**2.1 The genesis of chaotic systems with hyperbolic sine nonlinearity**

In 2011, Sprott and Munmuangsaen proposed an exponential chaotic system [14], which happens to be an example of the simplest chaotic system [15]. In the same year, Sprott used common resistors, capacitors, operational amplifiers, and a diode to successfully implement this system in a circuit [16]. Few years later, the simplest hyperbolic sine chaotic system is proposed [17]. Compared to the exponential chaotic system, the hyperbolic sine chaotic system changed the nonlinearity from exponential function (asymmetric function) to hyperbolic sine function (symmetric function), which can exhibit symmetry breaking, and offers the possibility that attractors will split or merge as some bifurcation parameter is changed [18].

The simplest chaotic system with a hyperbolic sine is described as follows:

$$\ddot{\mathbf{x}} + c\ddot{\mathbf{x}} + \mathbf{x} + \rho \ast \sinh\left(\rho \dot{\mathbf{x}}\right) = \mathbf{0} \tag{1}$$

**Figure 1.** *The corresponding circuit schematic diagram of Eq. (1).*

*Chaotic Systems with Hyperbolic Sine Nonlinearity DOI: http://dx.doi.org/10.5772/intechopen.94518*

Where c is considered as the bifurcation parameter, sinh ð Þ¼ *<sup>φ</sup>x*\_ *<sup>e</sup>φx*\_ �*e*�*φx*\_ <sup>2</sup> , *<sup>ρ</sup>* <sup>¼</sup> <sup>1</sup>*:*<sup>2</sup> <sup>∗</sup> <sup>10</sup>�<sup>6</sup> and *<sup>φ</sup>* <sup>¼</sup> <sup>1</sup> 0*:*026, which have been chosen to facilitate circuit implementation using diodes. The corresponding circuit schematic diagram of Eq. (1) is shown as **Figure 1**.

When *c* ¼ 0*:*75, the Eq. (1) can exhibit chaotic behavior, which is shown as **Figure 2**.

**Figure 2.** *Numerical and actual circuit state space plot in x* � *x plane.* €
