Research Article | Open Access

# Analysis of Memreactance with Fractional Kinetics

**Academic Editor:**Pietro Bia

#### 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.

#### 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 Section 3. 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]:where *L*_{M}(*t*) denotes the memductance. Since we assume the electromechanical model of the meminductor and the current controlled operation [39] in this work, *L*_{M}(*t*) can be given in terms of its minimum and maximum values denoted by *L*_{min} and *L*_{max} and the state variable (*x*_{L}(*t*)) aswhere *x*_{L}(*t*) can be given in terms of the meminductor’s current (*i*(*t*)) and mobility factor (*k*_{L}) by (3) if the linear dopant drift model has been assumed. Note also that stands for the window function which has been used for modelling the boundary effect of the device.

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]:where *D*_{M}(*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, *D*_{M}(*t*) can be in terms of the minimum and maximum of *D*_{M}(*t*), i.e., *D*_{min} and *D*_{max}, and the memcapacitor’s state variable (*x*_{C}(*t*)) aswhere *x*_{C}(*t*) can be given in terms of *q*(*t*) and the memcapacitor’s mobility factor (*k*_{C}) based on the linear dopant drift model by (6). Similarly to , denotes the window function of the memcapacitor.

#### 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:wherewhere, *J*^{a}[ ] 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* = *k*_{L}, *x*(*t*) = *x*_{L}(*t*), *y*(*t*) = , *y*_{min} = , and *y*_{max} = if the memreactance under consideration is a meminductor. For the memcapacitor on the contrary, *a* = 1, *k* = *k*_{C}, *x*(*t*) = *x*_{C}(*t*), *y*(*t*) = *D*_{M}(*t*), *y*_{min} = *D*_{min}, and *y*_{max} = *D*_{max}.

For the memreactance with fractional-order kinetics, we replace d/d*t* in (8) by the fractional-order derivative of arbitrary real order *α*, i.e., *D*^{α}[ ]. Therefore, we obtainwhere .

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

Therefore, (9) becomes

By taking *D*^{a}[ ] for both sides of this equation and keeping the above definition of *z*(*t*) in mind, we have

Since *a* can be either 0 or 1, we finally obtainwhere *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 havewhere 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 aswe obtainwhich 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:

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 and . 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].

As , (19) can be rearranged by following [3] aswhich is equivalent to

Since we define as the *α*^{th} order Riemann–Liouville fractional derivative [8] of *x*(*t*) with respect to *x*(*t*), we havewhich is equal to

If we assume , we obtain

Sinceas given in [5], the relationship between *x*(*t*) and *z*(*t*) can be obtained as

As it can be, respectively, seen from (7) that , at , the relationship between *y*(*t*) and *z*(*t*) can be finally obtained as follows:

When which means that the dopant drift is highly nonlinear [5] as the degree of nonlinearity is inversely proportional to and , (26) becomeswhich is equivalent to

Therefore, we have the following expression of *x*(*t*):

By combining (7) and (30) and keeping the above expression of *x*(0) in mind, the following expression of *y*(*t*) can be obtained:

However, it is very hard to obtain the exact expression of *y*(*t*) like (31) when because it is hard to derive the exact expression of *x*(*t*) like (30). As an example for illustration, we let . Thus by using (26), we havewhich is equivalent to

Obviously, it can be seen that *x*(*t*) is very hard to be analytically obtained. More complexity can be expected if larger values of 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 as *i*(*t*) = *I*_{DC}*u*(*t*), where *I*_{DC} 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 and *D*_{M}(*t*) can be obtained by using (17) and by keeping in mind that *β* = *α*, *k* = *k*_{L}, *y*(*t*) = , *y*_{min} = , and *y*_{max} = for the meminductor and *β* = *α* + 1, *k* = *k*_{C}, *y*(*t*) = *D*_{M}(*t*), *y*_{min} = *D*_{min}, and *y*_{max} = *D*_{max} for the memcapacitor as follows:

