Globally Exponentially Attracting Sets and Heteroclinic Orbits Revealed

Abstract

Motivated by the open problems on the global dynamics of the generalized four-dimensional Lorenz-like system, this paper deals with the existence of globally exponentially attracting sets and heteroclinic orbits by constructing a series of Lyapunov functions. Specifically, not only is a family of mathematical expressions of globally exponentially attracting sets derived, but the existence of a pair of orbits heteroclinic to $S_0$ and $S_{\pm }$ is also proven with the aid of a Lyapunov function and the definitions of both the $\alpha$-limit set and $\omega$-limit set. Moreover, numerical examples are used to justify the theoretical analysis. Since the obtained results improve and complement the existing ones, they may provide support in chaos control, chaos synchronization, the Hausdorff and Lyapunov dimensions of strange attractors, etc.

1. Introduction

In this paper, we consider the problems of a generalized four-dimensional Lorenz-like system [1], extending the existing results reported in [2,3,4,5]. The studied system includes many Lorenz-like systems as special cases [2,3,4,5,6,7,8,9], but it generates the coexistence of various attractors. Aiming to shed some light on its global dynamics, we rigorously prove the existence of globally exponentially attracting sets and heteroclinic orbits, which is particularly significant both for theoretical research and practical applications, such as cell signaling [10], neurons [11], biomathematics and mechanics [12,13,14], space missions [15,16,17], etc.

Referring also to [18,19], if $x_1$ and $x_2$ (resp., $L_1$ and $L_2$) are two hyperbolic equilibrium points (resp., periodic trajectories) of a system such that the stable (unstable) manifold $W^s{x}_1$ ($W^u{x}_1$) (resp., $W^s{L}_1$ ($W^u{L}_1$)) intersects the unstable (stable) manifold $W^u{x}_2$ ($W^s{x}_2$) (resp., $W^u{L}_2$ ($W^s{L}_2$)), then the orbit belonging to their intersection $W^s{x}_1\bigcap W^u{x}_2\ne \varnothing$ ($W^u{x}_1\bigcap W^s{x}_2\ne \varnothing$) (resp., $W^s{L}_1\bigcap W^u{L}_2\ne \varnothing$ ($W^u{L}_1\bigcap W^s{L}_2\ne \varnothing$)) is called a heteroclinic orbit (or homoclinic orbit if $x_1=x_2$ (resp., $L_1=L_2$)). However, the study of heteroclinic/homoclinic orbits is difficult. Hitherto, researchers have only proposed a few methods, such as the Melnikov method [20], the boundary value and contraction map [19], the Poincaré map [18], the fishing principle [21], the method of tracing the stable and unstable manifolds [22], etc. In 2006, Li et al. combined the Lyapunov function and the definitions of both the $\alpha$-limit set and $\omega$-limit set to prove the existence of heteroclinic orbits in the Chen system [6], which did not need to consider the mutual disposition of the stable and unstable manifolds of a saddle equilibrium. In light of this, the method has been applied in many other Lorenz-like systems [2,3,4,5,7,8,9,23,24].

Formally, a chaotic system is bounded, meaning that its dynamics remain inside an orbit, rather than escaping to infinity. As is known, a global attracting set occupies an important position in qualitative analysis, i.e., in global asymptotic/exponential stability, the uniqueness of equilibria, the existence of periodic/quasi-periodic solutions or various attractors, chaotic control, synchronization, Hausdorff and Lyapunov dimensions, etc. In addition, once one has proven the existence of a global attracting set for any one chaotic/hyperchaotic system, one can exclude the aforementioned dynamics outside the global attracting set. This also plays an important role in engineering applications associated with the existence of hidden attractors, which are not only difficult to predict, but also bring about crashes [25]. Moreover, the ultimate boundedness may also explain the forming mechanism of the chaotic motions of a continuous system, i.e., attracting orbits from outside to inside. The other factor is represented by positive Lyapunov exponents, which push orbits from the inside to outside [26,27,28].

Therefore, it is worthwhile to consider the globally exponentially attracting sets and heteroclinic orbits of system (1), which are also open problems.

In this endeavor, we reinvestigated system (1) and further obtained the following two results, i.e., the main contributions of the present work:

(1)

The globally exponentially attracting sets are located using a series of Lyapunov functions.

