# On the Dimensional Consistency Aware Fractional Domain Generalization of Simplest Chaotic Circuits

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

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## Abstract

In this research, we generalize the simplest Chua’s chaotic circuit which is even more simpler than the four-element Chua’s circuit in terms of number of elements and the novel simplest chaotic circuit in the fractional domain by using the fractional circuit elements. Unlike the previous works, the time dimensional consistency aware generalization has been performed for the first time in this work. The dynamics of the generalized fractional nonlinear circuits have been analyzed by means of the fractional calculus based on the modified Riemann–Liouville fractional derivative where the Lyapunov exponents and dimensions have also been numerically calculated. We have found that including the dimensional consistency significantly alters the dynamic of the obtained fractional domain Chua’s circuit from that of the previous dimensional consistency ignored counterpart as different Lyapunov exponents and dimensions can be obtained. The conditions for both fractional domain circuits which cease to be chaotic have also been determined where such condition of Chua's circuit presented in this study is different from that of the previous work. This is because the time dimensionalconsistency has been included. The dynamical analyses of these circuits have also been performed where their conditions for being nonchaotic have been verified. Moreover, their emulators have also been realized.


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## Body

## 1. Introduction

The fractional calculus and its related differential equation, i.e., FDE, which are the extensions of the conventional integer calculus and the ordinary differential equation (ODE), have been extensively utilized in various research areas, e.g., signal processing, biomedical engineering, electronics, robotics, and control theory [1–11]. The FDE has been used in the fractional domain analysis of electrical circuits in which their orders are fractional instead of being strictly integer, as proposed in the previous research studies [12–16]. Such fractional domain analysis has been found to be necessary because the electrical circuit components in practice have irreversible dissipative effects, e.g., Ohmic friction, thermal memory, and electromagnetic field-induced nonlinearities, which cannot be precisely analyzed by using the conventional integer domain methods. Mostly, the linear electrical circuits have been considered despite the fact that there exists nonlinear ones which employ various interesting properties, e.g., chaotic behaviors. Among various nonlinear circuits, Chua’s chaotic circuits [17–20], which employ interesting applications, e.g., secure communication system [21] and brain dynamic simulator [22], have been found to be often cited as they employ chaotic behaviors even though realizable by using very simple devices [19]. Therefore, their fractional domain generalizations have been proposed in many literatures [23–25]. Unfortunately, the dimensional consistency [26], which is often cited in many recent studies on the fractional domain linear circuit generalizations [27–29], has been neglected.By this motivation, we perform the fractional domain generalization of Chua’s chaotic circuit in this work by also considering such formerly ignored dimensional consistency. Similar to [23], the original simplest Chua’s chaotic circuit [20], which is composed of only three electrical components, i.e., capacitor, inductor, and memristor, has been considered. In addition, the novel simplest chaotic circuit proposed by Jin et.al. [30], which can be thought of as the parallel structured counterpart of Chua’s simplest chaotic circuit, has also been considered. The analyses of the generalized circuits have been performed based on Jumarie’s modified Riemann–Liouville fractional derivative [31] and nonlinear transformation [32] where the Lyapunov exponents and dimensions have also been calculated. Such derivative has been chosen despite the fact that there exist recent fractional derivatives, e.g., Caputo and Fabrizio [33] and Atangana and Baleanu fractional derivatives [34], because these new derivatives have been found to be controversial. As an example, it has been stated in [35] that these nonsingular kernel fractional derivatives are actually not derivatives. Moreover, they are also less accurate than the conventional fractional derivative in practice [36].We have found that the dimensional consistency awareness significantly alters the dynamic; thus, the chaotic behavior of the fractional domain Chua's circuit from that of the previous dimensional consistency ignored the counterpart [23]. This is because different Lyapunov exponents and dimension can be obtained. The conditions for both fractional domain circuits which cease to be chaotic have also been determined where such condition of Chua's study is different from that of the previous work. This is because the time dimensionalconsistency has been included. The dynamical analyses of these circuits including Hopf bifurcation analyses have also been performed where their conditions for being nonchaotic have been verified, and it has been found that both circuits have undergone Hopf bifurcation through their equilibrium points. In addition, their emulators have also been realized.In the following section, overview of the modified Riemann–Liouville fractional derivative will be briefly given followed by the introduction of the simplest Chua’s chaotic circuit and the novel simplest one in Sections3 and 4. The proposed fractional domain generalizations of the circuit will be shown in Section 5 where the corresponding dynamical analysis will also be presented. Finally, the conclusion will be drawn in Section 6.


## 2. Overview of the Modified Riemann–Liouville Fractional Derivative

In 2006, Jumarie proposed a modified version of Riemann–Liouville fractional derivative [31]. The proposed derivative can be mathematically defined as follows [31, 32].Definition 1.
Letf(t) be arbitrary function of t where t∈ℜ and 0 ≤ α ≤ 1 where α∈ℜ, Dtαft can be given by(1)Dtαft=1Γ−α∫0tt−η−α−1fη−f0dη,α<0;1Γ1−αddt∫0tt−η−αfη−f0dη,0<α<1;fntα−n,n≤α≤n+1,n≥1,where Dtα stands for the fractional derivative operator of order α with respected to t. Obviously, the modified Riemann–Liouville fractional derivative obeys fundamental rules of conventional calculus, e.g., chain rule and product rule [32]. The Laplace transformation of this modified fractional derivative depends on α. As an example, LDtαft when 1 < α ≤ 2 can be given by LDtαft=sαLft−sα−1f0−sα−2f′0[31].


## 3. The Simplest Chua’s Chaotic Circuit

In 2010, Muthuswamy and Chua proposed a chaotic circuit which was composed of merely three circuit elements, i.e., a capacitor, an inductor, and a memristor [20], connected in a series fashion. The proposed circuit has been found to be the simplest one among Chua’s family of chaotic circuits. It is even simpler than its predecessor as proposed by Barboza and Chua in 2008 [19] which composes of four elements, i.e., two capacitors, an inductor, and a nonlinear resistor. Therefore, it has been found to be of interest. The schematic diagram of simplest Chua’s chaotic circuit is depicted in Figure 1.Figure 1
The simplest Chua’s chaotic circuit [20].Moreover, it has been assumed that the memristor employs the following state equation and memristance:(2)ddtxt=−Axt−iMtxt+iMt,Mxt=Bxt2−1,where x(t), M(x(t)), A, and B denote the state variable, the memristance, and the memristor’s parameters, respectively. Moreover, iM(t) stands for the memristor’s current.As a result, the following model of the simplest Chua’s chaotic circuit can be obtained after performing a rigorous circuit analysis:(3)ddtxt=−Axt+iLtxt−iLt,ddtiLt=−1LBxt2−1iLt+vCt,ddtvCt=iLtC,where C, L, iL(t), and vCt denote the capacitance value, inductance value, current flowing through the inductor, and voltage drop across the capacitor, respectively. Note also that iL(t) = iC(t) = −iM(t). Before proceeding further, it should be mentioned here that the simplest Chua’s chaotic circuit model exhibits the chaotic behaviors if and only if certain conditions on parameter values, e.g., A = 0.2A, B = 1.7 Ω, L = 3.3H, and C = 1F, have been met. Otherwise, other behaviors such as periodic and quasiperiodic will be encountered instead.


## 4. The Novel Simplest Chaotic Circuit

Later, Jin et.al. proposed a novel simplest chaotic circuit in 2018 [30]. Unlike that of Chua, this circuit is composed of a capacitor, an inductor, and a memristor connected in a parallel fashion as depicted in Figure 2; thus, it can be thought of as the parallel structured counterpart of the simplest Chua’s circuit.Figure 2
The novel simplest chaotic circuit [30].According to [30], the memristor employs the following state equation:(4)ddtxt=a1xt+a3x3t+b1vmt+c11vmtxt,(5)Wxt=kx2t−xt−1,where a1, a3, b1, c11, and vmt are the memristor parameters and voltage. In addition, W(x(t)) stands for the memductance of the memristor [30].Based on (3) and (4), the following model of the novel simplest chaotic circuit can be obtained:(6)ddtvCt=−1Ckx2t−xt−1vCt+iLt,ddtiLt=1LvCt,ddtxt=a1xt+a3x3t+b1vCt+c11vCtxt.Note thatvLt = vCt = E. Similar to Chua’s circuit, this novel circuit also exhibits chaotic behaviors if and only if certain conditions on parameter values have been met. Moreover, certain condition on the initial values must have been satisfied as well for obtaining the chaotic behavior [30].


