**3.2 Simple chaotic system with hyperbolic sine nonlinearity**

The other way to construct the derived chaotic systems is to simplify the known chaotic systems. For example, if we remove the parameter *ρ* and *φ*, search the parameter space, we will have the following chaotic system:

$$\begin{cases} \dot{\mathbf{x}}\_1 = \mathbf{6x}\_2 - \mathbf{x}\_1 \\\\ \dot{\mathbf{x}}\_2 = \mathbf{x}\_3 \\\\ \dot{\mathbf{x}}\_3 = \mathbf{x}\_4 \\\\ \dot{\mathbf{x}}\_4 = -\mathbf{x}\_4 - \sinh\left(\mathbf{x}\_3\right) - \mathbf{x}\_1 \end{cases} \tag{7}$$

When initial conditions are set to be ð Þ¼ *x*1, *x*2, *x*3, *x*<sup>4</sup> ð Þ 0*:*7, 0*:*9, 1*:*0, 1*:*3 , or ð Þ¼ � *x*1, *x*2, *x*3, *x*<sup>4</sup> ð Þ 0*:*7, �0*:*9, �1*:*0, �1*:*3 , the system exhibits period behavior. When the initial conditions are set to be ð Þ¼ *x*1, *x*2, *x*3, *x*<sup>4</sup> ð Þ 7, 9, 10, 13 and ð Þ¼ � *x*1, *x*2, *x*3, *x*<sup>4</sup> ð Þ 7, �9, �10, �13 , the system exhibits chaotic behavior. Therefore, this system has four coexistence attractors [19], as shown in **Figure 8**.
