**3. Circuit Design of the Proposed Chaotic Flow**

In recent years, the physical realizations of theoretical chaos forms have been investigated extensively for approving the feasibility and employing them in practical usages [47–50]. Therefore, in this section, a circuit realization with the hyperbolic sinusoidal nonlinearity is presented. For the reason of easiness, the general design methodology is applied according to the operational amplifiers [51,52]. The circuit is designed by using the common electronic components as displayed in Figure 10. There are an inverting amplifier (U4), three integrators (U1–U3), and two analog multipliers (U7, U8) of type AD633. The circuit for simulating the hyperbolic sinusoidal nonlinearity, in the dotted frame, includes three resistors (*R*S1–*R*S3), two operational amplifiers (U5, U6) and two diodes (*D*1, *D*2).

Based on Figure 8, via the Kirchhoff's laws, the circuital equation of the circuit is found as

$$\begin{array}{l} \dot{X} = \frac{1}{RC} \Big[ X - \frac{R}{R\_1 V} YZ + V\_b \Big]\_{\prime} \\ \dot{Y} = \frac{1}{RC} \Big[ \frac{XZ}{1V} - \frac{2I\_S R\_S R}{R\_4} \sinh\left(\frac{R\_{S2}}{nV T R s\_1}\right) + V\_c \Big]\_{\prime} \\ \dot{Z} = \frac{1}{RC} X \end{array} \tag{9}$$

where *IS*, *n* and *VT* are diode's reverse bias saturation current, the diode's ideality factor, and the thermal voltage, correspondingly. Normalizing the Equation (9) with τ = *t*/*RC*, the dimensionless structure can be designated by

$$\begin{array}{c} \dot{X} = X - \frac{R}{R\_1 \Gamma V} YZ + V\_{b\prime} \\ \dot{Y} = \frac{XZ}{\Gamma V} - \frac{2I\_S R\_{S1} R}{R\_4} \sinh\left(\frac{R\_{S2}}{nV\_T R\_{S1}}\right) + V\_{c\prime} \\ \dot{Z} = X \end{array} \tag{10}$$

The variables (*X*, *Y*, *Z*) are equivalent to output voltages of integrators (U1–U3), when the power supply is ±15 VDC. The system (10) corresponds to the suggested system with the hyperbolic sinusoidal nonlinear function (2). The electronic components are selected for *a* = 2, and *b* = *c* = 0; then we have *R* = *R*<sup>4</sup> = 30 kΩ, *R*<sup>1</sup> = 15 kΩ, *R*<sup>2</sup> = 10 kΩ, *R*<sup>3</sup> = 90 kΩ, *R*S1 = 100 kΩ, *R*S2 = 50.66 kΩ, *R*S3 = 18.65 MΩ and *C* = 10 nF. The planned circuit of Figure 10 has been executed in Multisim, and some PSpice results are presented in Figure 11. One can obviously confirm the consistency of the simulations (Figure 11) and numerical outcomes (Figure 2).

**Figure 10.** Schematic of circuit simulation for the system with hyperbolic sinusoidal nonlinear function (2).

**Figure 11.** *Cont.*

**Figure 11.** PSpice chaotic attractors of system with hyperbolic sinusoidal nonlinearity in (**a**) *X*–*Y* plane, (**b)** *X*–*Z* plane, and (**c**) *Y*–*Z* plane (*x*: 2V/Div, *y*: 2V/Div).