After obtaining and *D*_{M}(*t*), *L*_{M}(*t*) and memcapacitance (*C*_{M}(*t*)) can be, respectively, obtained as the square of and reciprocal of *D*_{M}(*t*). By using *L*_{min} = 1 mH, *L*_{max} = 20 mH, *L*(0) = 5 mH, *D*_{min} = 0.1 MF^{−1}, *D*_{max} = 0.1 GF^{−1}, and *D*(0) = 0.01 GF^{−1}, *L*_{M}(*t*)’s and *C*_{M}(*t*)’s with various *α*’s excited by the DC waveform can be numerically simulated, as depicted in Figures 1–4 where *I*_{DC} = 1 A has been assumed in Figures 1 and 3 and *I*_{DC} = −1 A has been assumed in Figures 2 and 4. Moreover, *L*_{M}(*t*)’s and *C*_{M}(*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 *L*_{M}(*t*)’s and *C*_{M}(*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 *L*_{M}(*t*) is an increasing and decreasing function of *t* when *I*_{DC} > 0 and *I*_{DC} < 0, respectively, and vice versa for *C*_{M}(*t*). However, both *L*_{M}(*t*) and *C*_{M}(*t*) are saturated at either *L*_{max} or *L*_{min} and *C*_{max} = 1/*D*_{min} or *C*_{min} = 1/*D*_{max} which are the maximum and minimum values of *C*_{M}(*t*), after certain saturation times, i.e., *t*_{sat,L} and *t*_{sat,C}, for the meminductor and memcapacitor, respectively. It has also been shown that *t*_{sat,L} and *t*_{sat,C} are directly proportional to *α*.

By using the aforesaid observation, (34) and (35), we have

Therefore, *t*_{sat,L} and *t*_{sat,C} can be immediately given bywhich 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 *t*_{sat,L} and *t*_{sat,C} reach their maximum values given, respectively, by *t*_{sat,L,M} and *t*_{sat,C,M}, if and only if *L*_{M}(0) and *D*_{M}(0) reach their possible peak values given by either *L*_{min} and *D*_{min} when *I*_{DC} > 0 or *L*_{max} and *D*_{max} when *I*_{DC} < 0. Thus, *t*_{sat,L,M} and *t*_{sat,C,M} can be found as

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 and *D*_{M}(*t*) with arbitrary can be obtained:

If we let , the resulting and *D*_{M}(*t*) can be obtained by using (31) as follows:

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*) = *x*_{on}, i.e., *y*(*t*) = *y*_{sat,on}, or *x*(*t*) = *x*_{off}, i.e., *y*(*t*) = *y*_{sat,off}. By using (44) and keeping in mind that *x*_{on} = *x*_{on,L}, *x*_{off} = *x*_{off,L}, *y*_{sat,on} = , and *y*_{sat,off} = for the meminductor, the following nonlinear dopant drift model-based saturation time of the meminductor can be obtained:

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 *x*_{on} = *x*_{on,C}, *x*_{off} = *x*_{off,C}, *y*_{sat,on} = , and *y*_{sat,off} = as

Since the maximum saturation time occurred when the memreactance is saturated at either *y*(*t*) = *y*_{on} given *y*(0) = *y*_{off} or *y*(*t*) = *y*_{off} given *y*(0) = *y*_{on}, the maximal values of *T*_{sat,L} and *T*_{sat,C} can be, respectively, given bywhich can be simplified as

At this point, we simulate *t*_{sat,L,M}, *t*_{sat,C,M}, *T*_{sat,L,M}, and *T*_{sat,C,M} with respect to *I*_{DC} by assuming that *x*_{on,C} = *x*_{on,L} = 0.01 and *x*_{off,C} = *x*_{off,L} = 0.99 where other parameters similar to those of the simulation of *L*_{M}(*t*) and *C*_{M}(*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 *I*_{DC}. 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.

Before we proceed to the subsequent section, it should be mentioned here that

##### 4.2. Periodic Waveforms

For the sinusoidal waveform with arbitrary phase (*θ*), i.e., *i*(*t*) = *I*_{m}sin(*ωt* + *θ*), where *I*_{m} and *ω*, respectively, denote its peak value and angular frequency, with the linear dopant drift can be given by using (17) with *β* = *α*, *k* = *k*_{L}, *y*(*t*) = , *y*_{min} = , and *y*_{max} = aswhere denotes a generalized hypergeometric function with and [42].

At the steady state, 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]:

By using (53) with *θ* = *π*/2 rad, *L*_{min} = 1 mH, *L*_{max} = 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 *I*_{m} = 10 mA has been assumed in Figures 7 and 9. On the contrary, *I*_{m} = −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.

However, despite the asymmetricities, the pinched hysteresis loop in the *φ*(*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 unless *α* = 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., *A*_{UL} and *A*_{LL}, must be firstly calculated. For doing so, we use the following equations:

Since *φ*(*t*) = *L*_{M}(*t*)*i*(*t*), where *i*(*t*) = *I*_{m}sin(*ωt* + *θ*), *A*_{UL} and *A*_{LL} can be obtained by using (53)–(55) as follows:

Therefore, *A*_{UL} + *A*_{LL} can be immediately found as

Since *A*_{UL} and *A*_{LL} are, respectively, referred to as the energy intake and energy dissipated during the positive and negative half cycles, *A*_{UL} + *A*_{LL} 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. *A*_{UL} + *A*_{LL} ≠ 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 *A*_{UL} + *A*_{LL} = 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 which *φ*(*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.

