Strange Fractal Attractors and Optimal Chaos of Memristor–Memcapacitor via Non-local Differentials

The multi-dimensional electronic devices are so called memory circuit elements (memristor or memcapacitor); such memory circuit elements usually rely on previous applied voltage, current, flux or charge based on memory capability with their resistance, capacitance or inductance. In view of above fact, this manuscript investigates the non-integer modeling of memristor–memcapacitor in discrete-time domain through non-singular kernels of fractal fractional differentials and integrals operators. The governing equations of memristor–memcapacitor have been developed for the sake of the dynamical characteristics of simple chaotic circuit. The fractal fractional differentials and integrals operators have been invoked for non-integer modeling of memristor–memcapacitor that can exhibit a combination of dynamical chaotic phenomena. The numerical schemes, numerical simulations, stability analysis and equilibrium points have been highlighted in detail. The comparative chaotic graphs have been discussed in three ways (i) by keeping fractal component fixed and varying fractional component distinctly, (ii) by keeping fractional component fixed and varying fractal component distinctly and (iii) by varying both fractal component and fractional component distinctly. Our results suggest that fractal-fractional model of memristor–memcapacitor retains the memory characteristics.


Introduction
It is well established fact that the built-in memory-properties of mem-elements in the circuit theory play significant role in the condensed matter and engineering instruments based on promising potential applications. Dr. Chua extended his theory to all circuit elements with memory namely memcapacitors, memristors, and meminductors [1][2][3]. Ventra et al. [4] considered pinched hysteretic loops in the two constitutive variables for memristive system to capacitive and inductive elements. They focused that the dynamical properties of electrons and ions depend on the history of the system and time scales. Pershin and Ventra [5] examined the functional effects of memristors, memcapacitors and meminductors (resistors, capacitors, inductors with memory). Wu et al. [6] explored the nonlinear characteristics of the memristor the nonlinear characteristics of the memristor from which a type of heart-shaped attractors was generated. Additionally, they investigated dissipation and the stability of the equilibrium point when the polarity of the memristor is changed. Xu et al. [7] examined meminductor model for exploring its characteristics based on complex nonlinear phenomena. The complex nonlinear phenomena lied on two different kinds of chaotic transients, coexisting attractors and coexisting bifurcation modes. Ye et al. [8] presented two fluxcontrolled memristors and a charge-controlled memristor within stable and unstable regions in which they emphasized the 2D and 3D complexity characteristics with multiple varying parameters were analyzed. Nariman et al. [9] suggested fractionalized order mem-capacitor and mem-inductor using different combinations. For the finding the hysteresis loop area and the location of the pinched point through fractional technique, they envisaged to validate the theoretical findings. Nowadays, various types of mathematical definitions of fractional calculus are utilized for physical modeling. Due to this reason, fractional modeling of boundary value problems through theory of fractional differentials and integrals have played significant role in science [10][11][12][13] and engineering [14][15][16].
In brevity, this manuscript investigates the non-integer modeling of memristor-memcapacitor in discrete-time domain through non-singular kernels of fractal fractional differentials and integrals operators. The governing equations of memristor-memcapacitor have been developed for the sake of the dynamical characteristics of simple chaotic circuit. The fractal fractional differentials and integrals operators have been invoked for non-integer modeling of memristor-memcapacitor that can exhibit a combination of dynamical chaotic phenomena. The numerical schemes, numerical simulations, stability analysis and equilibrium points have been highlighted in detail. The comparative chaotic graphs have been discussed in three ways (i) by keeping fractal component fixed and varying fractional component distinctly, (ii) by keeping fractional component fixed and varying fractal component distinctly and (iii) by varying both fractal component and fractional component distinctly. Our results suggest that fractal-fractional model of memristor-memcapacitor retains the memory characteristics.

Mathematical Definition of the Memristor
Chua and Kang summarized the definition of the memristor as expressed in Eqs. (1a, 1b) and the reciprocal expression of the memcapacitance consisting of ideal chargecontrolled memcapacitor is illustrated as: For Eqs. (1a) and (1b), x and y denote the memristor's state variable and (x, y, t) and H (x, y, t) are the functions of memristor's state variable and time. Here, i M and v M denote the current and voltage across the memristor and a, b, c, d and e are the constants.

Mathematical Definition of the Memcapacitor
The reciprocal expression of the memcapacitance, based on the definition of ideal charge-controlled memcapacitor, is defined While for Eq. (2a), q(t) and u(t) are the charge and the corresponding voltage of the memcapacitor at time t respectively. And α and β constant coefficient of the memcapacitor, the charge q passes the memcapacitor by σ . For designing a chaotic Here the functional parameters are described for Eq. (2b) as: a, b, c, d and e are the constants and i M and v M denotes the current and voltage across the memristor. In this context, assuming the voltage of the memristor is a sinusoidal signal and supposing that the memcapacitor's charge is a sinusoidal signal in simple chaotic circuit as shown in Fig. 1. Figure 1 is the configuration of memristor-memcapacitor-based simple chaotic circuit. The governing equations of the system based on variables q as a flux, δ as a state variable memcapacitor, y as a state variable of memristor and i L as a current can be written as: Applying non-dimensional parameters The functional numerical values for different embedded parameters of Eq. (4) are: The step size is taken as 0.001 and the initial conditions are (0, 3.1, 1.31, 57). Equation (4) is the classical and evolutionary system of differential equation for chaotic circuit based on memristor-memcapacitor that the dynamical characteristics of simple chaotic circuit as depicted in Fig. 1. In order to develop the governing Eq. (4), we invoke the newly defined fractal-fractional integral and differential operator as:

Stability Analysis and Equilibrium Points
It is established fact that performance and efficiency of fractal-fractionalized differential equation for chaotic circuit based on memristor-memcapacitor depends on the stability criteria. The system of fractal-fractionalized differential equation for chaotic circuit based on memristor-memcapacitor can be checked as dissipative system by the divergence of fractal-fractionalized differential equation for chaotic circuit based on memristor-memcapacitor. The divergence formula is: Invoking the rheological parameters F 0 0.01, F 1 1, F 2 1, F 6 0.001, F 7 0.005, F 4 0.5, F 4 2, F 5 4 and initial conditions η 1 (0) 0, η 2 (0) 3.1, η 3 (0) 1.31, η 4 (0) 57 in Eq. (23). After substituting such parameters in Eq. (23), we have result less than zero. This result represents that the system may have chaotic attractors because the system is dissipative. Now the set of equilibrium point for finding the eigenvalues depending upon Routh-Hurwitz stability criterion can be investigated through Jacobi matrix. Before discussing Jacobi matrix, we set fractal-fractionalized differential equations for chaotic circuit based on memristor-memcapacitor is equal to d K 3 , K 4 η 1 Now the Jacobian matrix is obtained from Eq. (24), Here the line equilibrium set is P(0, 0, 0, n); that indicates fractal-fractionalized differential equations for chaotic circuit based on memristor-memcapacitor has infinite equilibrium. Now computing the characteristic equation from Eq. (25), we arrive at: Here, λ 0 . Now evaluating non-linear algebraic Eq. (26) through synthetic division method, it is concluded that fractal-fractionalized differential equations for chaotic circuit based on memristor-memcapacitor has three nonzero eigenvalues and one zero eigenvalue. This represents that system is stable and system will be likely to create chaos.

