4. Numerical Results
In this section, we present complex dynamics generated by fractal-fractional-order system of memristor with interesting characteristics based on the graphical illustration so called chaotic behavior say figures (2-10).The various chaotic attractors have been demonstrated by fractal-fractional-order system of memristor by the numerical simulations based on the three types of fractal-fractional operators namely Atangana-Baleanu, Caputo-Fabrizio and Caputo. From comparison point of view, the fractal-fractional mathematical operators have played their important roles in capturing some hidden chaotic behaviors that could not be revealed by non-fractional operators. It is observed from Figs. (2-10) that the differences and similarities within the behavior of the solution of the attractors have generated rich dynamics for fractal-fractional-order system of memristor. Figures from (2-10) are prepared by invoking the control parameters as\(R0\ =1k\Omega,R1\ =500\Omega,R2\ =1k\Omega,R3\ =1\ k\Omega,\ C1\ =100\ nF\)and \(a=2,\ b=1,\ c=0.2,\ k=0.92,\ m=-0.002,\ n=0.04\)subject to initial conditions say\(x\left(0\right)=y\left(0\right)=z\left(0\right)=u\left(0\right)=0.01.\)Fig. 2 is depicted for chaotic behaviors of memristor given by numerical scheme of Caputo fractal-fractional operator keeping (fractional parameter) \(\xi_{1}=1\) and (fractal parameter) \(\eta_{1}=0.98\)at \(t=400\). Fig. 3 illustrates the chaotic behaviors of memristor given by numerical scheme of Caputo fractal-fractional operator keeping (fractional parameter) \(\xi_{1}=0.99\) and (fractal parameter)\(\eta_{1}=1\) at \(t=900\). In order to disclose the hidden phenomenon, we presented Fig. 4 for three dimensional chaotic behaviors of memristor given by numerical scheme of Caputo fractal-fractional operator keeping (fractional parameter) \(\xi_{1}=0.99\) and (fractal parameter) \(\eta_{1}=0.98\) at \(t=300\). Fig. 5-7 elucidates the chaotic behaviors of memristor given by numerical scheme of Caputo-Fabrizio fractal-fractional operator; in which we varied fractional parameter \((\xi_{1}=0.99)\) and kept fractal parameter equal to one \((\eta_{1}=1)\) while reciprocally, we kept fractional parameter equal to one \((\xi_{1}=1)\) and varied fractal parameter. Such chaotic behaviors can be seen in in Fig. 5 and 6. Meanwhile, we varied fractional parameter as well as fractal parameter\(\left(\xi_{1}=0.99,,\eta_{1}=0.98\right)\) in Fig. 7. The similar trend is employed in Figs. (8-10)
Which present chaotic behaviors of memristor given by numerical scheme of Atangana-Baleanu fractal-fractional operator.