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

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. /e 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./e 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. /is is because the time dimensionalconsistency has been included. /e 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.


Introduction
e 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][2][3][4][5][6][7][8][9][10][11]. e 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][13][14][15][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][18][19][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]. erefore, their fractional domain generalizations have been proposed in many literatures [23][24][25]. Unfortunately, the dimensional consistency [26], which is often cited in many recent studies on the fractional domain linear circuit generalizations [27][28][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. e 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]. is is because different Lyapunov exponents and dimension can be obtained. e 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. is is because the time dimensionalconsistency has been included. e 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 Sections 3 and 4. e 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.

Overview of the Modified Riemann-Liouville Fractional Derivative
In 2006, Jumarie proposed a modified version of Riemann-Liouville fractional derivative [31]. e proposed derivative can be mathematically defined as follows [31,32].
where D α t 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].
e Laplace transformation of this modified fractional derivative depends on α. As an example,

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. e 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. erefore, it has been found to be of interest. e schematic diagram of simplest Chua's chaotic circuit is depicted in Figure 1.
Moreover, it has been assumed that the memristor employs the following state equation and memristance: d dt where x(t), M(x(t)), A, and B denote the state variable, the memristance, and the memristor's parameters, respectively. Moreover, i M (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: 2 Mathematical Problems in Engineering where C, L, i L (t), and v C (t) denote the capacitance value, inductance value, current flowing through the inductor, and voltage drop across the capacitor, respectively. Note also that

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. According to [30], the memristor employs the following state equation: where a 1 , a 3 , b 1 , c 11 , and v m (t) 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: Note that v L (t) � v C (t) � 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].

e 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 D α t 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]: where 0 ≤ α ≤ 1 and i L (t) � i C (t) � − i M (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 of (d/dt) and the generalized fractional derivative must be consistent which means that both of them must be given by the physically measurable sec − 1 . erefore, the following operation must be used: 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. Figure 2: e novel simplest chaotic circuit [30].

Mathematical Problems in Engineering
Since the dimension of σ is sec, that of (1/σ 1− α )D α t is also sec as the dimension of D α t is sec − α as stated above. erefore, the dimensions of (d/dt) and generalized fractional derivative, i.e., (1/σ 1− α )D α t , 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 1 After some rearrangement, we obtain where Note also that C α and L α 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 where In this work, the fractional derivative has been interpreted in the modified Riemann-Liouville sense as given by (1); thus, we have 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 i.e., where Note also that x(t) � X(ξ) where ξ � (t α /Γ(1 + α)) [32]. At this point, it can be seen that (11) has been transformed to (14). erefore, the solution of (11) can be conveniently obtained by solving (14) and keeping the above relationships between x(t) and X(ξ) 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 x(t), i L (t), and v C (t) can be simulated by also keeping in mind that i L (t) � Au(t) and v C (t) � Av(t) as depicted in Figures 3-5, where the strange attractors, which refers to the chaotic behaviors, can be observed.
For the quantitative analysis, we evaluate the corresponding Lyapunov exponents and Lyapunov dimension (D L ) by employing the following definition.