(2)

The existence of a pair of heteroclinic orbits is proven with the aid of a Lyapunov function and the definitions of both the $\alpha$-limit set and $\omega$-limit set.

Our obtained results not only contain the existing ones as special cases, but also show the rates of the trajectories of the system moving from the exterior of the trapping set to the interior of the trapping set.

The remainder of this paper is structured as follows. Section 2 introduces the generalized four-dimensional Lorenz-like system and presents the main results. In Section 3 and Section 4, we study the existence of globally exponentially attracting sets and heteroclinic orbits, respectively. Lastly, the conclusions are drawn in Section 5.

2. Mathematical Model and Main Results

In 2019, Hong considered a generalized four-dimensional Lorenz-like system [1]:

$$\left\{\begin{array}{cc}\dot{x}\hfill & =a(y-x),\hfill \\ \dot{y}\hfill & =bx-xz-cy+w,\hfill \\ \dot{z}\hfill & =xy-dz+k,\hfill \\ \dot{w}\hfill & =-ex-fy-gw,\hfill \end{array}\right.$$

(1)

where $(x,y,z,w)\in {\mathbb{R}}^4$ are state variables, and $(a,b,c,d,k,e,f,g)\in {\mathbb{R}}^8$ are parameters. Obviously, many other Lorenz-type systems, e.g., the ones in [2,3,4,5,6,7,8,9], are special cases of system (1). Although Hong intensively studied the local and global dynamics of system (1), such as the distribution of the equilibrium points; the local stability of $S_{\pm }=(\pm \sqrt{\Delta },\pm \sqrt{\Delta },b-c-\frac{e+f}{g},\mp \frac{e+f}{g}\sqrt{\Delta })$ and $S_0=(0,0,\frac{k}{d},0)$, where $\Delta =d(b-c-\frac{e+f}{g})-k$; the generic pitchfork and Hopf bifurcation at $S_0$; the hidden chaotic/hyperchaotic attractors; the coexistence of various attractors (chaotic/hyperchaotic attractors, period cycles, quasi-periodic cycles, etc.); and so on, she also proposed the following open problems:

Open Problem 1.

Does system (1) have globally exponentially attracting sets?

Open Problem 2.

How can one construct appropriate Lyapunov functions to prove the existence of homoclinic and heteroclinic orbits?

To the best of our knowledge, Open Problems 1 and 2 are not considered in the published literature. Therefore, it is of relevance to address them in the present work.

In constructing a series of Lyapunov functions, the main contributions can be summarized with the following propositions.

Setting $X=(x,y,z,w)$, $X_0=(x_0,y_0,z_0,w_0)$, $\forall \rho >0$, $\forall \lambda >0$, $2a>d>0$, $g>0$, $c<0$, and $k\ge 0$, we have the following:

(i) $r=\frac{d-c}{3a}$, $\epsilon =g$, $3g<d+2c$, $g<d$, or (ii) $r=\frac{g-2c}{2a}$, $\epsilon =\frac{g}{2}$, $d>3g-2c$, or (iii) $r=\frac{d-2c}{4a}$, $\epsilon =\frac{d+2c}{4}$, $d<3g-2c$, $d>-2c$.

$P=b+r(a-c-ra)+\frac{a\lambda }{\rho }-\frac{rf+e}{g+a(r-1)}$, $M=\frac{1}{[g+a(r-1\left)\right][gf-a(e+f\left)\right]}>0$, $\lambda (\epsilon -a)-r\rho [b+r(a-c-ra)]+\frac{r(rf+e)}{ra+g-a}\le 0$,

$$V_{\lambda ,\rho }\left(X\right)=\frac{1}{2}\{\lambda x^2+\rho {(y-rx)}^2+\rho {(z-P)}^2+\rho M{[(2f+e)x+(ra+g-a)w]}^2\},$$

(2)

and $L_{\lambda ,\rho }=\frac{\rho [4(\epsilon +2ar-d)(\epsilon P^2-Pk)-{(k(1-r)+P(d-2\epsilon ))}^2]}{8\epsilon (\epsilon +2ar-d)}$.

Proposition 1.

