1 have also been assumed for ceteris paribus.Figure 23
C
M(t) of the memcapacitor with linear (normal curve) and nonlinear (dashed curve) dopant drifts under the periodic waveform with Im > 0: α = 0.75 (red), α = 1 (green), α = 1.25 (blue), and SPICE model (black dots).Figure 24
C
M(t) of the memcapacitor with linear (normal curve) and nonlinear (dashed curve) dopant drifts under the periodic waveform with Im < 0: α = 0.75 (red), α = 1 (green), α = 1.25 (blue), and SPICE model (black dots).Figure 25
v
t-q(t) of the memcapacitor with the nonlinear dopant drift under the periodic waveform with Im > 0: α = 0.75 (red), α = 1 (green), α = 1.25 (blue), and SPICE model (black dots).Figure 26
v
t-q(t) of the memcapacitor with the nonlinear dopant drift under the periodic waveform with Im > 0: α = 0.75 (red), α = 1 (green), α = 1.25 (blue), and SPICE model (black dots).From Figures23–26, where strong agreements between our results with the nonlinear dopant drift and those based on the SPICE model with Jocklecar’s window function and p=1 can be observed, and Figures 19 and 20, it can be seen that the nonlinear dopant drift reduces the effects of α on the variation in CM(t) and variation in nonlinearity of the device. Therefore, the memcapacitor with the nonlinear dopant drift is more robust to the effects of fractional kinetic transport. Similarly to the meminductor, the memcapacitor with the nonlinear dopant drift also becomes more nonlinear with increasing α and employs a more symmetric pinched hysteresis loop than that with the linear dopant drift.


## 5. The Emulation by Using Memristor

Despite employing the fractional-order kinetics, the constitutive relations of the meminductor and memcapacitor remain as given by (1) and (4). As a result, the emulation of memreactance by using the memristor and conventional memristor to meminductor/memcapacitor emulators [33, 46] remains possible. In this section, such possibility will be illustrated.Since the constitutive relation of the memristor can be given by [5, 7](72)Mt=dφtdqt,(1) can be obtained via the following transformation:(73)ρLt=aφMt,qLt=bqMt,where a and b are real constants. Moreover, the subscripts L and M refer to the meminductor and memristor, respectively.By taking differentiation with respect tot for both sides of the above transformation equations, we have(74)φLt=avMt,iLt=biMt.AsφLt=∫0tvLτdτ, the following transformation equations can be obtained:(75)vLt=addtvMt,iLt=biMt.Since we assume thatiLt and iMt flow in the opposite directions, iLt=biMt becomes iLt=−biMt. After taking the Laplace transformation for both sides of the equations, the following linear transformation that transforms the memristor to the meminductor can be obtained:(76)VLsILs=as00bVMs−IMs.By using (76), LMt can be given in terms of Mt as follows:(77)LMt=abMt.For obtainingLMt exactly as given by (2) and the fractional-order kinetics, the following Mt must be used:(78)Mt=baLmin+xMtLmax−Lmin,where xM(t) can be mathematically defined by (19) with x(t) = xM(t), z(t) = i(t), and k = kM. For the meminductor emulation, kM = kL must be satisfied.At this point, it can be seen that the meminductor can be emulated by using the memristor and the conventional memristor to the meminductor emulator [33] despite taking the fractional-order kinetics into account. In addition, the kinetics of the memristor must be of fractional order in order to obtain the meminductor employing such fractional-order kinetics. Note also that Jocklecar’s window function has been adopted for modelling the nonlinear dopant drift of the memristor for ceteris paribus.Now, the emulation of the memcapacitor with fractional-order kinetics by using the memristor will be discussed. It can be seen that (4) can be obtained by applying the following transformation to (72):(79)φCt=bφMt,σCt=aqMt,where the subscript C refers to the memcapacitor.After the differentiation with respect tot and rearranging by keeping the relationship between charge and current in mind, we have(80)vCt=bvMt,iCt=−addtiMt.Note that the opposite directions ofiCt and iMt have been assumed.By applying the Laplace transformation, the following linear transformation that transforms the memristor to the memcapacitor can be obtained:(81)VCsICs=b00asVMs−IMs,which implies that DMt can be given in terms of Mt as follows:(82)DMt=baMt−1.In order to obtainDMt exactly as given by (5), Mt must be given by(83)Mt=abDmin+xMtDmax−Dmin−1.Note thatxMt can be defined similarly to that of (78) as Jocklecar’s window function has been assumed for ceteris paribus where kM = kC must be satisfied. At this point, it can be seen that the memcapacitor can be emulated by using the memristor and the conventional memristor to the meminductor emulator [47] despite taking the fractional-order kinetics into account. The kinetics of the memristor must be of fractional order for obtaining the memcapacitor with fractional-order kinetics.