From Figures 11 and 12 where *θ* = *π*/2 rad, *L*_{min} = 1 mH, *L*_{max} = 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 *t*_{x0}. Since *i*(*t*) = *I*_{m}sin(*ωt* + *θ*), *t*_{x0} can be obtained bywhere *n* = {1, 2, 3, …}.

Now, the effects of the nonlinear dopant drift will be studied. By using (27), the resulting with arbitrary *p* can be found aswhere Г( ) denotes the incomplete gamma function [44].

On the contrary, with can be obtained by using (31) as follows:

If we let *α* approach 0, approaches . Thus by also, respectively, applying (27) and (31), we havewhere *Si*( ) and *Ci*( ) stand for sine and cosine integral functions [45].

If *α* approaches 1 on the contrary, approaches . As a result, we have

With these results, we can simulate *L*_{M}(*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 where *θ* = *π*/2 rad, *L*_{min} = 1 mH, *L*_{max} = 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 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 *L*_{M}(*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.

At this point, the memcapacitor with fractional-order kinetics will be considered where cosinusoidal with arbitrary *θ*, i.e., *i*(*t*) = *I*_{m}cos(*ωt* + *θ*), will be assumed. Assuming the linear dopant drift, *D*_{M}(*t*) can be given by using (17) with *β* = *α* + 1, *k* = *k*_{C}, *y*(*t*) = *D*_{M}(*t*), *y*_{min} = *D*_{min}, and *y*_{max} = *D*_{max} as

Similarly to , *D*_{M}(*t*) is reduced to a simplified version at the steady state given by

By using (65) with *θ* = 0 rad, *D*_{min} = 0.1 MF^{−1}, *D*_{max} = 0.1 GF^{−1}, and *D*(0) = 0.01 GF^{−1}, we can simulate the *φ*(*t*)-*σ*(*t*) and -*q*(*t*) characteristics with various *α*’s, as shown in Figures 17–20 where *I*_{m} = 0.1 *μ*A has been assumed in Figures 17 and 19. On the contrary, *I*_{m} = −0.1 *μ*A has been adopted in Figures 18 and 20. Moreover, the vertical axes of these figures have been scaled down by 10^{7} 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 2^{nd} basic fingerprint, i.e., the pinched hysteresis loop in the -*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 -*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.

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 the -*q*(*t*) pinched hysteresis loop, i.e., *A*_{UC} and *A*_{LC}, must be calculated by using the following equations:

Since = *D*_{M}(*t*)*q*(*t*) and , where *i*(*t*) = *I*_{m}cos(*ωt* + *θ*), *A*_{UC} and *A*_{LC} can be obtained by using (65)–(67) as follows:which show that *A*_{UC} + *A*_{LC} = 0. Since *A*_{UC} and *A*_{LC}, respectively, refer to the energy intake and energy dissipated during the positive and negative half cycles, *A*_{UC} + *A*_{LC} and its multiple can be interpreted in a similar manner to *A*_{UL} + *A*_{LL} and its multiple. So, *A*_{UC} + *A*_{LC} = 0 means that zero average power consumption is obtained, and therefore, the memcapacitor with fractional-order kinetics is balanced. Moreover, both *A*_{UC} and *A*_{LC} 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 3^{rd} basic fingerprint of the memcapacitor, i.e., identical time instants at which and *q*(*t*) cross zero levels. As illustrations, we simulate of the fractional-order kinetic memcapacitor with various *α*’s, as depicted in Figures 21 and 22 where *q*(*t*) has also been included, ’s have been divided by 10^{7} for visibility, and *θ* = 0 rad, *D*_{min} = 0.1 MF^{−1}, *D*_{max} = 0.1 GF^{−1}, and *D*(0) = 0.01 GF^{−1} have been assumed. Obviously, these figures show that and *q*(*t*) employ identical *t*_{x0}. Since *i*(*t*) = *I*_{m}cos(*ωt* + *θ*), as . Therefore, *t*_{x0} can be obtained as given by (59).

Finally, the effects of the nonlinear dopant drift will be analyzed. By using (27) and (31), the resulting with arbitrary and can be, respectively, found as

If *α* approaches 0, we haveas approaches .

But if *α* approaches 1 instead, we have

This is because approaches .

