**2.2 The general equations of generating chaotic systems with hyperbolic sine nonlinearity**

It is obvious that Eq. (1) can be written in the form with jerk equations:

$$\begin{cases}
\dot{\mathbf{x}}\_1 = \mathbf{x}\_2 \\
\dot{\mathbf{x}}\_2 = \mathbf{x}\_3 \\
\dot{\mathbf{x}}\_3 = -c\mathbf{x}\_3 - f(\mathbf{x}\_2) - \mathbf{x}\_1
\end{cases} \tag{2}$$

where *f x*ð Þ¼ <sup>2</sup> *ρ* ∗ sinh ð Þ *φx*<sup>2</sup> . Therefore, the higher order chaotic systems with hyperbolic sine nonlinearity can be generated by adding jerk cabins, which is described by:

$$\begin{cases} \dot{\mathbf{x}}\_1 = \mathbf{x}\_2 - \mathbf{x}\_1 \\\\ \dot{\mathbf{x}}\_2 = \mathbf{x}\_3 - \mathbf{x}\_2 \\\\ \dots \\\\ \dot{\mathbf{x}}\_{n-3} = \mathbf{x}\_{n-2} - \mathbf{x}\_{n-3} \\\\ \dot{\mathbf{x}}\_{n-2} = \mathbf{x}\_{n-1} \\\\ \dot{\mathbf{x}}\_{n-1} = \mathbf{x}\_n \\\\ \dot{\mathbf{x}}\_n = -c\mathbf{x}\_n - f(\mathbf{x}\_{n-1}) - n\mathbf{x}\_{n-2} - n\mathbf{x}\_{n-3} - \dots - \frac{1}{2n}\mathbf{x}\_1 \end{cases} \tag{3}$$

where *x*\_ *<sup>k</sup>*�<sup>1</sup> ¼ *xk* � *xk*�<sup>1</sup> is the jerk cabin. With Eq. (3), we can construct nth-order (n > 3) chaotic systems with hyperbolic sine nonlinearity.

When n = 4, the equations of fourth-order chaotic systems will be:
