Heteroclinic Orbits Seized from Caputo Fractional-Order Lorenz System with Memristor

Abstract

By constructing two distinct Lyapunov-like functions and employing concepts of $\alpha$-/$\omega$-limit sets, this article reinvestigates the Caputo fractional-order ($0<\alpha \le 1$) memristor-based Lorenz system, and rigorously proves that there exists a pair of heteroclinic orbits. The analytical findings are validated via numerical simulations, and as far as we know, these results have not been previously reported, thereby extending the relevant research outcomes in this field.

1. Introduction

In 2008, Strukov et al. first introduced a physical model of a memristor, marking a landmark breakthrough in both theoretical research and engineering applications of this device [1,2]. Along with capacitance, resistance, and inductance, memristors exhibit tremendous application potential not only in power electronics and artificial intelligence, but also in a wide range of research areas including nonlinear dynamical systems [3], neural networks [4,5], nonvolatile storage [6], and logic circuit design [7].

It is well established that embedding a memristor into chaotic dynamical systems readily gives rise to rich and complex dynamical behaviors, i.e., coexisting multiple attractors, which include equilibrium points, periodic orbits, almost-periodic orbits, chaos, quasi-periodicity, and hyperchaos. For instance, Ruan et al. introduced a memristor-based hyperchaotic Lorenz system and demonstrated the existence of infinitely many equilibrium points in the model [8]. Jiang et al. developed a 3D memristive Lorenz system and revealed chaotic attractors coexisting with periodic cycles [9]. Recently, by the translation transformation $y\to {\displaystyle \frac{y}{a}}+x$, Huang and Chen converted the Lorenz system:

$$\left\{\begin{array}{cc}\dot{x}\hfill & =a(y-x),\hfill \\ \dot{y}\hfill & =cx-y-xz,\hfill \\ \dot{z}\hfill & =-bz+xy,\hfill \end{array}\right.$$

into the resulting one:

$$\left\{\begin{array}{cc}\dot{x}\hfill & =y,\hfill \\ \dot{y}\hfill & =(c-1)ax-(1+a)y-axz,\hfill \\ \dot{z}\hfill & =-bz+{\displaystyle \frac{xy}{a}}+x^2.\hfill \end{array}\right.$$

Then, by virtue of the method [10] and introducing the flux, the electric charge q, and the memductance:

$$\left\{\begin{array}{cc}q(x)\hfill & =-\rho (rx+m{x}^3),\hfill \\ w(x)\hfill & ={\displaystyle \frac{dq(x)}{dx}}=-\rho (r+3m{x}^2),\hfill \end{array}\right.$$

Huang and Chen established an integer-order Lorenz-type chaotic system with a memristor:

$$\left\{\begin{array}{cc}\dot{x}\hfill & =y,\hfill \\ \dot{y}\hfill & =(c-1)ax-(1+a+\rho r)y-axz-3\rho m{x}^{2}y,\hfill \\ \dot{z}\hfill & =-bz+{\displaystyle \frac{xy}{a}}+x^2,\hfill \end{array}\right.$$

(1)

where $a\ne 0,\rho >0,b,c,r,m\in \mathbb{R}$. They conducted an in-depth analysis of its pitchfork bifurcation, Hopf bifurcation, zero-Hopf bifurcation, double-zero bifurcation, as well as dynamical behaviors at infinity [11,12]. Nevertheless, the global bifurcation of system (1), particularly its heteroclinic orbits, remain largely unexplored. In contrast, heteroclinic orbits have been extensively investigated in various Lorenz-family chaotic systems, including symmetric and asymmetric Chen systems [13,14,15], symmetric and asymmetric Yang systems [16,17,18,19], symmetric T and Lü systems [20], 3D extended Lorenz-like systems with sine functions [21,22], and 3D and 4D Lorenz-type families [23,24]. In 2014, Aguila-Camacho et al. introduced a useful lemma for the Caputo fractional derivatives when $0<\alpha <1$ [25], which was widely applied to determine the global stability and boundedness of many fractional-order Lorenz-like chaotic systems with the help of the fractional-order extension of Lyapunov direct method, as shown in [26,27,28,29] and the references therein. Notably, Ke et al. simultaneously proved that there exist six pairs of symmetrical heteroclinic orbits in the Caputo fractional-order coupled Lorenz systems, using an approach that integrates the concepts of $\alpha$-/$\omega$-limit sets and Lyapunov-like functions [30].