## 5. The Fractional Domain Generalization

### 5.1. The Simplest Chua’s Circuit

For generalizing the simplest Chua’s chaotic circuit in the fractional domain, its conventional electrical circuit components, i.e., capacitor, inductor, and memristor, must be replaced by the fractional ones, i.e., fractional capacitor, fractional inductor, and fractional memristor, which can be emulated by simply replacing the conventional capacitor of the memristor emulator [20] by the fractional one. Mathematically, this is to replace all conventional derivatives in (3) by the fractional domain counterparts. In [23], such replacement has been directly performed without any awareness of the dimensional consistency as the time dimension of Dtα is sec−α while that of d/dt is sec−1. Moreover, sec−α is not physically measurable unlike sec−1. As a result, the following model of the fractional domain generalized simplest Chua’s chaotic circuit can be obtained [23]:(7)Dtαxt=−Axt+iLtxt−iLt,DtαiLt=−1LBxt2−1iLt+vCt,DtαvCt=iLtC,where 0 ≤ α ≤ 1 and iL(t) = iC(t) = −iM(t). Obviously, (7) is simply (3) with fractional derivatives.However, this is not the case for this research as the dimensional consistency has been concerned unlike those previous works. For achieving the dimensional consistency, the time dimensions ofd/dt and the generalized fractional derivative must be consistent which means that both of them must be given by the physically measurable sec−1. Therefore, the following operation must be used:(8)ddt⟶1σ1−αDtα,where σ denotes the fractional time component or the cosmic time [26]. Note that σ > 0 always for preventing singularity. In addition, the introduction of σ scales only the Laplace transformed derivative term where the frequency scaling scales all complex frequency variables.Since the dimension ofσ is sec, that of 1/σ1−αDtα is also sec as the dimension of Dtα is sec−α as stated above. Therefore, the dimensions of d/dt and generalized fractional derivative, i.e., 1/σ1−αDtα, are now consistent. As a result, the dimensional consistency of the generalized fractional domain model can be now achieved. Note also that (8) which has been adopted in those previous works on the fractional domain generalization of linear circuit models has been found to be suitable for our work since the assumed modified Riemann–Liouville fractional derivative employs a power law kernel as can be seen from (1). By using (8), we have(9)1σ1−αDtαxt=−Axt+iLtxt−iLt,1σ1−αDtαiLt=−1LBxt2−1iLt+vCt,1σ1−αDtαvCt=iLtC,where 0 < α ≤ 1 and iL(t) = iC(t) = −iM(t).After some rearrangement, we obtain(10)Dtαxt=−Aαxt+utxt−ut,Dtαut=−1LαBxt2−1ut+vt,Dtαvt=utCα,where Aα=A/σα−1, Cα=Cσα−1, Lα=Lσα−1, ut=iLt/A, and vt=vCt/A. Note also that Cα and Lα are generally known as the pseudocapacitance [37] of the fractional capacitor and the inductivity [38] of the fractional inductor, respectively. Obviously, (10), which is resulted by the dimensional consistency aware generalization, is significantly different from (7) as it is not merely (3) with fractional derivatives.By some mathematical manipulation, (11) can be rewritten in a matrix-vector format as(11)Dtαxt=A^xt,txt,where(12)xt=xtutvt,A^xt,t=−Aα−Aαxt−100−BLαxt2−1−1Lα01Cα0.In this work, the fractional derivative has been interpreted in the modified Riemann–Liouville sense as given by (1); thus, we have(13)xt=1Γ1−α∫0tt−τ−α−1A^xτ,τxτ−A^x0,0x0dτ.For analyzing the generalized fractional dimensional consistency aware simplest Chua’s chaotic circuit via simulations, its model given by (11) must be solved in a numerical manner. In order to do so, we apply the nonlinear transformation [32] to (13). As a result, we have(14)ddξXξ=A^Xξ,ξXξ,i.e.,(15)Xξ=X0+∫0ξA^Xz,zXzdz,where(16)Xξ=XξUξVξ,A^Xξ,ξ=−Aα−AαXξ−100−BLαXξ2−1−1Lα01Cα0.Note also thatxt=Xξ where ξ=tα/Γ1+α [32]. At this point, it can be seen that (11) has been transformed to (14). Therefore, the solution of (11) can be conveniently obtained by solving (14) and keeping the above relationships between xt and Xξ in mind. Here, we let A = 0.2A, B = 1.7 Ω, L = 3.3H, and C = 1F similar to [20]. We also assume that α = 0.9, σ = 1 sec, X(0) = 0.1, U(0) = 0, and V(0) = 0.1 Ω and numerically solve (14) with MATHEMATICA with the above relationship between ξ and t in mind. As a result, the phase portraits of xt, iLt, and vCt can be simulated by also keeping in mind that iLt=Aut and vCt=Avt as depicted in Figures 3–5, where the strange attractors, which refers to the chaotic behaviors, can be observed.Figure 3
i
L(t) vs x(t) (Chua’s circuit).Figure 4
v
C
t vs x(t) (Chua’s circuit).Figure 5
v
C
t vs iLt (Chua’s circuit).For the quantitative analysis, we evaluate the corresponding Lyapunov exponents and Lyapunov dimension (DL) by employing the following definition.Definition 2.
For any modified Riemann–Liouville fractional derivative‐based fractional order dynamical system defined by(17)Dtαxt=gxt,where(18)xt=x1tx2t…xNtT,gxt=g1x1t,x2t,…,xNtg2x1t,x2t,…,xNt⋮gNx1t,x2t,…,xNt,its jth Lyapunov exponent (λj) can be found as(19)λj=limξ⟶∞1ξlnujξuj0,where {j} = {1, 2, …, N} and ujξ stands for the jth tangent vector of the system’s trajectory in a ξ domain N dimensional space defined by (X1(ξ), X2(ξ), …, XN(ξ)). Note also that Xjξ=xjt,  denotes the Euclidian norm operator, and ujξ can be obtained by simultaneously solving the nonlinear transformed of (17) and its variational equation.
Obviously, Definition2 states that all λj’s and DL of the modified Riemann–Liouville fractional derivative based system can be determined in a similar manner to that of the conventional integer order system after a nonlinear transformation. As a result, it has been found by using (14), the algorithm proposed by Sandri [39] and MATHEMATICA, that λ1 = 0.0473804, λ2 = −0.0255304, and λ3 = −0.550074. Since λ1 > 0 and λ1 + λ2 + λ3 = −0.528224 which is less than 0, both expansion in one direction and contracting volumes in the phase space of the attractor that indicate the chaotic behavior. We have also found that the contraction outweighs the expansion as λ1 + λ2 + λ3 < 0; therefore, our system is dissipative. Moreover, it has been found that DL = 2.03972 which is a fractional number. Therefore, the manifold in the phase space is a strange attractor which indicates the chaotic behavior.
In order to demonstrate the significance of time dimensional consistency awareness, we compare our quantitative analysis results to their dimensional consistency ignored counterparts analyzed by using (7). For ceteris paribus and the applicability of Definition 2 thus Sandri’s algorithm, all fractional derivative terms of (7) have also been defined in the modified Riemann–Liouville sense, unlike [23] in which the Caputo fractional derivative has been adopted. Moreover, the similar parameters except σ have been used. As a result, we have found that λ1 = 0.0202319, λ2 = −0.000902803, λ3 = −0.437888, i.e., λ1 + λ2 + λ3 = −0.418559, and DL = 2.04414. These Lyapunov exponents and DL are significantly different from those of the dimensional consistency aware scenario. Therefore, it can be seen that the fractional domain circuit employs different dynamic thus different chaotic behavior when the time dimensional consistency has been concerned. This is because different amounts of expansion and contraction along with the fractal dimension phase space manifolds of the circuit can be obtained. It can also be seen that the dimensional consistency included the fractional circuit which became more dissipative than its dimensional consistency neglected predecessor as the latter employs lower λ1 + λ2 + λ3. Also, unlike the previous dimensional consistency ignored circuit which is ceased to be chaotic if α < 0.715 [23], a different condition on α can be obtained when the dimensional consistency is concerned. For illustration, we derive such condition that our dimensional consistency aware circuit ceases to be chaotic based on the same parameters as those of [23]. Firstly, the Jacobian matrix at equilibrium of the circuit’s dynamical equation must be formulated by using the following definition.Definition 3.
For any fractional order dynamical system, its Jacobian matrix at arbitrary equilibrium point given byE=x1E,x2E,…,xNE (JE) can be defined as(20)JE=∂g1∂x1∂g1∂x2…∂g1∂xN∂g2∂x1∂g2∂x2…∂g2∂xN∂g3∂x1∂g3∂x2…∂g3∂xN∂g4∂x1∂g4∂x2…∂g4∂xNE.
Since this circuit has only one equilibrium point which can be given in terms of 3-tuples, i.e., (x(t), u(t), vt), by E = (0, 0, 0), the resulting JE can be obtained as follows:(21)JE=−AαAα00BLα−1Lα01Cα0.
After obtainingJE, the characteristic equation can be determined by using the following corollary.Corollary 1.
