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.