By keeping in mind that *C*_{M}(*t*) = 1/*D*_{M}(*t*) and applying the above equations, we can simulate *C*_{M}(*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 -*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 -*q*(*t*) depicted in Figures 19 and 20. Here, we choose where *θ* = 0 rad, *D*_{min} = 0.1 MF^{−1}, *D*_{max} = 0.1 GF^{−1}, and *D*(0) = 0.01 GF^{−1} have also been assumed for ceteris paribus.

From Figures 23–26, where strong agreements between our results with the nonlinear dopant drift and those based on the SPICE model with Jocklecar’s window function and 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 *C*_{M}(*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](1) can be obtained via the following transformation: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 to *t* for both sides of the above transformation equations, we have

As , the following transformation equations can be obtained:

Since we assume that and flow in the opposite directions, becomes . 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:

By using (76), can be given in terms of as follows:

For obtaining exactly as given by (2) and the fractional-order kinetics, the following must be used:where *x*_{M}(*t*) can be mathematically defined by (19) with *x*(*t*) = *x*_{M}(*t*), *z*(*t*) = *i*(*t*), and *k* = *k*_{M}. For the meminductor emulation, *k*_{M} = *k*_{L} 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):where the subscript *C* refers to the memcapacitor.

After the differentiation with respect to *t* and rearranging by keeping the relationship between charge and current in mind, we have

Note that the opposite directions of and have been assumed.

By applying the Laplace transformation, the following linear transformation that transforms the memristor to the memcapacitor can be obtained:which implies that can be given in terms of as follows:

In order to obtain exactly as given by (5), must be given by

Note that can be defined similarly to that of (78) as Jocklecar’s window function has been assumed for ceteris paribus where *k*_{M} = *k*_{C} 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:where , , , , , and . 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 *u*_{1}(*t*) and *u*_{2}(*t*), for ceteris paribus as these model and voltage symbols have been assumed in [36].

According to [36], this meminductor-based oscillator with integer kinetics generates the periodic output if we let , , , , , and and becomes chaotic when and other parameters remain the same. By using (77) with {*A*, *B*, *C*, *D*, *E*, *F*, *u*_{1}(0), *u*_{2}(0), *φ*(0), *ρ*_{α}(0)} = {1, 6, 4.6, 0.4, 4, 1, 2.5, 0.01, 0.01, 0.01} and {*A*, *B*, *C*, *D*, *E*, *F*, *u*_{1}(0), *u*_{2}(0), *φ*(0), *ρ*_{α}(0)} = {1, 6, 4.6, 0.7, 4, 1, 0.01, 0.01, 0.01, 0.01} [28], we can simulate *u*_{2}(*t*) with various α’s, as depicted in Figures 28–30 and Figures 31–33. These figures show that *u*_{2}(*t*) remains periodic and chaotic when {*A*, *B*, *C*, *D*, *E*, *F*, *u*_{1}(0), *u*_{2}(0), * φ*(0),

*ρ*

_{α}(0)} = {1, 6, 4.6, 0.4, 4, 1, 2.5, 0.01, 0.01, 0.01} and {

*A*,

*B*,

*C*,

*D*,

*E*,

*F*,

*u*

_{1}(0),

*u*

_{2}(0),

*φ*(0),

*ρ*

_{α}(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

*u*

_{2}(

*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

*u*

_{2}(

*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*.

In addition, it can be seen from these simulation results that *D* 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 *u*_{1}(*t*), *u*_{2}(*t*), and *φ*(*t*) by using (77). It should be mentioned here that 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*, *u*_{1}(0), *u*_{2}(0), *φ*(0), *ρ*_{α}(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 Figure 34 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 LE_{1} = 0.152939, LE_{2} = 0.0294933, LE_{3} = −0.00451014, and LE_{4} = −4.00792 where the Lyapunov dimension can be found as *D*_{L} = 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 LE_{1} *+* LE_{2} *+* LE_{3} *+* LE_{4} < 0 where LE_{1} > 0 and LE_{2} > 0. In addition, the obtained fractional number *D*_{L} 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.

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, , will be given by [38]where *D*_{a}(*t*) = *D*_{m2}(*t*) = *D*_{m3}(*t*) and *D*_{b}(*t*) = *D*_{m1}(*t*) = *D*_{m4}(*t*).

Since we take the fractional-order kinetics of the memcapacitor into account, *D*_{a}(*t*) and *D*_{b}(*t*) can be derived by using (45) which assumes , for ceteris paribus as [37, 38] assumed such a value of in their simulations, and keeping the antiseries connection of the devices in mind as