### 2.1 Invariance

The positive *z*-axis, *u*-axis, and *ω*-axis are invariant under the flow, that is, they are positively invariant under the flow generated by system (2). However, this is not the case on the positive *x*-axis, *y*-axis, and *v*-axis for system (2) since they are all not positively invariant under the flow generated by system (2).

### 2.2 Ultimate bound set and domain of attraction

In this section, we further investigate the ultimate bound set and global domain of attraction of the high-order Lorenz-Stenflo system (2). The main result is described by the following theorems, Theorems 1 and 2.

### Theorem 1

*For any*
\(\lambda_{1} > 0\), \(m > 0\), \(\sigma > 0\), \(s > 0\), \(r > 0\), \(b > 0\), *there exists a positive number*
\(M > 0\)
*such that*

$$ \Psi = \bigl\{ X \mid \lambda_{1} ( x - m_{2} ) ^{2} + my^{2} + m ( z - 2 \lambda_{2} ) ^{2} + \lambda_{1}s ( v - m_{3} ) ^{2} + mu^{2} + m ( \omega - \lambda_{2} ) ^{2} \le M \bigr\} $$

*is the ultimate bound and positively invariant set of the high*-*order Lorenz*-*Stenflo system* (2), *where*
\(X ( t ) = ( x ( t ) ,y ( t ) ,z ( t ) ,v ( t ) ,u ( t ) ,\omega ( t ) )\).

### Proof

Define the following Lyapunov-like function

$$ V ( X ) = \lambda_{1} ( x - m_{2} ) ^{2} + my^{2} + m ( z - 2\lambda_{2} ) ^{2} + \lambda_{1}s ( v - m_{3} ) ^{2} + mu^{2} + m ( \omega - \lambda_{2} ) ^{2}, $$

(3)

where \(\forall \lambda_{1} > 0\), \(\forall m > 0\), \(\lambda_{2} = \frac{ \lambda_{1}\sigma + mr}{2m}\), \(X ( t ) = ( x ( t ) , y ( t ) , z ( t ) , v ( t ) , u ( t ) , \omega ( t ) ) \), and \(m_{2} \in R\), \(m_{3} \in R\) are arbitrary constants.

We have

$$\begin{aligned}& \frac{dV( X( t ) ) }{dt}\bigg| _{(2)} \\& \quad = 2\lambda_{1} ( x - m_{2} ) \frac{dx}{dt} + 2my \frac{dy}{dt} + 2m ( z - 2\lambda_{2} ) \frac{dz}{dt} + 2 \lambda_{1}s ( v - m_{3} ) \frac{dv}{dt} \\& \quad \quad {}+ 2mu\frac{du}{dt} + 2m ( \omega - \lambda_{2} ) \frac{d \omega }{dt}, \\& \quad = 2\lambda_{1} ( x - m_{2} ) ( \sigma y - \sigma x + sv ) + 2my ( - xz + rx - y ) + 2m ( z - 2\lambda_{2} ) ( xy - xu - bz ) \\& \quad \quad {}+ 2\lambda_{1}s ( v - m_{3} ) ( - x - \sigma v ) + 2mu \bigl[ xz - 2x\omega - ( 1 + 2b ) u \bigr] + 2m ( \omega - \lambda_{2} ) ( 2xu - 4b\omega ) \\& \quad = - 2\lambda_{1}\sigma x^{2} + 2\lambda_{1} \sigma m_{2}x + 2\lambda _{1}sm_{3}x - 2my^{2} - 2\lambda_{1}\sigma m_{2}y - 2bmz^{2} + 4bm \lambda_{2}z - 2\lambda_{1}s\sigma v^{2} \\& \quad \quad {}- 2\lambda_{1}sm_{2}v + 2\lambda_{1}sm_{3} \sigma v - 2m ( 1 + 2b ) u ^{2} - 8bm\omega^{2} + 8bm \lambda_{2}\omega. \end{aligned}$$

Let \(\frac{dV ( X ( t ) ) }{dt} = 0\). Then, we get that the surface

$$ \Gamma : \left \{ \textstyle\begin{array}{l} X \mid - 2\lambda_{1}\sigma x^{2} + 2\lambda_{1}\sigma m _{2}x + 2\lambda_{1}sm_{3}x - 2my^{2} \\ \quad {}- 2\lambda_{1}\sigma m_{2}y - 2bmz ^{2} + 4bm\lambda_{2}z - 2\lambda_{1}s\sigma v^{2} \\ \quad {}- 2\lambda_{1}sm_{2}v + 2\lambda_{1}sm_{3}\sigma v - 2m ( 1 + 2b ) u ^{2} - 8bm\omega^{2} + 8bm\lambda_{2}\omega = 0 \end{array}\displaystyle \right \} $$

