**2. Model Description**

A 3-D Lorenz-like chaotic system is proposed by Cang et al [39], which is,

$$\begin{cases} \dot{\mathbf{x}} = \ -ay - xz, \\ \dot{y} = \ -x + xz, \\ \dot{z} = \ -d - xy. \end{cases} \tag{1}$$

System (1) has a simple rotational symmetric structure with six terms. Based on system (1), a new hyperchaotic system is proposed as,

$$\begin{cases} \dot{\mathbf{x}} = \ -ay - \mathbf{x}\mathbf{z} - \mathbf{u},\\ \dot{\mathbf{y}} = \ -\mathbf{c}\mathbf{x} + \mathbf{x}\mathbf{z},\\ \dot{\mathbf{z}} = \ -b - \mathbf{m}\mathbf{x}y,\\ \dot{\mathbf{u}} = \ k\mathbf{x} - \mathbf{y}. \end{cases} \tag{2}$$

where *x*, *y*, *z*, *u* are system variables, and *a*, *b*, *c*, *k* are bifurcation parameters of system (2). When *a* = 5, *b* = 4, *c* = 1, *k* = 0.5 and *m* = 1, system (2) has a hyperchaotic attractor with Lyapunov exponents (0.3606, 0.1222, 0, −1.4827) and a Kaplan-Yorke dimension of *D*KY = 3.3256 under initial conditions (1, −1, −1, 1), as shown in Figure 2.

**Figure 2.** Hyperchaotic attractor of system (2) with *a* = 5, *b* = 4, *c* = 1, *k* = 0.5, *m* = 1 and initial conditions [1, −1, −1, 1]: (**a**) *x-y* plane, (**b**) *x-z* plane, (**c**) *y-z* plane, (**d**) *x-u* plane.

The hyper-volume contraction is

$$
\nabla V = \frac{\partial \dot{\mathbf{x}}}{\partial \mathbf{x}} + \frac{\partial \dot{y}}{\partial y} + \frac{\partial \dot{z}}{\partial z} + \frac{\partial \dot{u}}{\partial u} = -z \tag{3}
$$

When *a* = 5, *b* = 4, *c* = 1 and *k* = 0.5, the dissipative curve of Equation (3) is as shown in Figure 3. The negative average of ∇*V* proves that system (2) is dissipative.

**Figure 3.** Dissipative curve of system (2).