If $V_{\lambda ,\rho }\left(X\right)>L_{\lambda ,\rho }$ and $V_{\lambda ,\rho }\left(X_0\right)>L_{\lambda ,\rho }$, then one arrives at the following exponential inequality:

$$V_{\lambda ,\rho }\left(X\right)-L_{\lambda ,\rho }\le [V_{\lambda ,\rho }\left(X_0\right)-L_{\lambda ,\rho }]e^{-2\epsilon (t-t_0)}.$$

(3)

In a word, the sets

$$\begin{array}{cc}\hfill {\mathsf{\Psi }}_{\lambda ,\rho }& =\left\{X\right|V_{\lambda ,\rho }\left(X\right)\le L_{\lambda ,\rho }\}\hfill \\ \hfill & =\left\{(x,y,z,w)|\lambda x^2+\rho {(y-rx)}^2+\rho {(z-P)}^2\right.\hfill \\ \hfill & \phantom{=}\left.+\rho M{[(2f+e)x+(ra+g-a)w]}^2\le 2L_{\lambda ,\rho },\forall \lambda >0,\rho >0\right\}\hfill \end{array}$$

are globally exponentially attracting sets of system (1).

Proposition 2.

Consider $d\ge 2a>0$, $g>0$, $k-d(b-c-\frac{e+f}{g})<0$, $g-(e+f)a>0$, and $c+a>0$. Then, the following statements hold:

(i) 

Orbits that are neither homoclinic nor heteroclinic to $S_{+}$ and $S_{-}$ exist in system (1).

(ii) 

System (1) has a pair of orbits heteroclinic to $S_0$ and $S_{\pm }$.

Outlines of the proofs of Propositions 1 and 2 are presented in Section 3 and Section 4.

3. Globally Exponentially Attracting Sets and Proof of Proposition 1

Aiming to prove the existence of globally exponentially attracting sets of system (1), one has to derive the following result:

Proposition 3.

If $2a>d>0$ and $k\ge 0$, then we have the following inequality:

$$\underset{t\to \infty }{lim}[2az-x^2-k]\ge 0.$$

(4)

Proof. 

Write

$$V(x,z)=2az-x^2-k$$

(5)

and calculate the derivative of this along the orbits of system (1):

$$\dot{V}=2a\dot{z}-2x\dot{x}=-2adz+2ak+2a{x}^2,$$

(6)

i.e., $\dot{V}+dV=-(d-2a)(x^2+k).$

Since $d-2a<0$ and $k\ge 0$, we arrive at $\dot{V}+dV\ge 0$, which leads to

$$V\left(t\right)\ge V_0{e}^{-d(t-t_0)}\to 0,(t\to \infty ),\forall V\left(t_0\right)=V_0,$$

(7)

based on the comparison principle.

In brief, when $d-2a<0$ and $k\ge 0$, the following inequality holds:

$$\underset{t\to \infty }{lim}V\left(t\right)=\underset{t\to \infty }{lim}[2az-x^2-k]\ge 0.$$

(8)

The proof is complete. □

According to Proposition 3, the outline of the proof of Proposition 1 follows.

Proof of Proposition 1.

Based on Equation (2), one computes its derivative along the trajectories of system (1):

$$\begin{array}{ccc}\frac{d{V}_{\lambda ,\rho }}{dt}{|}_{\left(1\right)}\hfill & =\hfill & \lambda x\frac{dx}{dt}+\rho (y-rx)(\frac{dy}{dt}-r\frac{dx}{dt})+\rho M[(2f+e)x+(ra+g-a)w]\hfill \\ \hfill & \hfill & [(2f+e)\frac{dx}{dt}+(ra+g-a)\frac{dw}{dt}]+\rho (z-P)\frac{dz}{dt}\hfill \\ \hfill & =\hfill & \lambda a(y-x)x+\rho (y-rx)(bx-xz-cy+w-ra(y-x\left)\right)+\rho (z-P)(xy-dz+k)\hfill \\ \hfill & \hfill & +\rho M\left[\right(2f+e)x+(ra+g-a\left)w\right]\left[\right(2f+e\left)a\right(y-x)+(ra+g-a\left)\right(-ex-fy-gw\left)\right]\hfill \\ \hfill & =\hfill & \{-\lambda a-r\rho [b+r(a-c-ra)]+\frac{r(rf+e)}{[g+a(r-1\left)\right]}\}{x}^2-g\rho M{[(2f+e)x+(ra+g-a)w]}^2\hfill \\ \hfill & \hfill & -\rho (c+ra){(y-rx)}^2+r\rho x^{2}z-d\rho z^2+\rho (k+Pd)z-\rho Pk.\hfill \end{array}$$