For arbitrary dynamical system, its characteristic equation can be given by(22)detlI−JE=0,where l and I denote the eigenvalue symbol and the identity matrix.As a result, the characteristic equation can be obtained based on (22) as(23)detl+Aα−Aα00l−BLα1Lα0−1Cαl=0,which yields(24)l1=−Aα,l2=B2Lα+j4LαCα−BCα2LαCα,l3=B2Lα−j4LαCα−BCα2LαCα.For obtaining the chaotic behavior,E must be a saddle point of index 2 which generates the chaos in any three-dimensional system [40]. Thus, it can be seen from (24) that the following equation [41] (25)α≤2πtan−14LαCα−BCα2BCα,must be satisfied for our circuit ceases to be chaotic. This is because the stable region of the complex plane has been enlarged so that it covers l1, l2, and l3; thus, E becomes locally asymptotically stable. Based on the assumed parameters, we have α≤0.690015 for our dimensional consistency aware scenario which is different from the above condition of the dimensional consistency ignored circuit.Apart from satisfying (25), the circuit also ceases to be chaotic if B ≤ 0 because both formerly unstable complex conjugate eigenvalues, i.e., l2 and l3l2 and l3, will be resided in the stable region as Re[l2,3] ≤ 0 regardless of α. Moreover, the conditions on Lα and Cα with which the circuit ceases to be chaotic can be obtained by using (24) as follows:(26)Lα≥B2Cα41+tan2απ2,(27)Cα≤4LαB21+tan2απ/2,where α can be arbitrary. By satisfying either of these conditions, E becomes locally asymptotically stable as both l2 and l3 will be moved to the stable region. Therefore, all eigenvalues now reside in the stable region (as long as A > 0, which yields Aα > 0, is satisfied, l1 will always be located on the positive real axis of the complex plane) and the circuit ceases to be chaotic. If we let A = 0.2A, B = 1.7 Ω, α = 0.9, and σ = 1 sec, (26) and (27) will become Lα≥29.5238 Hsecα−1 for C = 1F and Cα≤0.111774 Fsecα−1 for L = 3.3H, respectively.In order to verify the above conditions and study the dynamic ofE which governs that of the circuit with respect to the changes in circuit parameters, we formulate(28)mi=απ2−argli,where {i} = {1, 2, 3}. It should be mentioned here that α, Lα, and Cα have been chosen as the bifurcation parameters and the effect of mi to the location of li which governs the system dynamic is similar to that of the real part of li if li is an eigenvalue of the integer system [42]. Since the plots of real parts of eigenvalues with respect to bifurcation parameters have been adopted for studying the dynamic of conventional integer system in previous works [43, 44], the plots of mi can be similarly used for studying the dynamic of the circuit despite that it is of fractional order. By using (24) and (28), we have(29)m1=α2−1π,(30)m2,3=απ2−tan−14LαCα−BCα2BCα.As a result, the dynamics ofm1 and m2,3 can be simulated as follows.It can be seen from Figures6–8 that m1<0 always unlike m2 and m3 depicted in Figures 9–11. Thus, the circuit’s stability is solely governed by l2 and l3 as l1 will always be in the stable region of the complex plane. The circuit becomes asymptotically stable if and only if α<0.690015, Lα>29.5238 Hsecα−1, and Cα<0.111774 Fsecα−1 have been assured because m2,3<0 which implies that l2 and l3 are in the stable region and can be observed. Otherwise, m2,3>0 which implies that l2 and l3 are in the unstable region; thus, the circuit becomes unstable and can be seen instead. The circuit becomes marginally stable when either α=0.690015, Lα=29.5238 Hsecα−1 or Cα=0.111774 Fsecα−1 has been satisfied since m2,3=0 can be found. This verifies (25)–(27) as the circuit becomes nonchaotic in these scenarios. In addition, we have found that the transversality condition is established at α=0.690015, Lα=29.5238 Hsecα−1, and Cα=0.111774 Fsecα−1 because(31)∂∂αm2,3α=0.690015=π2≠0,∂∂Lαm2,3Lα=29.5238=−0.00268232≠0,∂∂Cαm2,3Cα=0.111774=0.708053≠0,where m2,3=0 as stated above. Moreover, we have also found that the real eigenvalue can be given by l1 = −0.2 A/secα−1 ≠ 0 in these circumstances. As a result, the Hopf bifurcation condition for arbitrary three-dimensional fractional order system [42] has been satisfied. Thus, it can be stated that the circuit’s stability switches and the circuit undergo a Hopf bifurcation through E when either α = 0.690015, Lα = 29.5238 Hsecα−1 or Cα = 0.111774 Fsecα−1 has been satisfied. As an illustration, the phase portraits of iLt and vCt with Lα = 29.5238 Hsecα−1 can be simulated as depicted in Figure 12, where a limit cycle, which indicates a periodic solution, can be observed. It should be mentioned here that such Hopf bifurcation cannot occur when α ≠ 0.690015, Lα ≠ 29.5238 Hsecα−1, and Cα ≠ 0.111774 Fsecα−1 despite the fact that l1 ≠ 0 is satisfied (since l1 = −0.2 A/secα−1 is always based on the assumed parameter according to (24)). This is because m2,3≠0 as can be seen from Figures 7, 9, and 11.Figure 6
m
1 vs α.Figure 7
m
1 vs Lα.Figure 8
m
1 vs Cα..Figure 9
m
2,3 vs α.Figure 10
m
2,3 vs Lα.Figure 11
m
2,3 vs Cα.Figure 12
v
C
t vs iL(t) (Lα = 29.5238 Hsecα−1).Before proceeding further, it should be mentioned here that our dimensional consistency aware fractional domain Chua’s circuit cannot be realized by simply replacing the capacitor, the inductor, and the memristor by the fractional ones. This is because (10) is not merely (3) with fractional derivatives. However, (10) can be rewritten in terms of fractional integrals as(32)xt=−AαJtαxt+utxt−ut,ut=−1LβJtβBxt2ut−ut+vt,vt=1CγJtγut,where Jtα, Jtβ, and Jtγ, respectively, denote the fractional integral operators of order α, β, and γ with respect to t. Note that the order of the fractional derivative terms has been allowed to be incommensurate for obtaining full degree of freedom; thus, Lα and Cα must become Lβ and Cγ, which can be, respectively, given by Lβ=Lσβ−1 and Cγ=Cσγ−1, where 0 < β ≤ 1 and 0 < γ ≤ 1 for maintaining the dimensional consistency. By using (32), the emulator of our dimensional consistency aware circuit can be obtained generically without referring to any specific physical system as depicted in Figure 13. Based on this generic emulator, the specific circuit emulator can be obtained by using merely the shelf components. An example of such circuit emulator is depicted in Figure 14. Moreover, x(t), u(t), and vt are in terms of voltages. Note also that this circuit emulates the conventional simplest Chua’s chaotic circuit if all fractional capacitors have been replaced by the conventional ones. From Figure 14, it can be seen that the OPAMPs and fractional capacitors have been used for realizing the fractional integrators; AD633, which is off the shelf analog voltage multiplier, has been adopted for performing multiplication and the OPAMP-based unity gain inverting amplifiers have been used as the invertors. Here, we choose TL084 as our OPAMP.Figure 13
The generic emulator of dimensional consistency aware fractional domain Chua’s circuit.
(a)
(b)
(c)Figure 14
An example of specific circuit implementation of the emulator depicted in Figure13.
(a)
(b)
(c)For realizing the fractional capacitors, we firstly approximate the impedance function of fractional capacitor generally given byZδs=Cδ−1s−δ where Cδ=Cσδ−1 and 0 < δ ≤ 1 as follows [45]:(33)Cδ−1s−δ≈K+∑i=1NLis+Mi,where N can be arbitrary positive integer. Thus, the circuit that approximates the fractional capacitor which is solely composed of resistors and capacitors which are off the shelf components can be obtained as depicted in Figure 15, where R0 = K, Ri = Li/Mi, and Ci = 1/Li. It should be mentioned here that K, Li’s, and Mi’s can be computed by using the method of undetermined coefficients [45]. As a practical example, a circuit that approximates a fractional capacitor with Cδ = 1 μFsecδ−1 and δ = 0.9 can be realized by assuming that N = 3 as the RC approximated circuit  with 3 R//C stages composed ofR0 = 1 MΩ, R1 = 62.84 MΩ, R2 = 250 kΩ, R3 = 2.5 kΩ, C1 = 1.23 μF, C2 = 1.835 μF and C3 = 1.1 μF. If we let σ = 1 sec, A = 0.2A, L = 3.3H, C = 1F, B = 1.7S, x(0) = 0.1 V, u(0) = 0 V, v0 = 0.1 V, α = 0.8, and β = 0.95, γ = 0.85, the phase portraits of xt, 0.2ut, and 0.2vt can be simulated by using PSPICE which has been adopted in many previous works [40, 45] as depicted in Figures 16–18. Note that N = 7 has been assumed because the resulting approximated RC circuit with such N yields the results which are as accurate as the 7th order Oustaloup approximation-based results [45]. The parameter values of this circuit emulator can be summarized in Table 1 where the OPAMP’s supply voltage (Vsup) of ±15V has been adopted.Figure 15
The RC circuit approximation of fractional capacitor [45].Figure 16
0.2u(t) vs x(t).Figure 17
0.2
v
t vs x(t).Figure 18
0.2
v
t vs 0.2u(t).Table 1
Parameter values of the circuit emulator depicted in Figure14.
Parameter
Value
r
588 Ω
C
1α
5 mFsecα−1
C
2β
3.3 mFsecβ−1
C
3γ
1 mFsecγ−1
α
0.8
β
0.95
γ
0.85
V
supply
±
15
V


