1. Introduction
A memristor is the fourth passive or circuit element that has the capability for remembering its state history in power-off modes due to its nonlinear nature and plasticity properties. There are four categories of memristor (i) ionic thin film memristor, (ii) spin memristor, (iii) molecular memristor and (iv) magnetic memristor; each types of memristor has its own significance as hysteresis under the application of charge is detected by ionic thin film memristor, degree of freedom in electron is relied by spin memristor, anomalous current-voltage is exhibited by molecular memristor and a bilayer-oxide films substrate is perceived by magnetic memristor respectively [1-2]. The first fabricated physical memristor was found by Strukov et al. [2] as a missing memristor so called fourth fundamental circuit element in 2008. Bao et al. [3] presented dimensionless mathematic model based on a fifth-order chaotic circuit with two memristors. They discussed stability analysis, dynamical analysis methods and the memristor initial states. They also described transient hyperchaos state transitions with excellent nonlinear dynamical phenomena. The two memristors connected in antiparallel has been observed by Buscarino et al. [4] when a sinusoidal input is applied. Their setting for two memristors was consisted of two capacitors, an inductor, one negative resistor, two memristors connected in antiparallel in which characterization of the four embedded circuit parameters was also analyzed within dynamical behaviors. Adhikari et al. [5] exhibited role of memristors on the basis of three conditions as (i) pinched hysteresis loop when frequency tends to infinity, (ii) critical frequency decreases monotonically when excitation frequency increases and (iii) bipolar periodic signal in the voltage-current is assumed to be periodic. The three-dimensional chaotic system has been modified by Li et al. [6] in terms of four-dimensional memristive system on the basis of dissipativity and symmetry. Their main focus was to investigate the complex dynamics includes as hyperchaos, limit cycles, chaos, torus and few others. Chen et al. [7] studied classical memristive chaotic circuit with a first-order memristive diode bridge in which theoretical and numerical investigation has been displayed for complex nonlinear phenomena coexisting attractors and bifurcation modes. Zhou et al. [8] perceived the effective role of hyperchaotic multi-wing attractor in a 4D memristive circuit within complicated dynamics. Here they presented interesting controller parameters for 4D memristive circuit includes Lyapunov exponents, phase portrait, bifurcation diagram and Poincaré maps. The dynamical illustrations can be continued on memristors, we include here few latest attempts subject to chaos analysis [9-11].
Although fractional calculus is a burning field of mathematics that studies the generalization of classical concepts in mathematics and engineering via differential and integral operators yet the fractal calculus is relatively a new science of differential and integral operators based on two parameters. The fractal-fractional differentiation consists two dimensions namely one for fractional order and other for fractal order. The main significance of the fractal-fractional differentiation is to describe fractal kinetics effectively in which the fractal time is replaced into the continuous time. The fractal-fractional differentiation provides the fractal dimension through which the model can capture preferential paths for capturing the flow in fractured aquifers. It plays an extremely effective role in the phenomena of hierarchical or porous media, for instance, fractal gradient of temperature in a fractal medium [12-15]. Very recently an African Professor Atangana presented his concept of fractal-fractional differentiation based on Mittage-Leffler, exponential decay and power-law memories in which he described that fractal-fractional differentiation attracts more non-local natural problems that display at the same time fractal behaviors [16]. Atangana and Qureshi [17] captured self-similarities in the chaotic attractors based on the basis of three numerical schemes for systems of nonlinear differential equations. Their investigated dynamical systems containing the general conditions for the existence and the uniqueness have been explored. Gomez-Aguilar [18] presented the Shinriki’s oscillator model for the prediction of chaotic behaviors related to the fractal derivative in convolution with power-law, exponential decay law and the Mittag-Leffler function in which the Adams-Bashforth-Moulton scheme has been invoked for the numerical simulations at symmetric and asymmetric cases. Sania et al. [19] employed the concept of fractal-fractional operators presented in [16] for investigating the chaotic behaviors the Thomas cyclically symmetric attractor, the King Cobra attractor, Rossler attractor, the Langford attractor, the Shilnikov attractor. They claimed that new strange behaviors of the attractors have been which were impossible by fractional and classical differentiations. In short, the study can be continued for the charming and effective role of fractional calculus but we include here recent attempt therein [20-31]. Motivating by above discussion, our aim is to propose the controlling analysis and coexisting attractors provided by memristor through highly non-linear for mathematical relationships of governing differential equations. The mathematical model of memristor is established in terms of newly defined fractal-fractional differential operator so called Caputo-Fabrizio fractal-fractional differential operator. A novel numerical approach is developed for the governing differential equations of memristor on the basis Caputo-Fabrizio fractal-fractional differential operator. We discussed chaotic behavior of memristor under three criteria as (i) varying fractal order, we fixed fractional order, (ii) varying fractional order, we fixed fractal order and (ii) varying fractal and fractional orders simultaneously. Our investigated graphical illustrations and simulated results via MATLAB for the chaotic behaviors of memristor suggest that newly presented Caputo-Fabrizio fractal-fractional differential operator has generates significant results as compared with classical approach.