(9)

Since ${lim}_{t\to \infty }[2az-x^2-k]>0$ in Proposition 3, $\exists T_0>0$, such that $t>T_0$, we arrive at

$$\begin{array}{ccc}\frac{d{V}_{\lambda ,\rho }}{dt}{|}_{\left(1\right)}\hfill & \le \hfill & \{-\lambda a-r\rho [b+r(a-c-ra)]+\frac{r(rf+e)}{[g+a(r-1\left)\right]}\}{x}^2-g\rho M{[(2f+e)x+(ra+g-a)w]}^2\hfill \\ \hfill & \hfill & -\rho (c+ra){(y-rx)}^2+(2ar-d)\rho z^2+\rho (k+Pd-rk)z-\rho Pk\hfill \\ \hfill & =\hfill & -\epsilon \{\lambda x^2+\rho {(y-rx)}^2+\rho {(z-P)}^2+\rho M{[(2f+e)x+(ra+g-a)w]}^2\}\hfill \\ \hfill & \hfill & +\{-\lambda (a-\epsilon )-r\rho [b+r(a-c-ra)]+\frac{r(rf+e)}{[g+a(r-1\left)\right]}\}{x}^2+\rho (-c-ra+\epsilon ){(y-rx)}^2\hfill \\ \hfill & \hfill & +\rho M(\epsilon -g){[(2f+e)x+(ra+g-a)w]}^2\hfill \\ \hfill & \hfill & +\epsilon \rho {(z-P)}^2+(2ar-d)\rho z^2+\rho (k+Pd-rk)z-\rho Pk\hfill \\ \hfill & =\hfill & -2\epsilon V_{\lambda ,\rho }+\{-\lambda (a-\epsilon )-r\rho [b+r(a-c-ra)]+\frac{r(rf+e)}{[g+a(r-1\left)\right]}\}{x}^2\hfill \\ \hfill & \hfill & +\rho (-c-ra+\epsilon ){(y-rx)}^2+\rho M(\epsilon -g){[(2f+e)x+(ra+g-a)w]}^2\hfill \\ \hfill & \hfill & +\rho [(\epsilon +2ar-d)z^2+(k(1-r)+P(d-2\epsilon ))z+\epsilon P^2-Pk].\hfill \end{array}$$

(10)

When $2a>d>0$, $g>0$, $c<0$, and $k\ge 0$,

(i) $r=\frac{d-c}{3a}$, $\epsilon =g$, $3g<d+2c$, $g<d$, or (ii) $r=\frac{g-2c}{2a}$, $\epsilon =\frac{g}{2}$, $d>3g-2c$, or (iii) $r=\frac{d-2c}{4a}$, $\epsilon =\frac{d+2c}{4}$, $d<3g-2c$, $d>-2c$.

One then arrives at $-c-ra+\epsilon \le 0$, $\epsilon -g\le 0$ and $\epsilon +2ar-d<0$.

Because $M=\frac{1}{[g+a(r-1\left)\right][gf-a(e+f\left)\right)]}>0$ and $-\lambda (a-\epsilon )-r\rho [b+r(a-c-ra\left)\right]+$$\frac{r(rf+e)}{[g+a(r-1\left)\right]}\le 0$, we arrive at

$$\begin{array}{ccc}\frac{d{V}_{\lambda ,\rho }}{dt}{|}_{\left(1\right)}\hfill & \le \hfill & -2\epsilon V_{\lambda ,\rho }+\rho F\left(z\right)\hfill \\ \hfill & \le \hfill & -2\epsilon V_{\lambda ,\rho }+\rho \underset{z\in \mathbb{R}}{max}F\left(z\right)\hfill \\ \hfill & =\hfill & -2\epsilon V_{\lambda ,\rho }+\frac{\rho [4(\epsilon +2ar-d)(\epsilon P^2-Pk)-{(k(1-r)+P(d-2\epsilon ))}^2]}{4(\epsilon +2ar-d)}\hfill \\ \hfill & =\hfill & -2\epsilon [V_{\lambda ,\rho }-\frac{\rho [4(\epsilon +2ar-d)(\epsilon P^2-Pk)-{(k(1-r)+P(d-2\epsilon ))}^2]}{8\epsilon (\epsilon +2ar-d)}]\hfill \\ \hfill & =\hfill & -2\epsilon [V_{\lambda ,\rho }\left(X\right)-L_{\lambda ,\rho }],\hfill \end{array}$$

(11)

which yields Equation (3).

If $V_{\lambda ,\rho }\left(X\right)>L_{\lambda ,\rho }$ and $V_{\lambda ,\rho }\left(X_0\right)>L_{\lambda ,\rho }$, then taking the upper limit on both sides of (Equation (3)) as $t\to +\infty$ leads to

$$\overline{\underset{t\to +\infty }{lim}}{V}_{\lambda ,\rho }\left(X\right)\le L_{\lambda ,\rho }.$$

In other words,

$$\begin{array}{cc}\hfill {\mathsf{\Omega }}_{\lambda ,\rho }& =\left\{X\left(t\right)|\overline{\underset{t\to +\infty }{lim}}{V}_{\lambda ,\rho }\left(X\right)\le L_{\lambda ,\rho }\right\}\hfill \\ \hfill & =\left\{(x,y,z,w)|\lambda x^2+\rho {(y-rx)}^2+\rho {(z-P)}^2+\rho M{[(2f+e)x+(ra+g-a)w]}^2\le 2L_{\lambda ,\rho }\right\}\hfill \end{array}$$

are globally exponentially attracting sets of system (1). The proof is complete. □

Remark 1.

Proposition 1 includes the main results on the globally exponentially attracting sets of the hyperchaotic Lorenz-like systems [2,3,4,5].

For convenience in the discussion of the existence of heteroclinic orbits of system (1), in Section 4, i.e., the proof of Proposition 2, we provide the following notations:

(1)

${\varphi }_t\left(q_0\right)=(x(t;x_0),y(t;y_0),z(t;z_0),w(t;w_0))$ is expressed as a solution of system (1) through the initial point $q_0=(x_0,y_0,z_0,w_0)$.

(2)

$W_{+}^u$ (resp. $W_{-}^u$) is represented as the positive (resp. negative) branch of the unstable manifold $W^u\left(S_0\right)$ corresponding to $x>0$ (resp. $x<0$) for a large negative t.

(3)

${\gamma }^{\pm }=\left\{{\varphi }_t^{\pm }\left(q_0\right)\right|{\varphi }_t^{\pm }\left(q_0\right)=(\pm x_{+}(t;x_0),\pm y_{+}(t;y_0),z_{+}(t;z_0),\pm w_{+}(t;w_0))\in W_{\pm }^u,t\in \mathbb{R}\}$.

4. Existence of Heteroclinic Orbit and Proof of Proposition 2

In this section, we study the existence of heteroclinic orbits of system (1) with the aid of a Lyapunov function and the definitions of the $\alpha$-limit set and $\omega$-limit set [2,3,4,5,6,7,8,9]. First of all, we write the first Lyapunov function

$$\begin{array}{cc}{V}_1\left({\varphi }_t\left(q_0\right)\right)=V_1(x,y,z,w)=\hfill & \frac{1}{2}{\{g[g-a(e+f)](y-x)}^2+\frac{g[g-a(e+f\left)\right]}{d(d-2a)}{[x^2-dz+k]}^2\hfill \\ \hfill & +\frac{g[g-a(e+f\left)\right]}{2ad}{[k-d(b-c-\frac{e+f}{g})+x^2]}^2+{[(e+f)x+gw]}^2\}.\hfill \end{array}$$

(12)

for $-2a+d>0$, and then write the second one

$$\begin{array}{cc}{V}_2\left({\varphi }_t\left(q_0\right)\right)=V_2(x,y,z,w)=\hfill & \frac{1}{2}{\{2a(y-x)}^2+\frac{2a}{g[g-a(e+f\left)\right]}{[(e+f)x+gw]}^2\hfill \\ \hfill & +\frac{1}{2a}{[k-2a(b-c-\frac{e+f}{g})+x^2]}^2\}.\hfill \end{array}$$

(13)

for $-2a+d=0$.