## 6. The Influence of Fractional-Order Kinetics on Memreactance-Based Circuits

For studying the influence of the fractional-order kinetics on the meminductor-based circuit, the meminductor-based oscillator [36] depicted here in Figure 27 has been chosen. By taking the fractional-order kinetics of the meminductor into account, the dynamical equation of this circuit can be given as follows:(84)du1tdt=Aφt+Bφtραt−Cu1t,du2tdt=Du2t−Eφt−Fφtραt,dφtdt=u2t−u1t,Dαραt=φt,where A=Lmax/C1, B=kLmin−Lmax/C1, C=1/C1R, D=G/C2, E=Lmax/C2, and F=kLmin−Lmax/C2. Note also that the linear dopant drift model has been assumed in the formulation of (84) in which the voltage terms have been denoted by u1(t) and u2(t), for ceteris paribus as these model and voltage symbols have been assumed in [36].Figure 27
The candidate meminductor-based oscillator [28].According to [36], this meminductor-based oscillator with integer kinetics generates the periodic output if we let A=1, B=6, C=4.6, D=0.4, E=4, and F=1 and becomes chaotic when D=0.7 and other parameters remain the same. By using (77) with {A, B, C, D, E, F, u1(0), u2(0), φ(0), ρα(0)} = {1, 6, 4.6, 0.4, 4, 1, 2.5, 0.01, 0.01, 0.01} and {A, B, C, D, E, F, u1(0), u2(0), φ(0), ρα(0)} = {1, 6, 4.6, 0.7, 4, 1, 0.01, 0.01, 0.01, 0.01} [28], we can simulate u2(t) with various α’s, as depicted in Figures 28–30 and Figures 31–33. These figures show that u2(t) remains periodic and chaotic when {A, B, C, D, E, F, u1(0), u2(0), φ(0), ρα(0)} = {1, 6, 4.6, 0.4, 4, 1, 2.5, 0.01, 0.01, 0.01} and {A, B, C, D, E, F, u1(0), u2(0), φ(0), ρα(0)} = {1, 6, 4.6, 0.7, 4, 1, 0.01, 0.01, 0.01, 0.01}, respectively, despite the incorporation of the meminductor’s fractional-order kinetics. Note that the strong agreements between u2(t) with α = 1 and that based on the conventional integer kinetics which can be obtained by allowing all derivatives to be of the conventional type [36] can be observed. It can also be seen that u2(t) becomes more oscillatory with the increasing α. Therefore, it can be stated that incorporating the fractional-order kinetics can offer more degrees of freedom as the amount of oscillation becomes now controllable via α.Figure 28
u
2(t) when {A, B, C, D, E, F, u1(0), u2(0), φ(0), ρα(0)} = {1, 6, 4.6, 0.4, 4, 1, 2.5, 0.01, 0.01, 0.01}: α = 0.95.Figure 29
u
2(t) when {A, B, C, D, E, F, u1(0), u2(0), φ(0), ρα(0)} = {1, 6, 4.6, 0.4, 4, 1, 2.5, 0.01, 0.01, 0.01}: α = 1 and integer kinetics (dotted line).Figure 30
u
2(t) when {A, B, C, D, E, F, u1(0), u2(0), φ(0), ρα(0)} = {1, 6, 4.6, 0.4, 4, 1, 2.5, 0.01, 0.01, 0.01}: α = 1.05.Figure 31
u
2(t) when {A, B, C, D, E, F, u1(0), u2(0), φ(0), ρα(0)} = {1, 6, 4.6, 0.7, 4, 1, 0.01, 0.01, 0.01, 0.01}: α = 0.95.Figure 32
