Fig. 1. A chaotic circuit
The chaotic circuit designed in Fig. 1 includes fewer dynamic components such as four state variables are \(i_{1}\), \(i_{2}\), \(v_{C}\), and\(q\) respectively and a capacitor, two inductors and emulator with a resistor and a negative conductance. The circuit depicted in Fig. 1 is followed by volt-ampere characteristic of elements and Kirchhoff’s laws for generating the state equations containing nonlinear terms as written below:
\begin{equation} \frac{\text{dq}}{\text{dt}}-i_{1}=0,\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (1)\nonumber \\ \end{equation}\begin{equation} \frac{dv_{c}}{\text{dt}}+\frac{i_{1}}{C}-\frac{i_{2}}{C}=0,\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (2)\nonumber \\ \end{equation}\begin{equation} \frac{di_{2}}{\text{dt}}+\frac{v_{c}}{L_{2}}+\frac{i_{2}}{\text{GL}_{2}}=0,\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (3)\nonumber \\ \end{equation}\begin{equation} \frac{di_{1}}{\text{dt}}-\frac{v_{c}}{L_{1}}+\frac{v_{m}\left(q\right)}{L_{1}}+\frac{Ri_{1}}{L_{1}}=0.\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (4)\nonumber \\ \end{equation}
Here, the terminal voltage is symbolized by \(v_{m}\left(q\right)\). For replacing the terminal voltage say \(v_{m}\left(q\right)\) by applying the voltage across the memristor on the state equations (1-4) containing nonlinear terms; take place as
\begin{equation} \frac{\text{dq}}{\text{dt}}-i_{1}=0,\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (5)\nonumber \\ \end{equation}\begin{equation} \frac{dv_{c}}{\text{dt}}+\frac{i_{1}}{C}-\frac{i_{2}}{C}=0,\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (6)\nonumber \\ \end{equation}\begin{equation} \frac{di_{2}}{\text{dt}}+\frac{v_{c}}{L_{2}}+\frac{i_{2}}{\text{GL}_{2}}=0,\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (7)\nonumber \\ \end{equation}\begin{equation} \frac{di_{1}}{\text{dt}}-\frac{v_{c}}{L_{1}}+\frac{Ri_{1}}{L_{1}}+\frac{1}{L_{1}}\left(\frac{R_{2}\ R_{0}^{2}\text{\ q}}{10C_{1}R_{3}R_{1}}-R_{3}^{-1}R_{2}R_{0}\right)=0.\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (8)\nonumber \\ \end{equation}
In order to bring the dynamic equation of the system, the time scale transformation is being carried out by keeping in mind the following transformations as
\begin{equation} \begin{matrix}a=L_{2}L_{1}^{-1},b=L_{2}C^{-1},c=R,k=G^{-1},m=-R_{3}^{-1}R_{2}R_{0},n=R_{2}R_{0}^{2}\left(C_{1}R_{3}R_{1}\right)^{-1}\\ u=L_{2}^{-1}q,z=i_{2},t=d\tau L_{2},x=i_{1},\ y=v_{c}\text{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }\\ \end{matrix},\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (9)\nonumber \\ \end{equation}
The parameters involved in equation (9) for implementing the memristor are specified as\(R0\ =1k\Omega,R1\ =500\Omega,R2\ =1k\Omega,R3\ =1\ k\Omega,\ C1\ =100\ nF\). Invoking equation (9) among equations (5-8), we arrive at the simplified system of evolutionary differential equations containing nonlinear terms:
\begin{equation} \begin{matrix}\frac{\text{dx}}{\text{dt}}-a\left(nux+mx+cx-y\right)=0,\\ \begin{matrix}\frac{\text{dy}}{\text{dt}}+bx-bz=0,\\ \begin{matrix}\frac{\text{dz}}{\text{dt}}+y-kz=0,\\ \frac{\text{du}}{\text{dt}}-x=0,\\ \end{matrix}\\ \end{matrix}\\ \end{matrix},\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (10)\nonumber \\ \end{equation}
subject to the initial conditions,
\begin{equation} x\left(0\right)=y\left(0\right)=z\left(0\right)=u\left(0\right)=0.01,\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (11)\nonumber \\ \end{equation}
The chaotic phenomena and the phase portraits can be obtained by specification of embedded parameters in equation (10), as\(a=2,\ b=1,\ c=0.2,\ k=0.92,\ m=-0.002,\ n=0.04\). Developing the system of evolutionary differential equation containing nonlinear terms say (10) in terms of the new idea of fractal-fractional differential operator, we transferred governing nonlinear differential equation (10) of memristor as:
\begin{equation} \begin{matrix}\mathfrak{D}_{t}^{\alpha,\ \ \beta}x(t)-a\left(nux+mx+cx-y\right)=0,\\ \begin{matrix}\mathfrak{D}_{t}^{\alpha,\ \ \beta}y(t)+bx-bz=0,\\ \begin{matrix}\mathfrak{D}_{t}^{\alpha,\ \ \beta}z(t)+y-kz=0,\\ \mathfrak{D}_{t}^{\alpha,\ \ \beta}u(t)-x=0,\\ \end{matrix}\\ \end{matrix}\\ \end{matrix},\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (12)\nonumber \\ \end{equation}
Here, \(\mathfrak{D}_{t}^{\alpha,\ \ \beta}x(t)\),\(\mathfrak{D}_{t}^{\alpha,\ \ \beta}y(t)\),\(\mathfrak{D}_{t}^{\alpha,\ \ \beta}z(t)\) and\(\mathfrak{D}_{t}^{\alpha,\ \ \beta}u(t)\) represent the fractal-fractional differential operators.