Apparently, the derivatives of $V_{1,2}$ along the trajectories of system (1) are computed as follows:

$$\begin{array}{ccc}\frac{d{V}_1\left({\varphi }_t\left(q_0\right)\right)}{dt}{|}_{\left(\right)}\hfill & =\hfill & -g[g-a(e+f)](c+a){(y-x)}^2-\frac{g[g-a(e+f\left)\right]}{d-2a}{[x^2-dz+k]}^2\hfill \\ \hfill & \hfill & -g{[(e+f)x+gw]}^2\le 0\hfill \end{array}$$

(14)

and

$$\begin{array}{cc}\frac{d{V}_2\left({\varphi }_t\left(q_0\right)\right)}{dt}{|}_{\left(1\right)}=\hfill & -2a(c+a){(y-x)}^2-\frac{2a}{g-a(e+f)}{[(e+f)x+gw]}^2\le 0.\hfill \end{array}$$

(15)

Taking a similar approach to the ones in [2,3,4,5,6,7,8,9], we have to prove the following statement.

Proposition 4.

Assume that $d\ge 2a>0$, $g>0$, $k-d(b-c-\frac{e+f}{g})<0$, $g-(e+f)a>0$, and $c+a>0$. The following two assertions are true:

(i) 

If ∃$t_{1,2}$ such that $t_1<t_2$ and $V_{1,2}\left({\varphi }_{t_1}\left(q_0\right)\right)=V_{1,2}\left({\varphi }_{t_2}\left(q_0\right)\right)$, then $q_0$ is one of the equilibria of system (1).

(ii) 

If $lim{\varphi }_t{\left(q_0\right)}_{t\to -\infty }=S_0$, and, for some $t\in \mathbb{R}$, $x(t;x_0)<0$, then $V_{1,2}\left(S_0\right)>V_{1,2}\left({\varphi }_t\left(q_0\right)\right)$ and $x(t;x_0)<0$ for all $t\in \mathbb{R}$. Therefore, $q_0\in W_{-}^u$.

Proof. 

(i) Since $\frac{d{V}_{1,2}\left({\varphi }_t\left(q_0\right)\right)}{dt}{|}_{\left(1\right)}\le 0$ for $d\ge 2a>0$, $g>0$, $k-d(b-c-\frac{e+f}{g})<0$, $g-(e+f)a>0$, and $c+a>0$, Equations (14) and (15) lead to $\frac{d{V}_{1,2}\left({\varphi }_t\left(q_0\right)\right)}{dt}{|}_{\left(1\right)}=0$ for all $t\in (t_1,t_2)$, which thus yields that $q_0$ is an equilibrium point, i.e.,

$$\dot{x}(t;x_0)\equiv \dot{y}(t;y_0)\equiv \dot{z}(t;z_0)\equiv \dot{w}(t;w_0)\equiv 0.$$

(16)

In fact, $\dot{x}(t;x_0)=a(y-x)=0$ yields $x\left(t\right)=x_0$ and $\dot{y}(t,y_0)=0$, $\forall t\in \mathbb{R}$.

In addition, based on Proposition 3, we have $x^2+k\equiv 2az$ when $-2a+d=0$.

Specifically, ${\varphi }_t\left(q_0\right)\in \left\{a(y-x)=0\right\}\cap \left\{k-d(b-c-\frac{e+f}{g})+x^2=0\right\}\cap$$\left\{(e+f)x+gw=0\right\}$ leads to (16).

(ii) As $lim{\varphi }_t{\left(q_0\right)}_{t\to -\infty }=S_0$ and $x(t;x_0)<0$ for some $t\in \mathbb{R}$, $q_0$ cannot be an equilibrium point at all. Without loss of generality, if $\exists t_0\in \mathbb{R}$, such that $0<V_{1,2}\left(S_0\right)\le V_{1,2}\left({\varphi }_{t_0}\left(q_0\right)\right)$, there exists a $t_1$ such that $V_{1,2}\left({\varphi }_{t_1}\left(q_0\right)\right)\le V_{1,2}\left({\varphi }_{t_0}\left(q_0\right)\right)$. Since $\frac{d{V}_{1,2}\left({\varphi }_t\left(q_0\right)\right)}{dt}{|}_{\left(1\right)}\le 0$, the condition $V_{1,2}\left({\varphi }_{t_1}\left(q_0\right)\right)=V_{1,2}\left({\varphi }_{t_0}\left(q_0\right)\right)$ and assertion (i) yield that $q_0$ is one of the equilibria.

