**3. Derived chaotic systems/torus-chaotic system with hyperbolic sine nonlinearity**

#### **3.1 Multi-nonlinearities hyperbolic sine chaotic system**

One way to construct the derived chaotic systems is to add more nonlinear terms of the equations. For example, the new chaotic system can be constructed by Eq. (4), which is described as follows:

$$\begin{cases} \dot{\mathbf{x}}\_1 = \mathbf{x}\_2 - \rho \sinh(\rho \mathbf{x}\_1) \\\\ \dot{\mathbf{x}}\_2 = \mathbf{x}\_3 - \mathbf{0}.3\mathbf{x}\_2 - \rho \sinh(\rho \mathbf{x}\_2) \\\\ \dot{\mathbf{x}}\_3 = \mathbf{x}\_4 \\\\ \dot{\mathbf{x}}\_4 = -\mathbf{0}.25\mathbf{x}\_4 - \rho \sinh(\rho \mathbf{x}\_3) - \mathbf{0}.5\mathbf{x}\_2 - 4\mathbf{x}\_1 \end{cases} \tag{6}$$

Where *<sup>ρ</sup>* <sup>¼</sup> <sup>1</sup>*:*<sup>2</sup> <sup>∗</sup> <sup>10</sup>�6, *<sup>φ</sup>* <sup>¼</sup> <sup>1</sup> <sup>0</sup>*:*<sup>026</sup> . These equations can exhibit chaotic behavior as shown in **Figure 7**.

**Figure 7.** *Numerical phase space plot of Eq. (6).*