u
2(t) when {A, B, C, D, E, F, u1(0), u2(0), φ(0), ρα(0)} = {1, 6, 4.6, 0.7, 4, 1, 0.01, 0.01, 0.01, 0.01}: α = 1 and integer kinetics (dotted line).Figure 33
u
2(t) when {A, B, C, D, E, F, u1(0), u2(0), φ(0), ρα(0)} = {1, 6, 4.6, 0.7, 4, 1, 0.01, 0.01, 0.01, 0.01}: α = 1.05.In addition, it can be seen from these simulation results thatD is the most prominent control parameter of this circuit because it solely determines whether the circuit is periodic or chaotic. So, it is worthy to further explore the effect of D on the circuit’s dynamics. In order to do so, we simulate the 3D phase portraits of u1(t), u2(t), and φ(t) by using (77). It should be mentioned here that ραt is not of our interest because it is merely a fractional integral of φ(t) as can be seen from (77). Here, we assume {A, B, C, α, E, F, u1(0), u2(0), φ(0), ρα(0)} = {1, 6, 4.6, 0.95, 4, 1, 2.5, 0.01, 0.01, 0.01} and let D be the bifurcation parameter. As a result, the following phase portraits can be obtained.From Figure34 in which D = 0.7 has been assumed, a chaotic attractor can be observed, thus confirming the above simulation result depicted in Figure 31. In addition, we have also found by using the numerical computation with MATHEMATICA that the corresponding Lyapunov exponents are given by LE1 = 0.152939, LE2 = 0.0294933, LE3 = −0.00451014, and LE4 = −4.00792 where the Lyapunov dimension can be found as DL = 3.0463. From the Lyapunov exponents, it can be seen that a contracting volume with expansions in two directions in the phase space of the attractor indicates that the chaotic dissipative behavior can be observed because LE1 + LE2 + LE3 + LE4 < 0 where LE1 > 0 and LE2 > 0. In addition, the obtained fractional number DL states that the manifold in the phase space is a strange attractor which also indicates the chaotic behavior. From Figures 35–38, it can be seen that the circuit becomes marginally and asymptotically stable by reducing D where the asymptotic stability can be obtained if D ≤ 0.1 has been satisfied due to the observed stable focus.Figure 34
3D phase portraits ofu1(t), u2(t), and φ(t) when {A, B, C, α, E, F, u1(0), u2(0), φ(0), ρα(0)} = {1, 6, 4.6, 0.95, 4, 1, 0.01, 0.01, 0.01, 0.01}: D = 0.7.Figure 35
3D phase portraits ofu1(t), u2(t), and φ(t) when {A, B, C, α, E, F, u1(0), u2(0), φ(0), ρα(0)} = {1, 6, 4.6, 0.95, 4, 1, 0.01, 0.01, 0.01, 0.01}: D = 0.5.Figure 36
3D phase portraits ofu1(t), u2(t), and φ(t) when {A, B, C, α, E, F, u1(0), u2(0), φ(0), ρα(0)} = {1, 6, 4.6, 0.95, 4, 1, 0.01, 0.01, 0.01, 0.01}: D = 0.3.Figure 37