The precondition ${lim}_{t\to -\infty }{\varphi }_t\left(q_0\right)=S_0$ results in $q_0=S_0$ and $x(t;x_0)=0$, $\forall t\in \mathbb{R}$, which contradicts the hypothesis. Therefore, $V_{1,2}\left(S_0\right)>V_{1,2}\left({\varphi }_t\left(q_0\right)\right)$, $\forall t\in \mathbb{R}$.

Moreover, we prove $x(t,x_0)<0$, $\forall t\in \mathbb{R}$. Otherwise, there is at least a $t^{\prime }\in \mathbb{R}$ such that $x(t^{\prime },x_0)\ge 0$, and, using $x(t^{″},x_0)<0$ for some $t^{″}\in \mathbb{R}$ from the hypothesis of (ii), we obtain that there exists a $\tau \in \mathbb{R}$ such that $x(\tau ,x_0)=0$. Due to $V_{1,2}\left(S_0\right)>V_{1,2}\left({\varphi }_t\left(q_0\right)\right)$, $\forall t\in \mathbb{R}$, we have ${\varphi }_{\tau }\left(q_0\right)\in \mathsf{\Omega }\cap P$, where $\mathsf{\Omega }=\{(x,y,z,w):V_{1,2}\left(S_0\right)>V_{1,2}(x,y,z,w)\}$, and P is the plane $\{x=0\}$. However, $\mathsf{\Omega }\cap P$ is expressed by

$$\begin{array}{c}\frac{1}{2}\{g[g-a(e+f)]y^2+\frac{g[g-a(e+f\left)\right]}{d(d-2a)}{[-dz+k]}^2+\frac{g[g-a(e+f\left)\right]}{2ad}{[k-d(b-c-\frac{e+f}{g})]}^2+g^2{w}^2\}\hfill \\ <\frac{g[g-a(e+f\left)\right]}{4ad}{[k-d(b-c-\frac{e+f}{g})]}^2\hfill \end{array}$$

(17)

for $V_1$ and

$$\begin{array}{c}\frac{1}{2}\{2a{y}^2+\frac{2ag}{[g-a(e+f\left)\right]}{w}^2+\frac{1}{2a}{[k-2a(b-c-\frac{e+f}{g})]}^2\}<\frac{1}{4a}{[k-2a(b-c-\frac{e+f}{g})]}^2\hfill \end{array}$$

(18)

for $V_2$.

In either case, one obtains $\mathsf{\Omega }\cap P=\varnothing$, and a contradiction occurs. Hence, $x(t,x_0)<0$, $\forall t\in \mathbb{R}$. The proof is complete. □

With the help of Proposition 4, we present the proof of Proposition 2 as follows.

Proof of Proposition 2:

(i) When $d\ge 2a>0$, $g>0$, $k-d(b-c-\frac{e+f}{g})<0$, $g-(e+f)a>0$, and $c+a>0$, one can discuss the fact that orbits that are neither homoclinic nor heteroclinic to $S_{\pm }$ exist in system (1). Otherwise, let $\gamma (t,q_0)$ be an orbit homoclinic or heteroclinic to $S_{+}$ and $S_{-}$ of system (1) through an initial point $q_0\notin \{S_0,S_{-},S_{+}\}$, i.e.,

$$\underset{t\to -\infty }{lim}\gamma (t,q_0)=s_{-},\underset{t\to \infty }{lim}\gamma (t,q_0)=s_{+},$$

(19)

where $s_{-}$ and $s_{+}$ satisfy either $s_{-}=s_{+}\in \{S_0,S_{-},S_{+}\}$ or $\{s_{-},s_{+}\}=\{S_{-},S_{+}\}$. From Equations (14) and (15), one has

$$V_{1,2}\left(s_{-}\right)\ge V_{1,2}\left(\gamma (t,q_0)\right)\ge V_{1,2}\left(s_{+}\right).$$

(20)

In either case, the relationship $V_{1,2}\left(s_{-}\right)=V_{1,2}\left(s_{+}\right)$ is true, which suggests that $V_{1,2}\left(\gamma (t,q_0)\right)=V_{1,2}\left(s_{+}\right)$. Assertion (i) of Proposition 4 leads to the fact that $q_0$ is one of the equilibria of system (1). Therefore, there exist orbits that are neither homoclinic nor heteroclinic to $S_{+}$ and $S_{-}$ of system (1).