Definition 2.
For any modified Riemann-Liouville fractional derivative-based fractional order dynamical system defined by where its jth Lyapunov exponent (λj) can be found as where {j} = {1, 2, . . ., N} and u j (ξ) stands for the j th tangent vector of the system's trajectory in a ξ domain N dimensional space defined by (X 1 (ξ), X 2 (ξ), . . ., X N (ξ)). Note also that X j (ξ) � x j (t), ‖‖ denotes the Euclidian norm operator, and u j (ξ) can be obtained by simultaneously solving the nonlinear transformed of (17) and its variational equation.
Obviously, Definition 2 states that all λ j 's and D L 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 λ 1 = 0.0473804, λ 2 = − 0.0255304, and λ 3 = − 0.550074. Since λ 1 > 0 and λ 1 + λ 2 + λ 3 = − 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 λ 1 + λ 2 + λ 3 < 0; therefore, our system is dissipative. Moreover, it has been found that D L = 2.03972 which is a fractional number. erefore, 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 i.e., λ 1 + λ 2 + λ 3 = − 0.418559, and D L = 2.04414.
ese Lyapunov exponents and D L are significantly different from those of the dimensional consistency aware scenario. erefore, it can be seen that the fractional domain circuit employs different dynamic thus different chaotic behavior when the time dimensional consistency has been concerned.
is is because x(t) Mathematical Problems in Engineering 5 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 λ 1 + λ 2 + λ 3 . 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 by Since this circuit has only one equilibrium point which can be given in terms of 3-tuples, i.e., (x(t), u(t), v(t)), by E � (0, 0, 0), the resulting J E can be obtained as follows: After obtaining J E , the characteristic equation can be determined by using the following corollary.

Corollary 1. For arbitrary dynamical system, its characteristic equation can be given by
where l and I denote the eigenvalue symbol and the identity matrix.
As a result, the characteristic equation can be obtained based on (22) which yields For obtaining the chaotic behavior, E must be a saddle point of index 2 which generates the chaos in any threedimensional system [40]. us, it can be seen from (24) that the following equation [41] α must be satisfied for our circuit ceases to be chaotic. is is because the stable region of the complex plane has been enlarged so that it covers l 1 , l 2 , and l 3 ; 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., l 2 and l 3 l 2 and l 3 , will be resided in the stable region as Re[l 2 , 3 ] ≤ 0 regardless of α. Moreover, the conditions on L α and C α with which the circuit ceases to be chaotic can be obtained by using (24) as follows: where α can be arbitrary. By satisfying either of these conditions, E becomes locally asymptotically stable as both l 2 and l 3 will be moved to the stable region. erefore, all eigenvalues now reside in the stable region (as long as A > 0, which yields A α > 0, is satisfied, l 1 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 L α ≥ 29.5238 Hsec α− 1 for C � 1F and C α ≤ 0.111774 Fsec α− 1 for L � 3.3H, respectively.
In order to verify the above conditions and study the dynamic of E which governs that of the circuit with respect to the changes in circuit parameters, we formulate where {i} = {1, 2, 3}. It should be mentioned here that α, L α , and C α have been chosen as the bifurcation parameters and the effect of m i to the location of l i which governs the system dynamic is similar to that of the real part of l i if l i 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 m i 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 As a result, the dynamics of m 1 and m 2,3 can be simulated as follows.