Owing to its unique memory hereditary property, the Caputo fractional calculus has found widespread applications across scientific and engineering fields, including ecology [31], biochemistry [32], diffusion Equations [33,34,35], and optimization [36]. Meanwhile, heteroclinic orbits play a critical role in numerous research domains, such as space exploration [37,38,39], and biomathematics [40,41].

Given the facts above, a natural question arises: whether the Caputo fractional system (1) possesses heteroclinic orbits similar to those of the aforementioned coupled Lorenz systems [30]. To the best of the authors’ knowledge, even for the integer-order Lorenz system with a memristor (1), its heteroclinic orbits have not been considered at all, let alone those of its Caputo fractional-order counterpart. In this effort, we address the question of the existence of a pair of heteroclinic orbits in the Caputo fractional-order system (1), which aligns with the second principle of Sprott [42] and thus constitutes the main contribution of this paper.

What we do in this work not only exhibits the existence of heteroclinic orbits of a 3D memristor-based Caputo fractional-order Lorenz system, but also offers insight into the ones of other Lorenz-like systems, such as coupled memristor-based Lorenz systems, and hyperchaotic memristor-based Lorenz systems.

The remainder of this paper is structured as follows. Section 2 introduces the unified memristor-based Lorenz-type system, which encompasses Caputo fractional-order forms as special cases, along with our main theoretical result. In Section 3, we provide the detailed proof of heteroclinic orbits for the proposed theorem. Finally, Section 4 summarizes the main conclusions of this work and discusses future work.

2. The Model and the Primary Result

In this section, we first formulate a unified Caputo fractional-order memristor-based Lorenz system, given by

$$\left\{\begin{array}{cc}{}_{t_0}{}^C\mathcal{D}_t^{\alpha }x\hfill & =y,\hfill \\ {}_{t_0}{}^C\mathcal{D}_t^{\alpha }y\hfill & =(c-1)ax-(1+a+\rho r)y-axz-3\rho m{x}^{2}y,\hfill \\ {}_{t_0}{}^C\mathcal{D}_t^{\alpha }z\hfill & =-bz+{\displaystyle \frac{xy}{a}}+x^2,\hfill \end{array}\right.$$

(2)

where $a\ne 0,\rho >0,b,c,r,m\in \mathbb{R}$, the $\alpha$-order Caputo differential operator ${}_{t_0}{}^C\mathcal{D}_t^{\alpha }$ is denoted as follows:

$${}_{t_0}{}^C\mathcal{D}_t^{\alpha }f(t)={\displaystyle \frac{1}{\Gamma (n-\alpha )}}{\int }_{t_0}^t{(t-\tau )}^{n-\alpha -1}{f}^{(n)}(\tau )d\tau ,n=min\{k\in \mathbb{N}|k>\alpha \},$$

and we present the main result as the following theorem.

Theorem 1.

For $0\le 2a\le b$, $c>1$, $\rho m>0$, $a+1+\rho r>0$, and $0<\alpha \le 1$, system (2) has a pair of symmetrical orbits heteroclinic to $E_0=(0,0,0)$ and $E_{1,2}=(\pm \sqrt{b(c-1)},0,c-1)$, but no orbits homoclinic to $E_0$, or $E_{1,2}$, or heteroclinic to $E_1$ and $E_2$.

Remark 1.

In contrast, we must emphasize that system (2) is not topologically equivalent to the existing systems, especially the coupled Lorenz systems [30]. Indeed, the degree and dimension of the former are both three, while the ones of the latter are two and six, respectively. The numbers of equilibria and heteroclinic orbits of the former are three and two, which the ones of latter are nine and twelve. From the point of system formulation, the former is a memristor-based Lorenz system, while the latter is only a linear coupling of two 3D Lorenz systems. Although the method adopted in the present work and reference [30] is the same, i.e., the combination of Lyapunov function and α-/ω-limit sets, how to construct a Lyapunov function is unique. Further, to the best of our knowledge, other researchers may never consider the Lyapunov function of system (2) before. Therefore, the study of heteroclinic orbits of the fractional-order memristor-based system (2) is of theoretical and practical significance in its own right.

Remark 2.