### 5.2. The Novel Simplest Circuit

Now, the novel simplest chaotic circuit will be considered. After the fractional domain generalization, we have(34)1σ1−αDtαvCt=−1Ckx2t−xt−1vCt+iLt,1σ1−αDtαiLt=1LvCt,1σ1−αDtαxt=a1xt+a3x3t+b1vCt+c11vCtxt.By defininga1α=a1σ1−α, a3α=a3σ1−α, b1α=b1σ1−α, and c11α=c11σ1−α the following dimensional consistency aware fractional domain model of the simplest paralleled structured chaotic circuit can be obtained.(35)DtαvCt=−1Cαkx2t−xt−1vCt+iLt,DtαiLt=1LαvCt,Dtαxt=a1αxt+a3αx3t+b1αvCt+c11αvCtxt.After some rearrangement, the matrix vector formatted of (35) can be given as follows:(36)Dtαyt=B^yt,tyt,where(37)yt=vCtiLtxt,B^yt,t=−kCαx2t−xt−1−1Cα01Lα00b1α+c11αxt0a1α+a3αx2t.Since the fractional derivative has been interpreted in the modified Riemann–Liouville sense, we have(38)yt=1Γ1−α∫0tt−τ−α−1B^yτ,τyτ−B^y0,0y0dτ.For solving (36) in a numerical manner, the nonlinear transformation has also been applied. As a result, we have(39)ddξYξ=B^Yξ,ξYξ,i.e.,(40)Yξ=Y0+∫0ξB^Yz,zYzdz,where(41)Yξ=VCξILξXξ,B^Yξ,ξ=−kCαX2ξ−Xξ−1−1Cα01Lα00b1α+c11αXξ0a1α+a3αX2ξ.Note also thatyt=Yξ. Similar to [30], we let a1 = 1.8, a3 = −3.9, b1 = 1.4, c11 = −1.5, k = 1, C = 130 mF, L = 50 mH, vC0 = 0.1 V, iL(0) = 0.1A, and x(0) = 0.2. We also assume that α = 0.9 and σ = 1 sec and numerically solve (36). By using the obtained solutions and keeping the above relationship between yt and Yξ along with that of ξ and t mentioned above in mind, the phase portraits of xt, iLt, and vCt can be simulated as depicted in Figures 19–21, where the strange attractors can be observed. In addition, we have found that λ1 = 1.37332, λ2 = −0.23593, λ3 = −12.1655, and DL = 2.09348. Since λ1 > 0, λ1 + λ2 + λ3 = −11.02811, which is less than 0, and DL is a fractional number, a dissipative chaotic behavior, which is even more dissipative than Chua’s circuit analyzed in the previous section, can be observed. The Lyapunov exponents and dimensions of this circuit and Chua’s circuit are summarized in Table 2.Figure 19
i
L(t) vs x(t) (novel circuit).Figure 20
v
C
t vs x(t) (novel circuit).Figure 21
v
C
t vs iL(t) (novel circuit).Table 2
Lyapunov exponents and dimensions.
Circuit
λ
1
λ
2
λ
3
λ
1 + λ2 + λ3
D
L
Chua’s circuit
0.0473804
0.0255304
0.550074
−0.528224
2.03972
Chua’s circuit (dimensional consistency ignored)
0.0202319
0.000902803
0.437888
−0.418559
2.04414
Novel circuit
1.37332
−0.23593
−12.1655
−11.02811
2.09348Similar to the simplest Chua's circuit considered in the previous subsection, we also determine the condition onα in which this novel simplest circuit ceased to be chaotic. Unlike Chua’s circuit, which employs only one equilibrium point, the novel simplest circuit employs three points which can be given in terms of (vCt, iL(t), x(t)) as E1=0,0,0, E2=0,0,−a1α/a3α, and E3=0,0,−−a1α/a3α. Among these equilibrium points, only E2 will be considered because such point implies that x(0) ≥ 0 which guarantees the chaotic behavior of the circuit as the local activity of the memristive device can be assured [30]. As a result, the Jacobian matrix at E2 can be given by following Definition 3 as(42)JE2=kCαa1αa3α+−a1αa3α+1−1Cα01Lα00b1α+c11α−a1αa3α0−2a1α.Therefore, the characteristic equation can be obtained by following Corollary1 as follows:(43)detl−kCαa1αa3α+−a1αa3α+11Cα0−1Lαl0−b1α−c11α−a1αa3α0l+2a1α=0,which yields(44)l1=−2a1α,l2=ka1α+a3α2a3αCα+jka1αLα+4a1αa3αCαLα+ka1α+a3αLα−a1α/a3α−ka1αLα22a1αa3αCαLα,l3=ka1α+a3α2a3αCα−jka1αLα+4a1αa3αCαLα+ka1α+a3αLα−a1α/a3α−ka1αLα22a1αa3αCαLα.By using (44), we have found that the novel simplest chaotic circuit ceases to be chaotic if a1α+a3α/2a3αα≤0 has been satisfied. This is because l2 and l3 will reside in the stable region of the complex plane. Moreover, this circuit also ceases to be chaotic if(45)α≤2πtan−1−4a3α2LαCα−a1α2+a1αa3α+a3α2+2a1α+a3α−a1αa3αk2Lα21/2a1α+a3α+−a1αa3αkLα,has been satisfied due to the aforementioned stable region enlargement. Based on the assumed parameters, we have found that α≤0.75317 must be satisfied for this novel circuit ceases to be chaotic.In addition, we have also found that satisfying either (46) or (47) makes the novel circuit ceases to be chaotic despite the fact that (45) has not been satisfied. This is because l2 and l3 will be moved to the stable region. Here, we let a1 = 1.8, a3 = −3.9, b1 = 1.4, c11 = −1.5, k = 1, σ = 1 sec, and α = 0.9; therefore, (46) and (47) become Lα≤8.58018mHsecα−1 if C = 130 mF and Cα≥757.56mFsecα−1 if L = 50 mH, respectively.(46)Lα≤2a3α2a1α2+a1αa3α+a3α2+2a3α−a1α−a1αa3α+a3α2C1+cosαπa1α2+3a1αa3α+a3α22k2,(47)Cα≥a1α2+a1α+a3αa3α+2−a1αa3αk2Lα4a3α2cos2απ/2.At this point, we will study the dynamic ofE2 and the novel circuit and also verify the above conditions. For this circuit, it has been found that m1 can also be given by (29) according to (44); thus, the dynamic of m1 with respect to α similar to that depicted in Figure 6 can be obtained. On the other hand, m2,3 can be found as(48)m2,3=απ2−tan−1−4a3α2LαCα−a1α2+a1αa3α+a3α2+2a1α+a3α−a1αa3αk2Lα21/2a1α+a3α+−a1αa3αkLα.As a result, the dynamics ofm1 with respect to other bifurcation parameters and those of m2,3 can be simulated as depicted in Figures 22–26 which also imply that the circuit’s stability is solely governed by l2 and l3. In this case, the circuit becomes asymptotically stable when either α<0.75317, Lα<8.58018Hsecα−1 or Cα>757.56 Fsecα−1 has been assured and vice versa and become marginally stable when either α=0.75317, Lα=8.58018Hsecα−1 or Cα=757.56Fsecα−1 has been satisfied. This verifies (45)–(47) as the circuit becomes nonchaotic in these scenarios. On the other hand, the real eigenvalue can be given by l1 = −3.6 A/secα−1 ≠ 0 along the entire ranges of bifurcation parameters because l1 is constant with respect to these parameters as can be seen from (44). If at least one of the above marginal stability conditions on bifurcation parameters has been satisfied, we have m2,3=0 where(49)∂∂αm2,3α=0.75317=π2≠0,∂∂Lαm2,3Lα=8.58018×10−3=9.22967≠0,∂∂Cαm2,3Cα=757.56×10−3=−0.104536≠0,Figure 22
m
2,3 vs α.Figure 23
m
1 vs Lα..Figure 24
m
2,3 vs Lα.Figure 25
m
2,3 vs Cα.Figure 26
m
2,3 vs Cα.and as a result, the transversality condition has been established; thus, the Hopf bifurcation condition has been satisfied. Therefore, the stability switches and the circuit undergoes a Hopf bifurcation throughE2 when either α=0.75317, Lα=8.58018Hsecα−1 or Cα=757.56Fsecα−1. As an illustration, the phase portraits of iLt and vCt with Cα=757.56Fsecα−1 can be simulated as depicted in Figure 27 where a limit cycle can be obtained. Note also that such bifurcation does not exist when α≠0.75317, Lα≠8.58018Hsecα−1, and Cα≠757.56Fsecα−1 despite the fact that l1 ≠ 0. This is because m2,3 ≠ 0 as can be seen from Figures 22, 24, and 26.Figure 27
v
C
t vs iL(t) (Cα = 757.56 Fsecα−1).For implementing the emulator of the fractional domain simplest paralleled structured chaotic circuit, (35) must be firstly rewritten in terms of fractional integration. Since we let the order of the fractional derivative terms be incommensurate, Lα and Cα become Lγ and Cβ. We also define y(t) = vCt and z(t) = iL(t). As a result, we have(50)xt=Jtαa1αxt+a3αx3t+b1αyt+c11αytxt,yt=−1CβJtβkx2tyt−xtyt−yt+zt,zt=1LγJtγyt.Therefore, the resulting generic emulator can be obtained as depicted in Figure28.Figure 28
The generic emulator of the dimensional consistency aware simplest paralleled structured fractional domain chaotic circuit.
(a)
(b)
(c)An example of circuit emulator implemented based on such generic one is depicted in Figure29 where x(t), y(t), and z(t) in terms of voltages and C1α = 1 mFsecα−1 have been assumed. In addition, NR1 and NR2 are floating negative resistors because a3 < 0 and c11 < 0. These negative resistors have been realized by using OTA which is off the shelf component, by following [46], where the transconductances of these OTAs have been given by gm1=0.001a3α and gm2=0.001c11α. The OPAMP-fractional capacitor-based fractional integrators, unity gain inverting amplifier, and AD633 multiplier have also been adopted. We also use TL084 as our OPAMP and the aforementioned RC approximated circuit with N = 7 for realizing the fractional capacitors. It should be mentioned here that this emulator emulates the novel simplest paralleled structured chaotic circuit when all fractional capacitors have been replaced by the conventional ones. If we let σ = 1 sec, α = 0.8, β = 0.95, γ = 0.85, a1 = 1.8, a3 = −3.9, b1 = 1.4, c11 = −1.5, k = 1, C = 130 mF, L = 50 mH, x(0) = 0.2 V, y(0) = 0.1 V, and z(0) = 0.1 V, the phase portraits of xt, yt, and zt can be simulated by PSPICE as depicted in Figures 30–32. The circuit parameter values for realizing this emulator are summarized in Table 3 (±15 V DC supply voltage for each OPAMP has also been assumed).Figure 29
An example of specific circuit implementation of the emulator depicted in Figure28.
(a)
(b)
(c)Figure 30
z(t) vs x(t).Figure 31
y(t) vs x(t).Figure 32
y(t) vs z(t).Table 3
Parameter values of the circuit emulator depicted in Figure29.
Parameter
Value
r
1
555.5 Ω
r
2
714.3 kΩ
r
3
1 kΩ
C
1α
1 mFsecα−1
C
2β
130μFsecβ−1
C
3γ
50μFsecγ−1
α
0.8
β
0.95
γ
0.85
g
m
1
3.9 ms
g
m
2
1.5 ms
V
supply
±15 V