3D phase portraits ofu1(t), u2(t), and φ(t) when {A, B, C, α, E, F, u1(0), u2(0), φ(0), ρα(0)} = {1, 6, 4.6, 0.95, 4, 1, 0.01, 0.01, 0.01, 0.01}: D = 0.1.Figure 38
3D phase portraits ofu1(t), u2(t), and φ(t) when {A, B, C, α, E, F, u1(0), u2(0), φ(0), ρα(0)} = {1, 6, 4.6, 0.95, 4, 1, 0.01, 0.01, 0.01, 0.01}: D = 0.For a similar study on the memcapacitor-based circuit, the memcapacitor-based synaptic network [37] which is depicted here in Figure 39 will be considered. If a current pulse has been applied as an input to this circuit, the resulting synaptic weight, ψt, will be given by [38](85)ψt=Dat−DbtDat+Dbt,where Da(t) = Dm2(t) = Dm3(t) and Db(t) = Dm1(t) = Dm4(t).Figure 39
Memcapacitor-based synaptic network [37].Since we take the fractional-order kinetics of the memcapacitor into account,Da(t) and Db(t) can be derived by using (45) which assumes p=1, for ceteris paribus as [37, 38] assumed such a value of p in their simulations, and keeping the antiseries connection of the devices in mind as(86)Dat=Dmin+tanhtanh−1Da0−DminDmax−Dmin+kCΓ2−αI0tα+1α+1Dmax−Dmin,(87)Dbt=Dmin+tanhtanh−1Db0−DminDmax−Dmin−kCΓ2−αI0tα+1α+1Dmax−Dmin,where I0 stands for the magnitude of the applied current pulse.Therefore,ψt with the fractional-order kinetics can be obtained as follows:(88)ψt=tanhtanh−1Da0−Dmin/Dmax−Dmin+kCΓ2−αI0tα+1/α+1−tanhtanh−1Dmax−Da0/Dmax−Dmin−kCΓ2−αI0tα+1/α+1Dmax−Dmin2Dmin+tanhtanh−1Da0−Dmin/Dmax−Dmin+kCΓ2−αI0tα+1/α+1+tanhtanh−1Dmax−Da0/Dmax−Dmin−kCΓ2−αI0tα+1/α+1Dmax−Dmin.By allowingDm2(0) = Dm3(0) = 0.009901 MF−1 and other parameters to be similar to those of [30], ψt can be simulated under the assumption that I0 = 1 μA and the current pulse duration is 2 sec [37, 38], as depicted in Figure 40. From this figure, a strong agreement between ψt with α = 1 and that with integer kinetics can be observed. It can also be seen that the circuit quality can be improved with increasing α as faster weight adjustment and more proportion of linear regions [37] can be obtained. This marks the circuit quality improvement of employing the fractional-order kinetics.Figure 40
ψ(t) (Dm2(0) = Dm3(0) = 0.009901 MF−1): α = 0.5 (red), α = 1 (green), α = 1.5 (blue), and integer kinetics (black dotted line).Now, we letDm2(0) = Dm3(0) = 0.005 MF−1. As a result, ψt can be simulated, as depicted in Figure 41 where a strong agreement between ψt with α = 1 and that with integer kinetics can also be observed. From this figure, it can be seen that the circuit quality can also be improved by decreasing Dm2(0) and Dm3(0) as the faster weight adjustment can be obtained.Figure 41
ψ(t) (Dm2(0) = Dm3(0) = 0.005 MF−1): α = 0.5 (red), α = 1 (green), α = 1.5 (blue), and integer kinetics (black dotted line).