From [11] (p. 861), the eigenvalues of $E_0$ are ${\lambda }_1=-b$ and ${\lambda }_{2,3}={\displaystyle \frac{1}{2}}(-(1+a+\rho r)\pm \sqrt{{(1+a+\rho r)}^2+4(c-1)a})$. Therefore, the fact ${\lambda }_2>0$ holds when $0\le 2a\le b$, $c>1$, $\rho m>0$, $a+1+\rho r>0$, and $0<\alpha \le 1$. As a result, for $0<\alpha \le 1$, $E_0$ is unstable on the basis of the Routh–Hurwitz stability criterion [26,43].

3. Proof of Theorem 1 and Heteroclinic Orbits to ${\mathit{E}}_{\mathbf{0}}$ and ${\mathit{E}}_{\mathbf{1},\mathbf{2}}$

In this section, following the analytical framework used for 3D Lorenz-like systems [13,17,18,19,22] and 6D coupled Lorenz systems [30], we formulate a rigorous proof of a pair of symmetrical orbits heteroclinic to $E_0$ and $E_{1,2}$ when $b\ge 2a>0$, $c>1$, $\rho m>0$, $a+1+\rho r>0$, and $0<\alpha \le 1$. For the convenience of subsequent derivation, we first introduce the symbols below:

(1)

$X(t;X_0)=(x(t;x_0),y(t;y_0),z(t;z_0))$: a solution of system (2) starting from initial value $X_0=(x_0,y_0,z_0)$.

(2)

$W_{\pm }^u$: positive and negative branches of $W^u(E_0)$ with $x_{+}>0$ and $-x_{+}<0$ as $t\to -\infty$.

(3)

${\gamma }^{\pm }=\left\{X_{\pm }(t;X_0)\right|X_{\pm }(t;X_0)=(\pm x_{+}(t;x_0),\pm y_{+}(t;y_0),z(t;z_0))\in W_{\pm }^u,t\in \mathbb{R}\}$.

First, we construct Lyapunov function

$$\begin{array}{cc}{V}_1(X(t;X_0))=\hfill & {\displaystyle \frac{1}{2}}\{{\displaystyle \frac{2a}{b-2a}}(-bz+x^2{)}^2+[-b(c-1)+x^2{]}^2+{\displaystyle \frac{2b}{a}}{y}^2\}\hfill \end{array}$$

for $0<2a<b$, and

$$\begin{array}{cc}{V}_2(X(t;X_0))=\hfill & {\displaystyle \frac{1}{2}}\{[-2a(c-1)+x^2{]}^2+4y^2\}\hfill \end{array}$$

for $0<2a=b$ and $2az=x^2$, with the fractional derivatives ${}_{t_0}{}^C\mathcal{D}_t^{\alpha }{V}_{1,2}$:

$$\begin{array}{ccc}{}_{t_0}{}^C\mathcal{D}_t^{\alpha }{V}_1(X(t;X_0)){|}_{(2)}\hfill & \le \hfill & {\displaystyle \frac{2a}{b-2a}}(-bz+x^2)(-b{}_{t_0}{}^C\mathcal{D}_t^{\alpha }z+2x{}_{t_0}{}^C\mathcal{D}_t^{\alpha }x)+[-b(c-1)+x^2](2x{}_{t_0}{}^C\mathcal{D}_t^{\alpha }x)\hfill \\ \hfill & \hfill & +{\displaystyle \frac{2b}{a}}y({}_{t_0}{}^C\mathcal{D}_t^{\alpha }y)\hfill \\ \hfill & =\hfill & {\displaystyle \frac{2a}{b-2a}}(-bz+x^2)[-b(-bz+{\displaystyle \frac{xy}{a}}+x^2)+2xy]+[-b(c-1)+x^2](2xy)\hfill \\ \hfill & \hfill & +{\displaystyle \frac{2b}{a}}y[(c-1)ax-(1+a+\rho r)y-axz-3\rho m{x}^{2}y]\hfill \\ \hfill & =\hfill & {\displaystyle \frac{2a}{b-2a}}(-bz+x^2)[-b(-bz+x^2)+(2+{\displaystyle \frac{b}{a}})xy]-2b(c-1)xy+2x^{3}y\hfill \\ \hfill & \hfill & +2b(c-1)xy-{\displaystyle \frac{2b(a+1+\rho r)}{a}}{y}^2-2bxyz-{\displaystyle \frac{6b\rho m}{a}}{x}^2{y}^2\hfill \\ \hfill & =\hfill & -{\displaystyle \frac{2ab}{b-2a}}(-bz+x^2{)}^2-{\displaystyle \frac{2b(a+1+\rho r)}{a}}{y}^2-{\displaystyle \frac{6b\rho m}{a}}{x}^2{y}^2\le 0,\hfill \end{array}$$