## 5.1. The Simplest Chua’s Circuit

For generalizing the simplest Chua’s chaotic circuit in the fractional domain, its conventional electrical circuit components, i.e., capacitor, inductor, and memristor, must be replaced by the fractional ones, i.e., fractional capacitor, fractional inductor, and fractional memristor, which can be emulated by simply replacing the conventional capacitor of the memristor emulator [20] by the fractional one. Mathematically, this is to replace all conventional derivatives in (3) by the fractional domain counterparts. In [23], such replacement has been directly performed without any awareness of the dimensional consistency as the time dimension of Dtα is sec−α while that of d/dt is sec−1. Moreover, sec−α is not physically measurable unlike sec−1. As a result, the following model of the fractional domain generalized simplest Chua’s chaotic circuit can be obtained [23]:(7)Dtαxt=−Axt+iLtxt−iLt,DtαiLt=−1LBxt2−1iLt+vCt,DtαvCt=iLtC,where 0 ≤ α ≤ 1 and iL(t) = iC(t) = −iM(t). Obviously, (7) is simply (3) with fractional derivatives.However, this is not the case for this research as the dimensional consistency has been concerned unlike those previous works. For achieving the dimensional consistency, the time dimensions ofd/dt and the generalized fractional derivative must be consistent which means that both of them must be given by the physically measurable sec−1. Therefore, the following operation must be used:(8)ddt⟶1σ1−αDtα,where σ denotes the fractional time component or the cosmic time [26]. Note that σ > 0 always for preventing singularity. In addition, the introduction of σ scales only the Laplace transformed derivative term where the frequency scaling scales all complex frequency variables.Since the dimension ofσ is sec, that of 1/σ1−αDtα is also sec as the dimension of Dtα is sec−α as stated above. Therefore, the dimensions of d/dt and generalized fractional derivative, i.e., 1/σ1−αDtα, are now consistent. As a result, the dimensional consistency of the generalized fractional domain model can be now achieved. Note also that (8) which has been adopted in those previous works on the fractional domain generalization of linear circuit models has been found to be suitable for our work since the assumed modified Riemann–Liouville fractional derivative employs a power law kernel as can be seen from (1). By using (8), we have(9)1σ1−αDtαxt=−Axt+iLtxt−iLt,1σ1−αDtαiLt=−1LBxt2−1iLt+vCt,1σ1−αDtαvCt=iLtC,where 0 < α ≤ 1 and iL(t) = iC(t) = −iM(t).After some rearrangement, we obtain(10)Dtαxt=−Aαxt+utxt−ut,Dtαut=−1LαBxt2−1ut+vt,Dtαvt=utCα,where Aα=A/σα−1, Cα=Cσα−1, Lα=Lσα−1, ut=iLt/A, and vt=vCt/A. Note also that Cα and Lα are generally known as the pseudocapacitance [37] of the fractional capacitor and the inductivity [38] of the fractional inductor, respectively. Obviously, (10), which is resulted by the dimensional consistency aware generalization, is significantly different from (7) as it is not merely (3) with fractional derivatives.By some mathematical manipulation, (11) can be rewritten in a matrix-vector format as(11)Dtαxt=A^xt,txt,where(12)xt=xtutvt,A^xt,t=−Aα−Aαxt−100−BLαxt2−1−1Lα01Cα0.In this work, the fractional derivative has been interpreted in the modified Riemann–Liouville sense as given by (1); thus, we have(13)xt=1Γ1−α∫0tt−τ−α−1A^xτ,τxτ−A^x0,0x0dτ.For analyzing the generalized fractional dimensional consistency aware simplest Chua’s chaotic circuit via simulations, its model given by (11) must be solved in a numerical manner. In order to do so, we apply the nonlinear transformation [32] to (13). As a result, we have(14)ddξXξ=A^Xξ,ξXξ,i.e.,(15)Xξ=X0+∫0ξA^Xz,zXzdz,where(16)Xξ=XξUξVξ,A^Xξ,ξ=−Aα−AαXξ−100−BLαXξ2−1−1Lα01Cα0.Note also thatxt=Xξ where ξ=tα/Γ1+α [32]. At this point, it can be seen that (11) has been transformed to (14). Therefore, the solution of (11) can be conveniently obtained by solving (14) and keeping the above relationships between xt and Xξ in mind. Here, we let A = 0.2A, B = 1.7 Ω, L = 3.3H, and C = 1F similar to [20]. We also assume that α = 0.9, σ = 1 sec, X(0) = 0.1, U(0) = 0, and V(0) = 0.1 Ω and numerically solve (14) with MATHEMATICA with the above relationship between ξ and t in mind. As a result, the phase portraits of xt, iLt, and vCt can be simulated by also keeping in mind that iLt=Aut and vCt=Avt as depicted in Figures 3–5, where the strange attractors, which refers to the chaotic behaviors, can be observed.Figure 3
i
L(t) vs x(t) (Chua’s circuit).Figure 4
v
C
t vs x(t) (Chua’s circuit).Figure 5
v
C
t vs iLt (Chua’s circuit).For the quantitative analysis, we evaluate the corresponding Lyapunov exponents and Lyapunov dimension (DL) by employing the following definition.Definition 2.
For any modified Riemann–Liouville fractional derivative‐based fractional order dynamical system defined by(17)Dtαxt=gxt,where(18)xt=x1tx2t…xNtT,gxt=g1x1t,x2t,…,xNtg2x1t,x2t,…,xNt⋮gNx1t,x2t,…,xNt,its jth Lyapunov exponent (λj) can be found as(19)λj=limξ⟶∞1ξlnujξuj0,where {j} = {1, 2, …, N} and ujξ stands for the jth tangent vector of the system’s trajectory in a ξ domain N dimensional space defined by (X1(ξ), X2(ξ), …, XN(ξ)). Note also that Xjξ=xjt,  denotes the Euclidian norm operator, and ujξ can be obtained by simultaneously solving the nonlinear transformed of (17) and its variational equation.
Obviously, Definition2 states that all λj’s and DL of the modified Riemann–Liouville fractional derivative based system can be determined in a similar manner to that of the conventional integer order system after a nonlinear transformation. As a result, it has been found by using (14), the algorithm proposed by Sandri [39] and MATHEMATICA, that λ1 = 0.0473804, λ2 = −0.0255304, and λ3 = −0.550074. Since λ1 > 0 and λ1 + λ2 + λ3 = −0.528224 which is less than 0, both expansion in one direction and contracting volumes in the phase space of the attractor that indicate the chaotic behavior. We have also found that the contraction outweighs the expansion as λ1 + λ2 + λ3 < 0; therefore, our system is dissipative. Moreover, it has been found that DL = 2.03972 which is a fractional number. Therefore, the manifold in the phase space is a strange attractor which indicates the chaotic behavior.
In order to demonstrate the significance of time dimensional consistency awareness, we compare our quantitative analysis results to their dimensional consistency ignored counterparts analyzed by using (7). For ceteris paribus and the applicability of Definition 2 thus Sandri’s algorithm, all fractional derivative terms of (7) have also been defined in the modified Riemann–Liouville sense, unlike [23] in which the Caputo fractional derivative has been adopted. Moreover, the similar parameters except σ have been used. As a result, we have found that λ1 = 0.0202319, λ2 = −0.000902803, λ3 = −0.437888, i.e., λ1 + λ2 + λ3 = −0.418559, and DL = 2.04414. These Lyapunov exponents and DL are significantly different from those of the dimensional consistency aware scenario. Therefore, it can be seen that the fractional domain circuit employs different dynamic thus different chaotic behavior when the time dimensional consistency has been concerned. This is because different amounts of expansion and contraction along with the fractal dimension phase space manifolds of the circuit can be obtained. It can also be seen that the dimensional consistency included the fractional circuit which became more dissipative than its dimensional consistency neglected predecessor as the latter employs lower λ1 + λ2 + λ3. Also, unlike the previous dimensional consistency ignored circuit which is ceased to be chaotic if α < 0.715 [23], a different condition on α can be obtained when the dimensional consistency is concerned. For illustration, we derive such condition that our dimensional consistency aware circuit ceases to be chaotic based on the same parameters as those of [23]. Firstly, the Jacobian matrix at equilibrium of the circuit’s dynamical equation must be formulated by using the following definition.Definition 3.
For any fractional order dynamical system, its Jacobian matrix at arbitrary equilibrium point given byE=x1E,x2E,…,xNE (JE) can be defined as(20)JE=∂g1∂x1∂g1∂x2…∂g1∂xN∂g2∂x1∂g2∂x2…∂g2∂xN∂g3∂x1∂g3∂x2…∂g3∂xN∂g4∂x1∂g4∂x2…∂g4∂xNE.
Since this circuit has only one equilibrium point which can be given in terms of 3-tuples, i.e., (x(t), u(t), vt), by E = (0, 0, 0), the resulting JE can be obtained as follows:(21)JE=−AαAα00BLα−1Lα01Cα0.
After obtainingJE, the characteristic equation can be determined by using the following corollary.Corollary 1.