(4)

is an ellipsoid in \(R^{6}\)
\(\forall \lambda_{1} > 0\), \(m > 0\), \(\sigma > 0\), \(s > 0\), \(r > 0\), \(b > 0\). Outside Γ, \(\frac{dV ( X ( t ) ) }{dt} < 0\), whereas inside Γ, \(\frac{dV ( X ( t ) ) }{dt} > 0\). Thus, the ultimate boundedness for system (2) can only be reached on Γ. Since the Lyapunov-like function \(V ( X ) \) is a continuous function and Γ is a bounded closed set, the function (3) can reach its maximum value \(\max_{X \in \Gamma } V ( X ) = M\) on the surface Γ. Obviously, \(\{ X \mid V ( X ) \le \max_{X \in \Gamma } V ( X ) = M,X\in \Gamma \} \) contains solutions of system (2). It is obvious that the set Ψ is the ultimate bound set and positively invariant set for system (2).

This completes the proof. □

Theorem 1 points that the trajectories of system (2) are ultimately bounded. However, Theorem 1 does not give the rate of the trajectories of system (2) going from the exterior of the trapping set to the interior of the trapping set. The rate of the trajectories rate of system (2) is studied in the next theorem, Theorem 2.

In the following section, we further investigate the globally attractive set of the high-order Lorenz-Stenflo system (2). We use the following Lyapunov-like function

$$ V ( X ) = \lambda_{1} ( x - m_{2} ) ^{2} + my^{2} + m ( z - 2\lambda_{2} ) ^{2} + \lambda_{1}s ( v - m_{3} ) ^{2} + mu^{2} + m ( \omega - \lambda_{2} ) ^{2} $$

(5)

which is obviously positive definite and radially unbounded. Here, \(\forall \lambda_{1} > 0\), \(\forall m > 0\), \(\lambda_{2} = \frac{\lambda_{1}\sigma + mr}{2m}\), and \(m_{2} \in R\), \(m_{3} \in R\) are arbitrary constants.

Let \(X ( t ) = ( x ( t ) ,y ( t ) ,z ( t ) ,v ( t ) ,u ( t ) ,\omega ( t ) ) \) be an arbitrary solution of system (2). We have the following results for system (2).

### Theorem 2

*Suppose that*
\(\forall \sigma > 0\), \(s > 0\), \(r > 0\), \(b > 0\), *and let*

$$\begin{aligned}& L^{2} = \frac{1}{\theta } \biggl[ \frac{\lambda_{1}s^{2} ( m_{3} ) ^{2}}{\sigma } + \frac{ ( \lambda_{1}\sigma m_{2} ) ^{2}}{m} + \frac{\lambda_{1}s ( m_{2} ) ^{2}}{\sigma } + \lambda_{1} \sigma ( m_{2} ) ^{2} + 8bm ( \lambda_{2} ) ^{2} + \lambda_{1}s\sigma ( m_{3} ) ^{2} \biggr] , \\& \theta = \min ( \sigma ,b ) > 0. \end{aligned}$$

*Then*, *for system* (2), *we have the estimate*

$$ \bigl[ V \bigl( X ( t ) \bigr) - L^{2} \bigr] \le \bigl[ V \bigl( X ( t_{0} ) \bigr) - L^{2} \bigr] e^{ - \theta ( t - t_{0} ) }. $$

(6)

*Thus*
\(\Omega = \{ X \mid V ( X ) \le L^{2} \} \)
*is a globally exponential attractive set of system* (2), *that is*, \(\overline{\lim }_{t \to + \infty } V ( X ( t ) ) \le L^{2}\).

### Proof

Define the following functions:

$$ f ( x ) = - \lambda_{1}\sigma x^{2} + 2\lambda_{1}sm_{3}x,\qquad h( y ) = - my^{2} - 2\lambda_{1}\sigma m_{2}y,\qquad g ( v ) = - \lambda_{1}s\sigma v^{2} - 2\lambda_{1}m_{2}sv. $$

Then we have

$$ \max_{x \in R}f ( x ) = \frac{\lambda_{1}s^{2} ( m_{3} ) ^{2}}{\sigma },\qquad \max _{y \in R}h ( y ) = \frac{ ( \lambda _{1}\sigma m_{2} ) ^{2}}{m},\qquad \max _{v \in R}g ( v ) = \frac{ \lambda_{1}s ( m_{2} ) ^{2}}{\sigma }. $$