Discussion of Results Through Chaotic Phenomenon
In this section, we describe complex dynamic behaviors for chaotic circuit based on memristor-memcapacitor that contain chaos phenomenon, attractors, and differ-  Figs. 2, 3, 4, 5, and 6 are depicted for fractional chaos of memristor-memcapacitor at K 1 0.899 and K 2 1 for current (η 1 ) and state variable memristor (η 2 ) via Caputo-Fabrizio (CF) and Atangana-Baleanu (AB) methods, fractal chaos of memristor-memcapacitor at K 1 1 and K 2 0.787 for current (η 1 ) and state variable memristor (η 2 ) via CF and AB methods, fractal-fractional chaos of memristor-memcapacitor at K 1 0.899andK 2 0.787 for current (η 1 ) and state variable memristor (η 2 ) via CF and AB methods, fractal-fractional chaos of memristor-memcapacitor at K 1 0.899 and K 2 0.787 for flux (η 3 ) and state variable memcapacitor (η 4 ) via CF and AB methods and fractal-fractional chaos of memristor-memcapacitor at K 1 0.899 and K 2 0.787 for flux (η 3 ) and state variable memcapacitor (η 4 ) via CF and AB methods. Additionally, initial condition for each parameter is the final value of the trajectory in the previous parameter. In order to examine the deformation of chaotic circuit based on memristor-memcapacitor, the outcomes have been discussed. Figure 2 reflects the comparative analysis of CF and AB fractional differential operators for current (η 1 ) and state variable memristor (η 2 ) in which the pinched hysteresis    Fractal-fractional chaos of memristor-memcapacitor at K 1 0.899 and K 2 0.787 for η 1 and η 2 via CF and AB methods at smaller interval of time curves of the non-integer orders have shown the hysteresis curve of fractional order memristor-memcapacitor under an excitation voltage. Such comparison lead that the local passive (or active) characteristics occur in non-integer modeling of memristor-memcapacitor for current (η 1 ) and state variable memristor (η 2 ). The CF and AB fractal differential operators have been contrasted in Fig. 3 for current (η 1 ) and state variable memristor (η 2 ). It is observed that chaotic states as seen in Fig. 3 are reciprocal periodic orbits that appear as the initial value changes declaring that the system has many different reciprocal coexisting attractors with respect to each fractal differential operator. The discrimination of CF and AB fractal-fractional differential operators for current (η 1 ) and state variable memristor (η 2 ) has been observed in Fig. 4 on the basis of accuracy of numerical method. The trends of CF and AB fractal-fractional differential operators are quite distinct; this is because they strongly depend on values of initial conditions and a memory ability manifested through a closed pinched hysteresis loop in the characteristics. On the contrary, the similar trends CF and AB fractal-fractional differential operators have been depicted for flux (η 3 ) and state variable memcapacitor (η 4 ) in Fig. 5 and state variable memcapacitor (η 4 ) and flux (η 3 ) in Fig. 6. Such trends reflect the dissimilarity information due to multidimensional scaling and pattern visualization. Additionally, Fig. 7 presents fractal-fractional chaos of memristor-memcapacitor at K 1 0.899andK 2 0.787 for η 1 and η 2 via CF and AB methods at smaller interval of time. On the other hand, Fig. 8 is depicted for Lyapunov exponent by means of CF and AB fractal-fractional differential operators, in which it is observed that the system exhibits chaotic state because all of the phase diagrams are extremely strong chaotic.

Conclusion
The new idea is proposed to develop the mathematical models of a memristor and a memcapacitor based on fractal and fractional differential and integral operators. The dynamical characteristics generated by fractal as well fractional operators are Fig. 8 Lyapunov exponent by means of CF and AB fractal-fractional differential operators highly complex and sensitive as the circuit parameters vary. A chaotic circuit containing a memristor and a memcapacitor is designed with different chaotic strange attractors. The numerical schemes, numerical simulations, stability analysis and equilibrium points have been highlighted in detail. The comparative chaotic graphs have been discussed in three ways (i) by keeping fractal component fixed and varying fractional component distinctly, (ii) by keeping fractional component fixed and varying fractal component distinctly and (iii) by varying both fractal component and fractional component distinctly. After above discussion and concluding remarks, the following results have been accumulated as: • The current and state variable memristor have pinched hysteresis curves at the non-integer orders that show the hysteresis curve of fractional order memristor-memcapacitor under an excitation voltage. • The chaotic states are reciprocal and have periodic orbits that appear as the initial value changes declaring that the system has many different reciprocal coexisting attractors with respect to each fractal differential operator. • The Caputo-Fabrizio fractal-fractional differential operator and Atangana-Baleanu fractal-fractional differential operator have quite distinct chaotic trends depending upon the values of initial conditions. • The flux (η 3 ) and state variable memcapacitor (η 4 ) reflect the dissimilarity information due to multidimensional scaling and pattern visualization.