For arbitrary dynamical system, its characteristic equation can be given by(22)detlI−JE=0,where l and I denote the eigenvalue symbol and the identity matrix.As a result, the characteristic equation can be obtained based on (22) as(23)detl+Aα−Aα00l−BLα1Lα0−1Cαl=0,which yields(24)l1=−Aα,l2=B2Lα+j4LαCα−BCα2LαCα,l3=B2Lα−j4LαCα−BCα2LαCα.For obtaining the chaotic behavior,E must be a saddle point of index 2 which generates the chaos in any three-dimensional system [40]. Thus, it can be seen from (24) that the following equation [41] (25)α≤2πtan−14LαCα−BCα2BCα,must be satisfied for our circuit ceases to be chaotic. This is because the stable region of the complex plane has been enlarged so that it covers l1, l2, and l3; thus, E becomes locally asymptotically stable. Based on the assumed parameters, we have α≤0.690015 for our dimensional consistency aware scenario which is different from the above condition of the dimensional consistency ignored circuit.Apart from satisfying (25), the circuit also ceases to be chaotic if B ≤ 0 because both formerly unstable complex conjugate eigenvalues, i.e., l2 and l3l2 and l3, will be resided in the stable region as Re[l2,3] ≤ 0 regardless of α. Moreover, the conditions on Lα and Cα with which the circuit ceases to be chaotic can be obtained by using (24) as follows:(26)Lα≥B2Cα41+tan2απ2,(27)Cα≤4LαB21+tan2απ/2,where α can be arbitrary. By satisfying either of these conditions, E becomes locally asymptotically stable as both l2 and l3 will be moved to the stable region. Therefore, all eigenvalues now reside in the stable region (as long as A > 0, which yields Aα > 0, is satisfied, l1 will always be located on the positive real axis of the complex plane) and the circuit ceases to be chaotic. If we let A = 0.2A, B = 1.7 Ω, α = 0.9, and σ = 1 sec, (26) and (27) will become Lα≥29.5238 Hsecα−1 for C = 1F and Cα≤0.111774 Fsecα−1 for L = 3.3H, respectively.In order to verify the above conditions and study the dynamic ofE which governs that of the circuit with respect to the changes in circuit parameters, we formulate(28)mi=απ2−argli,where {i} = {1, 2, 3}. It should be mentioned here that α, Lα, and Cα have been chosen as the bifurcation parameters and the effect of mi to the location of li which governs the system dynamic is similar to that of the real part of li if li is an eigenvalue of the integer system [42]. Since the plots of real parts of eigenvalues with respect to bifurcation parameters have been adopted for studying the dynamic of conventional integer system in previous works [43, 44], the plots of mi can be similarly used for studying the dynamic of the circuit despite that it is of fractional order. By using (24) and (28), we have(29)m1=α2−1π,(30)m2,3=απ2−tan−14LαCα−BCα2BCα.As a result, the dynamics ofm1 and m2,3 can be simulated as follows.It can be seen from Figures6–8 that m1<0 always unlike m2 and m3 depicted in Figures 9–11. Thus, the circuit’s stability is solely governed by l2 and l3 as l1 will always be in the stable region of the complex plane. The circuit becomes asymptotically stable if and only if α<0.690015, Lα>29.5238 Hsecα−1, and Cα<0.111774 Fsecα−1 have been assured because m2,3<0 which implies that l2 and l3 are in the stable region and can be observed. Otherwise, m2,3>0 which implies that l2 and l3 are in the unstable region; thus, the circuit becomes unstable and can be seen instead. The circuit becomes marginally stable when either α=0.690015, Lα=29.5238 Hsecα−1 or Cα=0.111774 Fsecα−1 has been satisfied since m2,3=0 can be found. This verifies (25)–(27) as the circuit becomes nonchaotic in these scenarios. In addition, we have found that the transversality condition is established at α=0.690015, Lα=29.5238 Hsecα−1, and Cα=0.111774 Fsecα−1 because(31)∂∂αm2,3α=0.690015=π2≠0,∂∂Lαm2,3Lα=29.5238=−0.00268232≠0,∂∂Cαm2,3Cα=0.111774=0.708053≠0,where m2,3=0 as stated above. Moreover, we have also found that the real eigenvalue can be given by l1 = −0.2 A/secα−1 ≠ 0 in these circumstances. As a result, the Hopf bifurcation condition for arbitrary three-dimensional fractional order system [42] has been satisfied. Thus, it can be stated that the circuit’s stability switches and the circuit undergo a Hopf bifurcation through E when either α = 0.690015, Lα = 29.5238 Hsecα−1 or Cα = 0.111774 Fsecα−1 has been satisfied. As an illustration, the phase portraits of iLt and vCt with Lα = 29.5238 Hsecα−1 can be simulated as depicted in Figure 12, where a limit cycle, which indicates a periodic solution, can be observed. It should be mentioned here that such Hopf bifurcation cannot occur when α ≠ 0.690015, Lα ≠ 29.5238 Hsecα−1, and Cα ≠ 0.111774 Fsecα−1 despite the fact that l1 ≠ 0 is satisfied (since l1 = −0.2 A/secα−1 is always based on the assumed parameter according to (24)). This is because m2,3≠0 as can be seen from Figures 7, 9, and 11.Figure 6
m
1 vs α.Figure 7
m
1 vs Lα.Figure 8
m
1 vs Cα..Figure 9
m
2,3 vs α.Figure 10
m
2,3 vs Lα.Figure 11
m
2,3 vs Cα.Figure 12
v
C
t vs iL(t) (Lα = 29.5238 Hsecα−1).Before proceeding further, it should be mentioned here that our dimensional consistency aware fractional domain Chua’s circuit cannot be realized by simply replacing the capacitor, the inductor, and the memristor by the fractional ones. This is because (10) is not merely (3) with fractional derivatives. However, (10) can be rewritten in terms of fractional integrals as(32)xt=−AαJtαxt+utxt−ut,ut=−1LβJtβBxt2ut−ut+vt,vt=1CγJtγut,where Jtα, Jtβ, and Jtγ, respectively, denote the fractional integral operators of order α, β, and γ with respect to t. Note that the order of the fractional derivative terms has been allowed to be incommensurate for obtaining full degree of freedom; thus, Lα and Cα must become Lβ and Cγ, which can be, respectively, given by Lβ=Lσβ−1 and Cγ=Cσγ−1, where 0 < β ≤ 1 and 0 < γ ≤ 1 for maintaining the dimensional consistency. By using (32), the emulator of our dimensional consistency aware circuit can be obtained generically without referring to any specific physical system as depicted in Figure 13. Based on this generic emulator, the specific circuit emulator can be obtained by using merely the shelf components. An example of such circuit emulator is depicted in Figure 14. Moreover, x(t), u(t), and vt are in terms of voltages. Note also that this circuit emulates the conventional simplest Chua’s chaotic circuit if all fractional capacitors have been replaced by the conventional ones. From Figure 14, it can be seen that the OPAMPs and fractional capacitors have been used for realizing the fractional integrators; AD633, which is off the shelf analog voltage multiplier, has been adopted for performing multiplication and the OPAMP-based unity gain inverting amplifiers have been used as the invertors. Here, we choose TL084 as our OPAMP.Figure 13
The generic emulator of dimensional consistency aware fractional domain Chua’s circuit.
(a)
(b)
(c)Figure 14
An example of specific circuit implementation of the emulator depicted in Figure13.
(a)
(b)
(c)For realizing the fractional capacitors, we firstly approximate the impedance function of fractional capacitor generally given byZδs=Cδ−1s−δ where Cδ=Cσδ−1 and 0 < δ ≤ 1 as follows [45]:(33)Cδ−1s−δ≈K+∑i=1NLis+Mi,where N can be arbitrary positive integer. Thus, the circuit that approximates the fractional capacitor which is solely composed of resistors and capacitors which are off the shelf components can be obtained as depicted in Figure 15, where R0 = K, Ri = Li/Mi, and Ci = 1/Li. It should be mentioned here that K, Li’s, and Mi’s can be computed by using the method of undetermined coefficients [45]. As a practical example, a circuit that approximates a fractional capacitor with Cδ = 1 μFsecδ−1 and δ = 0.9 can be realized by assuming that N = 3 as the RC approximated circuit  with 3 R//C stages composed ofR0 = 1 MΩ, R1 = 62.84 MΩ, R2 = 250 kΩ, R3 = 2.5 kΩ, C1 = 1.23 μF, C2 = 1.835 μF and C3 = 1.1 μF. If we let σ = 1 sec, A = 0.2A, L = 3.3H, C = 1F, B = 1.7S, x(0) = 0.1 V, u(0) = 0 V, v0 = 0.1 V, α = 0.8, and β = 0.95, γ = 0.85, the phase portraits of xt, 0.2ut, and 0.2vt can be simulated by using PSPICE which has been adopted in many previous works [40, 45] as depicted in Figures 16–18. Note that N = 7 has been assumed because the resulting approximated RC circuit with such N yields the results which are as accurate as the 7th order Oustaloup approximation-based results [45]. The parameter values of this circuit emulator can be summarized in Table 1 where the OPAMP’s supply voltage (Vsup) of ±15V has been adopted.Figure 15
The RC circuit approximation of fractional capacitor [45].Figure 16
0.2u(t) vs x(t).Figure 17
0.2
v
t vs x(t).Figure 18
0.2
v
t vs 0.2u(t).Table 1
Parameter values of the circuit emulator depicted in Figure14.
Parameter
Value
r
588 Ω
C
1α
5 mFsecα−1
C
2β
3.3 mFsecβ−1
C
3γ
1 mFsecγ−1
α
0.8
β
0.95
γ
0.85
V
supply
±
15
V