(ii) Next, let us prove the existence of an orbit heteroclinic to $S_0$ and $S_{-}$ of system (1). As $W_{-}^u$ is a negative branch with respect to x of the unstable manifold of $S_0$, ∃ a $t_1\in \mathbb{R}$ such that $x(t_1,x_0)<0$ for $x_0\in W_{-}^u$, which, by Proposition 4(ii), gives $x(t,x_0)<0$, $\forall t\in \mathbb{R}$ and $x_0\in W_{-}^u$, i.e., each trajectory on $W_{-}^u$ never approaches $S_{+}$, which lies in $x>0$. Based on Proposition 4(i), it tends toward one of the equilibria except $S_0$. Consequently, letting ${\gamma }^{-}\left(t\right)$ be an orbit on $W_{-}^u$, we arrive at ${lim}_{t\to \infty }{\gamma }^{-}\left(t\right)=S_{-}$, which suggests that ${\gamma }^{-}\left(t\right)$ is indeed an orbit heteroclinic to $S_{-}$ and $S_0$ lying in $x<0$. Further, one shows the uniqueness of ${\gamma }^{-}\left(t\right)$ in $x<0$. Assume that ${\varphi }_t\left(q_0\right)$ is any one solution of system (1) with $q_0$ being arbitrary, not necessarily on $W_{-}^u$, with $s_{-},s_{+}$ as above:

$$\underset{t\to -\infty }{lim}{\varphi }_t\left(q_0\right)=s_{-},\underset{t\to \infty }{lim}{\varphi }_t\left(q_0\right)=s_{+},$$

where $s_{-},s_{+}\in \{S_0,S_{-}\}$, i.e., ${\varphi }_t\left(q_0\right)$, is a second orbit heteroclinic to $S_0$ and $S_{-}$.

According to $\frac{d{V}_{1,2}\left({\varphi }_t\left(q_0\right)\right)}{dt}{|}_{\left(1\right)}\le 0$, one has

$$V_{1,2}\left(s_{-}\right)\ge V_{1,2}\left({\varphi }_t\left(q_0\right)\right)\ge V_{1,2}\left(s_{+}\right)$$

(21)

for all $t\in \mathbb{R}$. The fact that $V_{1,2}\left(S_0\right)>V_{1,2}\left(S_{-}\right)$ leads to $s_{-}=S_0$ and $s_{+}=S_{-}$ yields $q_0\in W_{-}^u$, by Proposition 4 (ii), i.e., the orbit ${\varphi }_t\left(q_0\right)$ is the same as ${\gamma }^{-}\left(t\right)$. Since the orbits of system (1) are symmetrical with respect to the z-axis, ${\gamma }^{+}\left(t\right)$ is a unique heteroclinic orbit that is symmetrical to ${\gamma }^{-}\left(t\right)$.

As shown in Figure 1 and Figure 2, several numerical examples verify the theoretical results. The proof is complete. □

Remark 2.

The results of the existence of heteroclinic orbits given in [2,3,4,5] are easily derived from Proposition 2.

5. Conclusions

This method of using Lyapunov functions can be implemented in many areas of application, e.g., in the context of locating global attracting sets or attractors, in problems of the existence of homoclinic and heteroclinic orbits, in the estimation of the dimensions of attractors, etc. Inspired by this, this study revisited an existing generalized four-dimensional Lorenz-like system and proved the existence of globally exponentially attracting sets and heteroclinic orbits by constructing a series of Lyapunov functions, which not only contain the ones described in [2,3,4,5] as special cases but also may shed light on the Hausdorff and Lyapunov dimensions of strange attractors, chaos control, synchronization, etc.

Notably, although the method involving the Lyapunov functions and the definitions of both the $\alpha$-limit set and $\omega$-limit set has been applied to sub-quadratic, cubic, and other Lorenz-like systems [23], the globally exponentially attracting sets of these are still unknown. In addition to this, one needs to consider other important problems, e.g., self-excited or hidden conservative Lorenz-like chaotic flows, homoclinic orbits, entropy [29], real-world applications, etc.