It can be seen from Figures 6-8 that m 1 < 0 always unlike m 2 and m 3 depicted in Figures 9-11. us, the circuit's stability is solely governed by l 2 and l 3 as l 1 will always be in the stable region of the complex plane. e circuit becomes asymptotically stable if and only if α < 0.690015, L α > 29.5238 Hsec α− 1 , and C α < 0.111774 Fsec α− 1 have been assured because m 2,3 < 0 which implies that l 2 and l 3 are in the stable region and can be observed. Otherwise, m 2,3 > 0 which implies that l 2 and l 3 are in the unstable region; thus, the circuit becomes unstable and can be seen instead. e circuit becomes marginally stable when either α � 0.690015, L α � 29.5238 Hsec α− 1 or C α � 0.111774 Fsec α− 1 has been satisfied since m 2,3 � 0 can be found. is 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, L α � 29.5238 Hsec α− 1 , and where m 2,3 � 0 as stated above. Moreover, we have also found that the real eigenvalue can be given by l 1 = − 0.2 A/ sec α− 1 ≠ 0 in these circumstances. As a result, the Hopf bifurcation condition for arbitrary three-dimensional fractional order system [42] has been satisfied. us, it can be stated that the circuit's stability switches and the circuit undergo a Hopf bifurcation through E when either α = 0.690015, L α = 29.5238 Hsec α− 1 or C α = 0.111774 Fsec α− 1 has been satisfied. As an illustration, the phase portraits of i L (t) and v C (t) with L α = 29.5238 Hsec α− 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, L α ≠ 29.5238 Hsec α− 1 , and C α ≠ 0.111774 Fsec α− 1 despite the fact that l 1 ≠ 0 is satisfied (since l 1 = − 0.2 A/sec α− 1 is always based on the assumed parameter according to (24)). is is because m 2,3 ≠ 0 as can be seen from Figures 7, 9, and 11.
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. is is because (10) is not merely (3) with fractional derivatives. However, (10) can be rewritten in terms of fractional integrals as where J α t , J β t , and J c t , respectively, denote the fractional integral operators of order α, β, and c 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, L α and C α must become L β and C c , which can be, respectively, given by L β � Lσ β− 1 and C c � Cσ c− 1 , where 0 < β ≤ 1 and 0 < c ≤ 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 v(t) 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.
For realizing the fractional capacitors, we firstly approximate the impedance function of fractional capacitor generally given by Z δ (s) � C − 1 δ s − δ where C δ � Cσ δ− 1 and 0 < δ ≤ 1 as follows [45]: where N can be arbitrary positive integer. us, 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 Mathematical Problems in Engineering R 0 = K, R i = L i /M i , and C i = 1/L i . It should be mentioned here that K, L i 's, and M i 's can be computed by using the method of undetermined coefficients [45]. As a practical example, a circuit that approximates a fractional capacitor with C δ = 1 μFsec δ− 1 and δ = 0.9 can be realized by assuming that     Mathematical Problems in Engineering 100 kΩ , and 0.2v(t) 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 7 th order Oustaloup approximation-based results [45]. e parameter values of this circuit emulator can be summarized in Table 1 where the OPAMP's supply voltage (V sup ) of ± 15 V has been adopted.