## 5.2. The Novel Simplest Circuit

Now, the novel simplest chaotic circuit will be considered. After the fractional domain generalization, we have(34)1σ1−αDtαvCt=−1Ckx2t−xt−1vCt+iLt,1σ1−αDtαiLt=1LvCt,1σ1−αDtαxt=a1xt+a3x3t+b1vCt+c11vCtxt.By defininga1α=a1σ1−α, a3α=a3σ1−α, b1α=b1σ1−α, and c11α=c11σ1−α the following dimensional consistency aware fractional domain model of the simplest paralleled structured chaotic circuit can be obtained.(35)DtαvCt=−1Cαkx2t−xt−1vCt+iLt,DtαiLt=1LαvCt,Dtαxt=a1αxt+a3αx3t+b1αvCt+c11αvCtxt.After some rearrangement, the matrix vector formatted of (35) can be given as follows:(36)Dtαyt=B^yt,tyt,where(37)yt=vCtiLtxt,B^yt,t=−kCαx2t−xt−1−1Cα01Lα00b1α+c11αxt0a1α+a3αx2t.Since the fractional derivative has been interpreted in the modified Riemann–Liouville sense, we have(38)yt=1Γ1−α∫0tt−τ−α−1B^yτ,τyτ−B^y0,0y0dτ.For solving (36) in a numerical manner, the nonlinear transformation has also been applied. As a result, we have(39)ddξYξ=B^Yξ,ξYξ,i.e.,(40)Yξ=Y0+∫0ξB^Yz,zYzdz,where(41)Yξ=VCξILξXξ,B^Yξ,ξ=−kCαX2ξ−Xξ−1−1Cα01Lα00b1α+c11αXξ0a1α+a3αX2ξ.Note also thatyt=Yξ. Similar to [30], we let a1 = 1.8, a3 = −3.9, b1 = 1.4, c11 = −1.5, k = 1, C = 130 mF, L = 50 mH, vC0 = 0.1 V, iL(0) = 0.1A, and x(0) = 0.2. We also assume that α = 0.9 and σ = 1 sec and numerically solve (36). By using the obtained solutions and keeping the above relationship between yt and Yξ along with that of ξ and t mentioned above in mind, the phase portraits of xt, iLt, and vCt can be simulated as depicted in Figures 19–21, where the strange attractors can be observed. In addition, we have found that λ1 = 1.37332, λ2 = −0.23593, λ3 = −12.1655, and DL = 2.09348. Since λ1 > 0, λ1 + λ2 + λ3 = −11.02811, which is less than 0, and DL is a fractional number, a dissipative chaotic behavior, which is even more dissipative than Chua’s circuit analyzed in the previous section, can be observed. The Lyapunov exponents and dimensions of this circuit and Chua’s circuit are summarized in Table 2.Figure 19
i
L(t) vs x(t) (novel circuit).Figure 20
v
C
t vs x(t) (novel circuit).Figure 21
v
C
t vs iL(t) (novel circuit).Table 2
Lyapunov exponents and dimensions.
Circuit
λ
1
λ
2
λ
3
λ
1 + λ2 + λ3
D
L
Chua’s circuit
0.0473804
0.0255304
0.550074
−0.528224
2.03972
Chua’s circuit (dimensional consistency ignored)
0.0202319
0.000902803
0.437888
−0.418559
2.04414
Novel circuit
1.37332
−0.23593
−12.1655
−11.02811
2.09348Similar to the simplest Chua's circuit considered in the previous subsection, we also determine the condition onα in which this novel simplest circuit ceased to be chaotic. Unlike Chua’s circuit, which employs only one equilibrium point, the novel simplest circuit employs three points which can be given in terms of (vCt, iL(t), x(t)) as E1=0,0,0, E2=0,0,−a1α/a3α, and E3=0,0,−−a1α/a3α. Among these equilibrium points, only E2 will be considered because such point implies that x(0) ≥ 0 which guarantees the chaotic behavior of the circuit as the local activity of the memristive device can be assured [30]. As a result, the Jacobian matrix at E2 can be given by following Definition 3 as(42)JE2=kCαa1αa3α+−a1αa3α+1−1Cα01Lα00b1α+c11α−a1αa3α0−2a1α.Therefore, the characteristic equation can be obtained by following Corollary1 as follows:(43)detl−kCαa1αa3α+−a1αa3α+11Cα0−1Lαl0−b1α−c11α−a1αa3α0l+2a1α=0,which yields(44)l1=−2a1α,l2=ka1α+a3α2a3αCα+jka1αLα+4a1αa3αCαLα+ka1α+a3αLα−a1α/a3α−ka1αLα22a1αa3αCαLα,l3=ka1α+a3α2a3αCα−jka1αLα+4a1αa3αCαLα+ka1α+a3αLα−a1α/a3α−ka1αLα22a1αa3αCαLα.By using (44), we have found that the novel simplest chaotic circuit ceases to be chaotic if a1α+a3α/2a3αα≤0 has been satisfied. This is because l2 and l3 will reside in the stable region of the complex plane. Moreover, this circuit also ceases to be chaotic if(45)α≤2πtan−1−4a3α2LαCα−a1α2+a1αa3α+a3α2+2a1α+a3α−a1αa3αk2Lα21/2a1α+a3α+−a1αa3αkLα,has been satisfied due to the aforementioned stable region enlargement. Based on the assumed parameters, we have found that α≤0.75317 must be satisfied for this novel circuit ceases to be chaotic.In addition, we have also found that satisfying either (46) or (47) makes the novel circuit ceases to be chaotic despite the fact that (45) has not been satisfied. This is because l2 and l3 will be moved to the stable region. Here, we let a1 = 1.8, a3 = −3.9, b1 = 1.4, c11 = −1.5, k = 1, σ = 1 sec, and α = 0.9; therefore, (46) and (47) become Lα≤8.58018mHsecα−1 if C = 130 mF and Cα≥757.56mFsecα−1 if L = 50 mH, respectively.(46)Lα≤2a3α2a1α2+a1αa3α+a3α2+2a3α−a1α−a1αa3α+a3α2C1+cosαπa1α2+3a1αa3α+a3α22k2,(47)Cα≥a1α2+a1α+a3αa3α+2−a1αa3αk2Lα4a3α2cos2απ/2.At this point, we will study the dynamic ofE2 and the novel circuit and also verify the above conditions. For this circuit, it has been found that m1 can also be given by (29) according to (44); thus, the dynamic of m1 with respect to α similar to that depicted in Figure 6 can be obtained. On the other hand, m2,3 can be found as(48)m2,3=απ2−tan−1−4a3α2LαCα−a1α2+a1αa3α+a3α2+2a1α+a3α−a1αa3αk2Lα21/2a1α+a3α+−a1αa3αkLα.As a result, the dynamics ofm1 with respect to other bifurcation parameters and those of m2,3 can be simulated as depicted in Figures 22–26 which also imply that the circuit’s stability is solely governed by l2 and l3. In this case, the circuit becomes asymptotically stable when either α<0.75317, Lα<8.58018Hsecα−1 or Cα>757.56 Fsecα−1 has been assured and vice versa and become marginally stable when either α=0.75317, Lα=8.58018Hsecα−1 or Cα=757.56Fsecα−1 has been satisfied. This verifies (45)–(47) as the circuit becomes nonchaotic in these scenarios. On the other hand, the real eigenvalue can be given by l1 = −3.6 A/secα−1 ≠ 0 along the entire ranges of bifurcation parameters because l1 is constant with respect to these parameters as can be seen from (44). If at least one of the above marginal stability conditions on bifurcation parameters has been satisfied, we have m2,3=0 where(49)∂∂αm2,3α=0.75317=π2≠0,∂∂Lαm2,3Lα=8.58018×10−3=9.22967≠0,∂∂Cαm2,3Cα=757.56×10−3=−0.104536≠0,Figure 22