Differentiating the Lyapunov-like function \(V ( X ) \) in (5) with respect time *t* along the trajectory of system (2) yields

$$\begin{aligned}& \frac{dV ( X ( t ) ) }{dt} \bigg| _{(2)} \\& \quad = 2 \lambda_{1} ( x - m_{2} ) \frac{dx}{dt} + 2my\frac{dy}{dt} + 2m ( z - 2 \lambda_{2} ) \frac{dz}{dt} + 2\lambda_{1}s ( v - m_{3} ) \frac{dv}{dt} \\& \quad \quad {} + 2mu\frac{du}{dt} + 2m ( \omega - \lambda_{2} ) \frac{d \omega }{dt} \\& \quad = 2\lambda_{1} ( x - m_{2} ) ( \sigma y - \sigma x + sv ) + 2my ( - xz + rx - y ) + 2m ( z - 2\lambda_{2} ) ( xy - xu - bz ) \\& \quad \quad {}+ 2\lambda_{1}s ( v - m_{3} ) ( - x - \sigma v ) + 2mu \bigl[ xz - 2x\omega - ( 1 + 2b ) u \bigr] + 2m ( \omega - \lambda_{2} ) ( 2xu - 4b\omega ) \\& \quad = - 2\lambda_{1}\sigma x^{2} + 2\lambda_{1}\sigma m_{2}x \\& \quad \quad {}+ 2\lambda _{1}sm_{3}x - 2my^{2} - 2\lambda_{1}\sigma m_{2}y - 2bmz^{2} + 4bm \lambda_{2}z - 2\lambda_{1}s\sigma v^{2} \\& \quad \quad {}- 2\lambda_{1}sm_{2}v + 2\lambda_{1}sm_{3} \sigma v - 2m ( 1 + 2b ) u ^{2} - 8bm\omega^{2} + 8bm \lambda_{2}\omega \\& \quad = - \lambda_{1}\sigma x^{2} + 2\lambda_{1}\sigma m_{2}x - \lambda_{1} \sigma x^{2} \\& \quad \quad {} + 2 \lambda_{1}sm_{3}x - my^{2} - my^{2} - 2 \lambda_{1} \sigma m_{2}y - 2bmz^{2} + 4bm \lambda_{2}z \\& \quad \quad {}- \lambda_{1}s\sigma v^{2} + 2\lambda_{1}sm_{3} \sigma v - \lambda_{1}s \sigma v^{2} - 2\lambda_{1}sm_{2}v - 2m ( 1 + 2b ) u^{2} - 8bm \omega^{2} + 8bm \lambda_{2}\omega \\& \quad = - \lambda_{1}\sigma x^{2} + 2\lambda_{1}\sigma m_{2}x + f ( x ) - my^{2} + h ( y ) - 2bmz^{2} + 4bm\lambda_{2}z - \lambda_{1}s\sigma v^{2} + 2 \lambda_{1}sm_{3}\sigma v \\& \quad \quad {}+ g ( v ) - 2m ( 1 + 2b ) u^{2} - 8bm\omega^{2} + 8bm \lambda_{2}\omega \\& \quad \le - \lambda_{1}\sigma x^{2} + 2\lambda_{1}\sigma m_{2}x + f ( x ) - my^{2} + h ( y ) - bmz^{2} + 4bm \lambda_{2}z - \lambda_{1}s\sigma v^{2} + 2 \lambda_{1}sm_{3}\sigma v \\& \quad \quad {}+ g ( v ) - 2m ( 1 + 2b ) u^{2} - 8bm\omega^{2} + 8bm \lambda_{2}\omega \\& \quad = - \lambda_{1}\sigma ( x - m_{2} ) ^{2} + \lambda_{1} \sigma ( m_{2} ) ^{2} + f ( x ) - my^{2} + h ( y ) - bm ( z - 2\lambda_{2} ) ^{2} + 4bm ( \lambda _{2} ) ^{2} \\& \quad \quad {}- \lambda_{1}s\sigma ( v - m_{3} ) ^{2} + \lambda_{1}s \sigma ( m_{3} ) ^{2} + g ( v ) - 2m ( 1 + 2b ) u^{2} - 8bm\omega^{2} + 8bm\lambda_{2} \omega \\& \quad = - \lambda_{1}\sigma ( x - m_{2} ) ^{2} -my^{2} - bm ( z - 2\lambda_{2} ) ^{2} \\& \quad \quad {}- \lambda_{1}s\sigma ( v - m_{3} ) ^{2} - 2m ( 1 + 2b ) u^{2} - 8bm\omega^{2} + 8bm\lambda_{2} \omega \\& \quad \quad {}+ f ( x ) + h ( y ) + g ( v ) + \lambda _{1}\sigma ( m_{2} ) ^{2} + 4bm ( \lambda_{2} ) ^{2} + \lambda_{1}s\sigma ( m_{3} ) ^{2} \\& \quad \le - \lambda_{1}\sigma ( x - m_{2} ) ^{2} - my^{2} - bm ( z - 2\lambda_{2} ) ^{2} - \lambda_{1}s\sigma ( v - m _{3} ) ^{2} - mu^{2} - 4bm\omega^{2} + 8bm\lambda_{2}\omega \\& \quad \quad {}+ f ( x ) + h ( y ) + g ( v ) + \lambda _{1}\sigma ( m_{2} ) ^{2} + 4bm ( \lambda_{2} ) ^{2} + \lambda_{1}s\sigma ( m_{3} ) ^{2} \\& \quad = - \lambda_{1}\sigma ( x - m_{2} ) ^{2} -my^{2} - bm ( z - 2\lambda_{2} ) ^{2} - \lambda_{1}s\sigma ( v - m_{3} ) ^{2} - mu^{2} - 4bm ( \omega - \lambda_{2} ) ^{2} \\& \quad \quad {}+ f ( x ) + h ( y ) + g ( v ) + \lambda _{1}\sigma ( m_{2} ) ^{2} + 4bm ( \lambda_{2} ) ^{2} + \lambda_{1}s\sigma ( m_{3} ) ^{2} + 4bm ( \lambda _{2} ) ^{2} \\& \quad \le - \lambda_{1}\sigma ( x - m_{2} ) ^{2} - my^{2} - bm ( z - 2\lambda_{2} ) ^{2} - \lambda_{1}s\sigma ( v - m _{3} ) ^{2} - mu^{2} - bm ( \omega - \lambda_{2} ) ^{2} \\& \quad \quad {}+ f ( x ) + h ( y ) + g ( v ) + \lambda _{1}\sigma ( m_{2} ) ^{2} + 8bm ( \lambda_{2} ) ^{2} + \lambda_{1}s\sigma ( m_{3} ) ^{2} \\& \quad \le - \theta V ( X ) + \max_{x \in R}f ( x ) + \max _{y \in R}h ( y ) + \max_{v \in R}g ( v ) + \lambda_{1}\sigma ( m_{2} ) ^{2} + 8bm ( \lambda_{2} ) ^{2} + \lambda_{1}s\sigma ( m_{3} ) ^{2} \\& \quad = - \theta V ( X ) + \frac{\lambda_{1}s^{2} ( m_{3} )^{2}}{\sigma } + \frac{ ( \lambda_{1}\sigma m_{2} ) ^{2}}{m} + \frac{\lambda_{1}s ( m_{2} ) ^{2}}{\sigma } \\& \quad \quad {} + \lambda_{1} \sigma ( m_{2} ) ^{2} + 8bm ( \lambda_{2} ) ^{2} + \lambda_{1}s\sigma ( m_{3} ) ^{2} \\& \quad = - \theta \bigl[ V \bigl( X ( t ) \bigr) - L^{2} \bigr] . \end{aligned}$$