## 7. Extension to the Fractional-Order Memreactance

Fractional-order memreactance or memfractance in short is a common term for both the fractional-order meminductor and the fractional-order memcapacitor. According to [30], the characteristic of the fractional-order meminductor is in between those of the meminductor and memristor, whereas that of the fractional-order memcapacitor is in between those of the memristor and memcapacitor as the results of applying the concept of fractional calculus to the port equations of both the meminductor and the memcapacitor. For illustrating this issue, consider the following voltage-current relationship of the meminductor:(89)vt=ddtLMtit.By applying the fractional calculus concept, we have(90)vt=DγLMtit,which is equivalent to(91)D−γvt=LMtit,where 0 ≤ γ ≤ 1.Sincevt=d/dtφt, the following port equation of the fractional-order meminductor can be finally obtained:(92)D1−γφt=LMtit.By keeping in mind that the memcapacitor and meminductor are dual elements and thus so do their fractional-order counterparts, we obtain the following port equation of the fractional-order memcapacitor by inspecting (92):(93)D1−γqt=CMtvt.Obviously, ifγ = 0, (92) and (93) become(94)vt=LMtit,it=CMtvt,which show that both the fractional-order meminductor and the fractional-order memcapacitor, respectively, behave in a similar manner to a memristor with memristance of LM(t) and memductance of CM(t) if γ = 0.On the contrary, ifγ = 1, (92) and (93) become(95)φt=LMtit,qt=CMtvt,which are the port equations of the meminductor and memcapacitor, respectively. Therefore, it can be seen that the fractional-order meminductor and fractional-order memcapacitor behave similarly to their conventional counterparts if γ = 1. This asserts the aforesaid statement on the interpolate characteristic of these fractional-order memreactive devices. From (92) and (93), the following port equation of the fractional-order memreactance can be generally obtained:(96)D1−γuot=XMtuit,where {ui(t), uo(t), XM(t)} can be either {i(t), φ(t), LM(t)} or {vt, q(t), CM(t)} for the fractional-order meminductor and fractional-order memcapacitor, respectively.Now, we will show that our mathematical model and its related results have been found to be applicable to the fractional-order memreactance with fractional-order kinetics as they determineLM(t) and CM(t). Thus, the effects of fractional-order kinetics on the fractional-order memreactance can be analyzed. By using (96) with appropriate sets of variables, φ(t) and vt due to arbitrary i(t) can be, respectively, found as(97)φt=φ0+1Γ1−γ∫0tt−τ−γLMτiτdτ,vt=CM−1tΓ1+γddt∫0tt−τγiτdτ,where LM(t) and CM(t) can be determined by using any of our equations of LMt and DM(t) derived in Section 4 depending on the input, drift model, and assumption on α. Here, we assume that γ = 0.5 as this value is the midpoint between 0 and 1. Note that two extreme cases, i.e., γ = 0 and γ = 1, have already been considered in [7] and the previous sections of this work, respectively. This is because the fractional-order memreactance, respectively, becomes the memristor when γ = 0 and memreactance when γ = 1 as mentioned above. In addition, we let i(t) = Imsin(ωt + θ). If we allow θ = π/2 rad, Lmin = 1 mH, Lmax = 20 mH, and L(0) = 5 mH similarly to the meminductor, the resulting asymptotic φ(t) and φ(t)-i(t) characteristics of the fractional-order meminductor can be simulated, as depicted in Figures 42–45 where φ(0) = 0 Wb and nonlinear dopant drift with p=1 have been assumed. From these figures, we have found that φ(t) of the fractional-order meminductor employs a DC term which is positive if Im > 0, and vice versa. Moreover, such a DC term has been found to be directly proportional to α. Note also that the fractional-order meminductor does not share the 2nd basic fingerprint of the meminductor as its φ(t)-i(t) hysteresis loop does not pinch but takes the elliptical shape which indicates that the fractional-order meminductor with the assumed γ behaves like an AC resonator on the φ(t)-i(t) plane, as can be seen from Figures 44 and 45 where the scaling has been applied to the vertical axis for visibility.Figure 42
φ(t) of the fractional-order meminductor with fractional kinetics: α = 0.75 (red), α = 1 (green), and α = 1.25 (blue) under the periodic input with Im > 0.Figure 43