m
2,3 vs α.Figure 23
m
1 vs Lα..Figure 24
m
2,3 vs Lα.Figure 25
m
2,3 vs Cα.Figure 26
m
2,3 vs Cα.and as a result, the transversality condition has been established; thus, the Hopf bifurcation condition has been satisfied. Therefore, the stability switches and the circuit undergoes a Hopf bifurcation throughE2 when either α=0.75317, Lα=8.58018Hsecα−1 or Cα=757.56Fsecα−1. As an illustration, the phase portraits of iLt and vCt with Cα=757.56Fsecα−1 can be simulated as depicted in Figure 27 where a limit cycle can be obtained. Note also that such bifurcation does not exist when α≠0.75317, Lα≠8.58018Hsecα−1, and Cα≠757.56Fsecα−1 despite the fact that l1 ≠ 0. This is because m2,3 ≠ 0 as can be seen from Figures 22, 24, and 26.Figure 27
v
C
t vs iL(t) (Cα = 757.56 Fsecα−1).For implementing the emulator of the fractional domain simplest paralleled structured chaotic circuit, (35) must be firstly rewritten in terms of fractional integration. Since we let the order of the fractional derivative terms be incommensurate, Lα and Cα become Lγ and Cβ. We also define y(t) = vCt and z(t) = iL(t). As a result, we have(50)xt=Jtαa1αxt+a3αx3t+b1αyt+c11αytxt,yt=−1CβJtβkx2tyt−xtyt−yt+zt,zt=1LγJtγyt.Therefore, the resulting generic emulator can be obtained as depicted in Figure28.Figure 28
The generic emulator of the dimensional consistency aware simplest paralleled structured fractional domain chaotic circuit.
(a)
(b)
(c)An example of circuit emulator implemented based on such generic one is depicted in Figure29 where x(t), y(t), and z(t) in terms of voltages and C1α = 1 mFsecα−1 have been assumed. In addition, NR1 and NR2 are floating negative resistors because a3 < 0 and c11 < 0. These negative resistors have been realized by using OTA which is off the shelf component, by following [46], where the transconductances of these OTAs have been given by gm1=0.001a3α and gm2=0.001c11α. The OPAMP-fractional capacitor-based fractional integrators, unity gain inverting amplifier, and AD633 multiplier have also been adopted. We also use TL084 as our OPAMP and the aforementioned RC approximated circuit with N = 7 for realizing the fractional capacitors. It should be mentioned here that this emulator emulates the novel simplest paralleled structured chaotic circuit when all fractional capacitors have been replaced by the conventional ones. If we let σ = 1 sec, α = 0.8, β = 0.95, γ = 0.85, a1 = 1.8, a3 = −3.9, b1 = 1.4, c11 = −1.5, k = 1, C = 130 mF, L = 50 mH, x(0) = 0.2 V, y(0) = 0.1 V, and z(0) = 0.1 V, the phase portraits of xt, yt, and zt can be simulated by PSPICE as depicted in Figures 30–32. The circuit parameter values for realizing this emulator are summarized in Table 3 (±15 V DC supply voltage for each OPAMP has also been assumed).Figure 29
An example of specific circuit implementation of the emulator depicted in Figure28.
(a)
(b)
(c)Figure 30
z(t) vs x(t).Figure 31
y(t) vs x(t).Figure 32
y(t) vs z(t).Table 3
Parameter values of the circuit emulator depicted in Figure29.
Parameter
Value
r
1
555.5 Ω
r
2
714.3 kΩ
r
3
1 kΩ
C
1α
1 mFsecα−1
C
2β
130μFsecβ−1
C
3γ
50μFsecγ−1
α
0.8
β
0.95
γ
0.85
g
m
1
3.9 ms
g
m
2
1.5 ms
V
supply
±15 V


## 6. Conclusion

The simplest Chua’s chaotic circuit and its novel parallel structured counterpart have been generalized in the fractional domain with the dimensional consistency awareness for the first time in this work. The analysis has been performed where the chaotic behaviors can be observed. The conditions for both which cease to be chaotic have also been determined. With these conditions, the behaviors of the circuits can be controlled. We have found that the inclusion of time dimensional consistency awareness significantly affects the behaviors of the fractional domain circuit. This is because different Lyapunov exponent spectra/dimension and condition for the circuit which cease to be chaotic can be obtained when such consistency awareness has been taken into account. The dynamical analyses including Hopf bifurcation analyses of these circuits have also been done where their conditions for being nonchaotic have been verified. It has been found that both circuits have undergone Hopf bifurcations through their equilibrium. Moreover, the their emulators have also been realized and simulated by PSPICE. At this point, it can be seen that this work has been found to be beneficial to the analysis and design of fractional chaotic circuit along with the related research areas.



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