e Novel Simplest
Circuit. Now, the novel simplest chaotic circuit will be considered. After the fractional domain generalization, we have 1 By defining a 1α � a 1 σ 1− α , a 3α � a 3 σ 1− α , b 1α � b 1 σ 1− α , and c 11α � c 11 σ 1− α the following dimensional consistency aware fractional domain model of the simplest paralleled structured chaotic circuit can be obtained.
After some rearrangement, the matrix vector formatted of (35) can be given as follows: where Figure 14: An example of specific circuit implementation of the emulator depicted in Figure 13. Figure 15: e RC circuit approximation of fractional capacitor [45]. 10 Mathematical Problems in Engineering Since the fractional derivative has been interpreted in the modified Riemann-Liouville sense, we have For solving (36) in a numerical manner, the nonlinear transformation has also been applied. As a result, we have d dξ i.e., where Note also that y(t) � Y(ξ). Similar to [30], we let 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 y(t) and Y(ξ) along with that of ξ and t mentioned above in mind, the phase portraits of x(t), i L (t), and v C (t) can be simulated as depicted in Figures 19-21, where the strange attractors can be observed. In addition, we have found that λ 1 � 1.37332, λ 2 � − 0.23593, λ 3 � − 12.1655, and D L � 2.09348. Since λ 1 > 0, λ 1 + λ 2 + λ 3 � − 11.02811, which is less than 0, and D L 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. e Lyapunov exponents and dimensions of this circuit and Chua's circuit are summarized in Table 2.
Similar to the simplest Chua's circuit considered in the previous subsection, we also determine the condition on α 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 (v C (t), i L (t), x(t)) as E 1 � 0, 0, 0 , 1α /a 3α ) ). Among these equilibrium points, only E 2 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 E 2 can be given by following Definition 3 as  Table 1: Parameter values of the circuit emulator depicted in Figure 14.
x(t) Figure 19: i L (t) vs x(t) (novel circuit). erefore, the characteristic equation can be obtained by following Corollary 1 as follows: which yields By using (44), we have found that the novel simplest chaotic circuit ceases to be chaotic if (a 1α + a 3α /2a 3αα ) ≤ 0 has been satisfied. is is because l 2 and l 3 will reside in the stable region of the complex plane. Moreover, this circuit also ceases to be chaotic if 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. is is because l 2 and l 3 will be moved to the stable region. Here, we let a 1 � 1.8, a 3 � − 3.9, b 1 � 1.4, c 11 � − 1.5, k � 1, σ � 1 sec, and α � 0.9; therefore, (46) and (47) become L α ≤ 8.58018 mHsec α− 1 if C � 130 mF and C α ≥ 757.56 mFsec α− 1 if L � 50 mH, respectively. At this point, we will study the dynamic of E 2 and the novel circuit and also verify the above conditions. For this circuit, it has been found that m 1 can also be given by (29) according to (44); thus, the dynamic of m 1 with respect to α similar to that depicted in Figure 6 can be obtained. On the other hand, m 2,3 can be found as As a result, the dynamics of m 1 with respect to other bifurcation parameters and those of m 2,3 can be simulated as depicted in Figures 22-26 which also imply that the circuit's stability is solely governed by l 2 and l 3 . In this case, the circuit becomes asymptotically stable when either α < 0.75317, L α < 8.58018 Hsec α− 1 or C α > 757.56 Fsec α− 1 has been assured and vice versa and become marginally stable when either α � 0.75317, L α � 8.58018 Hsec α− 1 or C α � 757.56 Fsec α− 1 has been satisfied.
is verifies (45)-(47) as the circuit becomes nonchaotic in these scenarios. On the other hand, the real eigenvalue can be given by l 1 � − 3.6 A/sec α− 1 ≠ 0 along the entire ranges of bifurcation parameters because l 1 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 m 2,3 � 0 where and as a result, the transversality condition has been established; thus, the Hopf bifurcation condition has been satisfied. erefore, the stability switches and the circuit undergoes a Hopf bifurcation through E 2 when either α � 0.75317, L α � 8.58018 Hsec α− 1 or C α � 757.56 Fsec α− 1 . As an illustration, the phase portraits of i L (t) and v C (t) with C α � 757.56 Fsec α− 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, L α ≠ 8.58018 Hsec α− 1 , and C α ≠ 757.56 Fsec α− 1 despite the fact that l 1 ≠ 0. is is because m 2,3 ≠ 0 as can be seen from Figures 22, 24, and 26.
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, L α and C α become L c and C β . We also define y(t) = v C (t) and z(t) = i L (t). As a result, we have (50) erefore, the resulting generic emulator can be obtained as depicted in Figure 28.
An example of circuit emulator implemented based on such generic one is depicted in Figure 29 where x(t), y(t), and z(t) in terms of voltages and C 1α � 1 mFsec α− 1 have been assumed. In addition, NR1 and NR 2 are floating negative resistors because a 3 < 0 and c 11 < 0. ese 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 g m1 � 0.001a 3α and g m2 � 0.001c 11α .
e 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, c � 0.85, a 1 � 1.8, a 3 � − 3.9, b 1 � 1.4, c 11 � − 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 x(t), y(t), and z(t) can be simulated by PSPICE as depicted in Figures 30-32. e circuit parameter values for realizing this emulator are summarized in Table 3 (±15 V DC supply voltage for each OPAMP has also been assumed).

Conclusion
e 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. e analysis has been performed where the chaotic behaviors can be observed. e 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. is 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. e 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 -+ C 3γ = 0.001C γ z (t) 1 kΩ y (t) -+ 100 kΩ (c) Figure 29: An example of specific circuit implementation of the emulator depicted in Figure 28.    found to be beneficial to the analysis and design of fractional chaotic circuit along with the related research areas.

Data Availability
e simulated data used to support the findings of this study are included within the article.

Conflicts of Interest
e author declares that there are no conflicts of interest regarding the publication of this article.  3.9 ms g m2 1.5 ms V supply ±15 V