(3)

and

$$\begin{array}{ccc}{}_{t_0}{}^C\mathcal{D}_t^{\alpha }{V}_2(X(t;X_0)){|}_{(2)}\hfill & \le \hfill & [-2a(c-1)+x^2](2x{}_{t_0}{}^C\mathcal{D}_t^{\alpha }x)+4y({}_{t_0}{}^C\mathcal{D}_t^{\alpha }y)\hfill \\ \hfill & =\hfill & [-2a(c-1)+x^2](2xy)+4y[(c-1)ax-(1+a+\rho r)y-axz-3\rho m{x}^{2}y]\hfill \\ \hfill & =\hfill & -4a(c-1)xy+2x^{3}y+4a(c-1)xy-4(a+1+\rho r)y^2-2x^{3}y-12\rho m{x}^2{y}^2\hfill \\ \hfill & =\hfill & -4(a+1+\rho r)y^2-2x^{3}y-12\rho m{x}^2{y}^2\le 0.\hfill \end{array}$$

(4)

Herein, we only discuss the scenario of $0<2a<b$ of Theorem 1, and the case of $b=2a>0$ is similar and thus omitted. We first establish the following results.

Lemma 1.

For $0<2a<b$, $c>1$, $\rho m>0$, $a+1+\rho r>0$, and $0<\alpha \le 1$, the following conclusions hold.

(i) 

If $\exists t_{1,2}$, $t_1<t_2$ and $V_1(X(t_1;X_0))=V_1(X(t_2;X_0))$, then $X_0\in \{E_0,E_{1,2}\}$, i.e., ${}_{t_0}{}^C\mathcal{D}_t^{\alpha }{V}_1=0$.

(ii) 

If $\exists t_3\in \mathbb{R}$, $t\to -\infty$, $X(t;X_0)\to E_0$ and $x(t_3;x_0)>0$, then $V_1(X(t;X_0))<V_1(E_0)$ and $x(t;x_0)>0$, for all $t\in \mathbb{R}$. Namely, $X_0\in {\gamma }^{+}$.

Proof. 

(i) If $0<2a<b$, $c>1$, $\rho m>0$, $a+1+\rho r>0$, and $0<\alpha \le 1$, then it follows from ${}_{t_0}{}^C\mathcal{D}_t^{\alpha }{V}_1(X(t;X_0))\le 0$ in Equation (3) that ${}_{t_0}{}^C\mathcal{D}_t^{\alpha }{V}_1(X(t;X_0))=0$, for all $t\in (t_1,t_2)$, and consequently $X_0\in \{E_0,E_{1,2}\}$, i.e.,

$${}_{t_0}{}^C\mathcal{D}_t^{\alpha }x(t;x_0)\equiv 0,\hspace{1em}{}_{t_0}{}^C\mathcal{D}_t^{\alpha }y(t;y_0)\equiv 0,\hspace{1em}{}_{t_0}{}^C\mathcal{D}_t^{\alpha }z(t;z_0)\equiv 0.$$

(5)

More precisely, ${}_{t_0}{}^C\mathcal{D}_t^{\alpha }x(t;x_0)=y=0$ implies $x(t;x_0)=x_0$, and ${}_{t_0}{}^C\mathcal{D}_t^{\alpha }y(t;y_0)=0$, for all $t\in \mathbb{R}$. Consequently, $X_0\in \{E_0,E_{1,2}\}$.