φ(t) of the fractional-order meminductor with fractional kinetics: α = 0.75 (red), α = 1 (green), and α = 1.25 (blue) under the periodic input with Im < 0.Figure 44
φ(t)-i(t) of the fractional-order meminductor with fractional kinetics: α = 0.75 (red), α = 1 (green), and α = 1.25 (blue) under the periodic input with Im > 0.Figure 45
φ(t)-i(t) of the fractional-order meminductor with fractional kinetics: α = 0.75 (red), α = 1 (green), and α = 1.25 (blue) under the periodic input with Im < 0.For the fractional-order memcapacitor on the contrary, we letθ = 0 rad, Dmin = 0.1 MF−1, Dmax = 0.1 GF−1, and D(0) = 0.01 GF−1 as well as for the memcapacitor. The simulated asymptotic vt and vt-q(t) characteristics of the fractional order can be depicted in Figures 46–49 where the nonlinear dopant drift with p=1 has been assumed. From Figures 46 and 47, we have found that the distortion of vt is directly proportional to α. From Figures 48 and 49 where the vertical axis has been scaled for visibility, it can be seen that the area of the hysteresis loop of the fractional-order memcapacitor is directly proportional to α; thus, less linearity can be obtained by using the fractional-order memcapacitor with higher α. This is in contrast to the behavior of the memcapacitor. Similarly to the fractional-order meminductor, the fractional-order memcapacitor does not share the 2nd basic fingerprint of the memcapacitor as its vt-q(t) hysteresis loop does not pinch but takes the elliptical shape which implies that the fractional-order memcapacitor with the assumed γ behaves like an AC resonator on the vt-q(t) plane.Figure 46
v
t of the fractional-order memcapacitor with fractional kinetics: α = 0.75 (red), α = 1 (green), and α = 1.25 (blue) under the periodic input with Im > 0.Figure 47
v
t of the fractional-order memcapacitor with fractional kinetics: α = 0.75 (red), α = 1 (green), and α = 1.25 (blue) under the periodic input with Im > 0.Figure 48
v
t-q(t) of the fractional-order memcapacitor with fractional kinetics: α = 0.75 (red), α = 1 (green), and α = 1.25 (blue) under the periodic input with Im > 0.Figure 49
v
t-q(t) of the fractional-order memcapacitor with fractional kinetics: α = 0.75 (red), α = 1 (green), and α = 1.25 (blue) under the periodic input with Im < 0.


## 8. Conclusion

We analyze the memreactance with fractional-order kinetic transport in this work by means of fractional calculus. Novel fractional-order kinetics involved analytical expressions of memreactance in general; meminductances, memcapacitances, and related parameters due to both DC and periodic input waveforms including those with the nonlinear dopant drift which are rather complicated have been derived. Moreover, the behavioral analysis has been thoroughly performed by using the derived equations and numerical simulations. As a result, the effects of fractional-order kinetics have been thoroughly explored where both linear and nonlinear dopant drift scenarios have been considered. In summary, it can be seen from the simulation results that the fractional-order kinetic transport significantly affects the behavior of memreactance due to both DC and periodic inputs, e.g., saturation time and nonlinearity, where the memreactance with the nonlinear dopant drift is more robust to the effects of fractional-order kinetics as stated above. For performing the numerical simulations in this work, higher computational effort compared to those of the previous works on the conventional memreactance with integer kinetics [5, 33, 34] is required due to more complexity of our equations, yet higher real memreactance modelling accuracy can be obtained in return because the real electronic device employs fractional-order kinetics in practice [22] and so does the memreactance. Moreover, the memreactance emulation by using the memristor with fractional-order kinetics, the effect of fractional-order kinetics on the memreactance-based circuits, and the application of our results to the fractional-order memreactance have also been presented. The results of this work have been found to be beneficial to those memelement-involved research areas.



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*Source: 2302604-2020-01-21.xml*