Thus, we have

$$ \bigl[ V \bigl( X ( t ) \bigr) - L^{2} \bigr] \le \bigl[ V \bigl( X ( t_{0} ) \bigr) - L^{2} \bigr] e^{ - \theta ( t - t_{0} ) }. $$

Therefore,

$$ \mathop{\overline{\lim}}_{t \to + \infty } V \bigl( X ( t ) \bigr) \le L^{2}, $$

which clearly shows that \(\Omega = \{ X \mid V ( X ) \le L^{2} \} \) is a globally exponential attractive set of system (2). The proof is complete. □

### Remark 2

(i) Let us take \(\lambda_{1} = 1\), \(m = 1\), \(m_{2} = 0\), \(m_{3} = 0\) in Theorem 2. Then we get that

$$ \Delta = \biggl\{ ( x,y,z,v,u,w ) \mid x^{2} + y^{2} + ( z - \sigma - r ) ^{2} + sv^{2} + u^{2} + \biggl( \omega - \frac{\sigma + r}{2} \biggr) ^{2} \le \frac{2b ( \sigma + r ) ^{2}}{\min ( \sigma ,b ) } \biggr\} $$

(7)

is a globally exponential attractive set of system (2) according to Theorem 2.

(ii) Taking \(\sigma = 10\), \(b = \frac{8}{3}\), \(r = 40\), \(s = 50\), we get that

$$ \Delta = \bigl\{ ( x,y,z,v,u,w ) \mid x^{2} + y ^{2} + ( z - 50 ) ^{2} + 50v^{2} + u^{2} + ( \omega - 25 ) ^{2} \le ( 50\sqrt{2} ) ^{2} \bigr\} $$

(8)

is a globally exponential attractive set of system (2) according to Theorem 2. Figure 5 shows chaotic attractors of system (2) in the \(( x,y,z ) \) space defined by Δ in (8). Figure 6 shows chaotic attractors of system (2) in the \(( x,y,v ) \) space defined by Δ in (8). Figure 7 shows chaotic attractors of system (2) in the \(( x,y,u ) \) space defined by Δ in (8). Figure 8 shows chaotic attractors of system (2) in the \(( x,y, \omega ) \) space defined by Δ in (8).