(ii) Let us first demonstrate $V_1(E_0)>V_1(X(t;X_0))$, $\forall t\in \mathbb{R}$. Otherwise, $\exists t^{*}\in \mathbb{R}$, the statement $0<V_1(E_0)\le V_1(X(t^{*};X_0))$ holds. Since $\underset{t\to -\infty }{lim}X(t;X_0)=E_0$ and $V_1$ is continuous in t, there exists a sequence $\left\{t_n\right\}$ and an integer $n_1>0$ with $\underset{n\to \infty }{lim}{t}_n=-\infty$, and we arrive at $|V_1(X(t_n;X_0))-V_1(E_0)| <\epsilon$, $\forall \epsilon >0$ when $n>n_1$. Due to $\underset{n\to \infty }{lim}{t}_n=-\infty$ and $t^{*}\in \mathbb{R}$, one can easily find an integer $n_2>0$ with $t_n<t^{*}$, for all $n>max\{n_1,n_2\}$. Take $max\{n_1,n_2\}=n_0$ and $\epsilon ={\displaystyle \frac{1}{2}}[V_1(X(t^{*};X_0))-V_1(E_0)]>0$. We derive $V_1(X(t_n;X_0))-V_1(X(t^{*};X_0))=V_1(X(t_n;X_0))-V_1(E_0)+V_1(E_0)-V_1(X(t^{*};X_0))<\epsilon +V_1(E_0)-V_1(X(t^{*};X_0))=-\epsilon \le 0$. On the other side, ${}_{t_0}{}^C\mathcal{D}_t^{\alpha }{V}_1(X(t;X_0)){|}_{(2)}\le 0$ in Equation (3) results in $V_1(X(t_n;X_0))\ge V_1(X(t^{*};X_0))$, $\forall t_n<t^{*}$, $n>n_0$. In summary, the fact $V_1(X(t_n;X_0))=V_1(X(t^{*};X_0))$ together with the first assertion (i) result in $X_0\in \{E_0,E_{\pm }\}$. The hypothesis $\underset{t\to -\infty }{lim}X(t;X_0)=E_0$ yields $X_0\equiv E_0$ and $x(t,x_0)\equiv 0$, $\forall t\in \mathbb{R}$, contradicting the hypothesis $x(t,x_0)>0$ for some t. Hence, $V_1(X(t;X_0))<V_1(E_0)$, for all $t\in \mathbb{R}$.

Next, we prove $x(t;x_0)>0$, for all $t\in \mathbb{R}$. Otherwise, $\exists t_4\in \mathbb{R}$, such that $x(t_4;x_0)\le 0$. Since $x(t_3;x_0)>0$, $\exists t_3\in \mathbb{R}$, we obtain $x(t_5;x_0)=0$, $\exists t_5\in \mathbb{R}$. Since $V_1(X(t;X_0))<V_1(E_0)$ for all $t\in \mathbb{R}$, we arrive at $X(t_5;X_0)\in \left\{X(t;X_0)|V_1(E_0)>V_1(X(t;X_0))\right\}\cap \left\{X(t;X_0)|x=0\right\}=\left\{(0,y,z)|{\displaystyle \frac{a{b}^2{z}^2}{b-2a}}+{\displaystyle \frac{b}{a}}{y}^2+{\displaystyle \frac{1}{2}}{b}^2(c-1{)}^2<\right.\left.{\displaystyle \frac{1}{2}}{b}^2(c-1{)}^2\right\}=\mathsf{\varnothing }$, which leads to a contradiction and thus the fact $x(t;x_0)>0$, $\forall t\in \mathbb{R}$. The proof is finished. □

Utilizing Lemma 1, we arrive at Theorem 1’s proof below.

Proof. 

(1) Firstly, for $0<2a<b$, $c>1$, $\rho m>0$, $a+1+\rho r>0$, and $0<\alpha \le 1$, let us show that orbits homoclinic to $E_{1,2}$, or $E_0$, or heteroclinic to $E_1$ and $E_2$ do not exist.

Assume, to the contrary, that $X(t)=(x,y,z)$ is a homoclinic orbit to $E_0$, or $E_{1,2}$, or a heteroclinic orbit to $E_1$ and $E_2$, i.e., $\underset{t\to \mp \infty }{lim}X(t)=s_{\mp }$, where $s_{-}=s_{+}\in \{E_1,E_0,E_2\}$ or $\{s_{-},s_{+}\}=\{E_1,E_2\}.$ From Equation (3), we get

$$V_1(s_{+})\le V_1(X(t))\le V_1(s_{-}),$$

(6)

from which we arrive at $V_1(s_{-})=V_1(s_{+})$ and thus $V_1(X(t))\equiv V_1(s_{+})$. Based on Lemma 1 (i), $X(t)\in \{E_0,E_{1,2}\}$. Namely, the orbits homoclinic to $E_0$, or $E_{1,2}$, or heteroclinic to $E_1$ and $E_2$ do not exist.

