2. Description of the Switched Generalized Function Projective Synchronization and Two New Hyperchaotic Systems

Consider the following drive and response systems:

$$\begin{cases} \dot{x} = f(x) \\ \dot{y} = g(y) + u(t, x, y) \end{cases} \tag{1}$$

where x, y <sup>∈</sup> <sup>R</sup><sup>n</sup> are the state vectors, <sup>f</sup>(x), g(x) : <sup>R</sup><sup>n</sup> <sup>→</sup> <sup>R</sup><sup>n</sup> are differentiable vector functions, and u(t, x, y) is the controller vector to be designed.

The error states between the drive and response systems are defined as

$$e\_i = y\_i - \phi\_i(x)x\_j, (i, j = 1, 2, \dots, n, \ i \neq j) \tag{2}$$

where <sup>φ</sup>i(x) : <sup>R</sup><sup>n</sup> <sup>→</sup> <sup>R</sup>(<sup>i</sup> = 1, <sup>2</sup>, ..., n) are scaling function factors, and are continuous differentiable bounded , which compose the scaling function matrix φ(x), φ(x) = diag{φ1(x), φ2(x), ..., φn(x)}.

Definition 1. For the two systems described in Equation (1), we say that they are switched generalized function projective synchronous with respect to the scaling function matrix φ(x) if there exists a controller vector u(t, x, y) such that

$$\lim\_{t \to \infty} \|e\_i\| = \lim\_{t \to \infty} \|y\_i - \phi\_i(x)x\_j\| = 0, (i, j = 1, 2, \dots, n, i \neq j) \tag{3}$$

which implies that the error dynamic system (2) between the drive and response systems is globally asymptotically stable.

Remark 1. For the SGFPS, we define i = j in the above Equation (3). If i = j, the SGFPS degenerates to the GFPS [25].

Recently, Li *et al*. [45] proposed a new hyperchaotic Lorenz-type system described by

$$\begin{cases} \dot{x} = a(y - x) \\ \dot{y} = bx - xz - cy + w \\ \dot{z} = xy - dz \\ \dot{w} = -ky - rw \end{cases} \tag{4}$$

where a, b, c, d, k and r are positive constant system parameters. When a = 12, b = 23, c = 1, d = 2.1, k = 6 and r = 0.2, and with the initial condition [1, 2, 3, 4]<sup>T</sup> , system (4) is hyperchaotic and its attractor is shown in Figure 1.

Lately, Dadras *et al*. [46] reported the following four-wing hyperchaotic system, which has only one unstable equilibrium

$$\begin{cases} \dot{x} = ax - yz + w \\ \dot{y} = xz - by \\ \dot{z} = xy - cz + xw \\ \dot{w} = -y \end{cases} \tag{5}$$

where a, b and c are positive constant system parameters. When a = 8, b = 40 and r = 14.9, and with the initial condition [10, 1, 10, 1]<sup>T</sup> , system (5) is hyperchaotic and its attractor is shown in Figure 2.

Figure 1. Hyperchaotic attractor of system (4) with a = 12, b = 23, c = 1, d = 2.1, k = 6 and r = 0.2: (a) x − y − z space; (b) x − y plane; (c) x − z plane; (d) x − w plane.

Figure 2. Hyperchaotic attractor of system (5) with a = 8, b = 40 and r = 14.9: (a) x − y − z space; (b) x − y plane; (c) x − z plane; (d) y − w plane.

For more information on the dynamical behaviors of these two systems, please refer to [45,46].