Next, we demonstrate that ${\gamma }^{+}$ is indeed an orbit heteroclinic to $E_0$ and $E_1$, that is $\underset{t\to +\infty }{lim}{X}_{+}(t;X_0)=E_1$. Based on Lemma 1 (ii) and the concept of ${\gamma }^{+}$, we obtain the fact $x_{+}(t)>0$, $\forall t\in \mathbb{R}$, i.e., $\underset{t\to +\infty }{lim}{X}_{+}(t;X_0)\ne E_2,E_0$. Therefore, $\underset{t\to +\infty }{lim}{X}_{+}(t;X_0)=E_1$.

Finally, we establish the uniqueness of ${\gamma }^{+}$. Otherwise, suppose $X_1(t;X_0)$ were another orbit heteroclinic to $E_0$ and $E_1$, i.e., $\underset{t\to \mp \infty }{lim}{X}_1(t;X_0)=s_1^{\mp },$ where $\{s_1^{-},s_1^{+}\}=\{E_0,E_1\}.$ Like Equation (6), for all $t\in \mathbb{R}$, we also obtain $V_1(s_1^{-})\ge V_1(X_1(t;X_0))\ge V_1(s_1^{+})$ based on Equation (5).

Since $V_1(E_1)<V_1(E_0)$, we only derive $s_1^{-}=E_0$, and $s_1^{+}=E_1$, i.e.,

$$\underset{t\to -\infty }{lim}{X}_1(t;X_0)=E_0,\underset{t\to +\infty }{lim}{X}_1(t;X_0)=E_1.$$

Based on Lemma 1 (ii), we conclude that $X_1(t;X_0)\in {\gamma }^{+}$. Due to the symmetry of system (2), ${\gamma }^{-}$ is another orbit heteroclinic to $E_0$ and $E_2$. The proof is finished. □

Using the software package nlfode_vec in the FOTF Toolbox of Matlab R2016b [44] (p. 283), and choosing the termination time $t_n=200$, different parameter values, initial conditions, step sizes, and memory durations, Figure 1, Figure 2, Figure 3, Figure 4, Figure 5 and Figure 6 verify Theorem 1, i.e., the existence of a pair of orbits heteroclinic to $E_0$ and $E_{1,2}$.

Corollary 1.

It should be emphasized that the above proof also yields a pair of symmetrical orbits heteroclinic to $E_{1,2}$ and $E_0$ in the integer-order counterpart, i.e., system (1), as depicted in Figure 1, Figure 2, Figure 3, Figure 4, Figure 5 and Figure 6.

Remark 3.

Theorem 1 and paper [30] indicate a fact: if we construct Lyapunov-like functions to show heteroclinic orbits in the integer-order Lorenz-like systems, then the corresponding Caputo fractional-order counterparts of them have the same heteroclinic orbits.

Remark 4.

Generally speaking, constructing a suitable Lyapunov function for proving the existence of heteroclinic orbits in the Lorenz chaotic family is unique. However, there are still rules to follow. As indicated in [13,14,15,16,17,18,19,20,21,22,23,24,30] and as demonstrated by the Lyapunov functions $V_{1,2}$ with their corresponding fractional derivatives ${}_{t_0}{}^C\mathcal{D}_t^{\alpha }{V}_{1,2}$, one observes that $V(X(t;X_0))={}_{t_0}{}^C\mathcal{D}_t^{\alpha }V(X(t;X_0))=0$ whenever $X(t;X_0)$ is an equilibrium point. Guided by this principle and by trial and error, one may find the targeted Lyapunov function if it exists.

4. Conclusions

Several fundamental open questions remain in the study of memristor-based Lorenz-type chaotic systems: do they have heteroclinic orbits? If so, how can their existence be rigorously proven? Can we adopt the method of Lyapunov-like functions and $\alpha$-/$\omega$-limit sets, and if so, how can we find suitable Lyapunov-like functions? To date, researchers have paid little attention to these questions. In this effort, we address these gaps by investigating a 3D Caputo fractional-order Lorenz-type chaotic system with a flux-controlled memristor. By constructing two suitable Lyapunov-like functions, we rigorously prove that there is a pair of heteroclinic orbits in the Caputo fractional-order forms of the system. The theoretical results are further validated through numerical simulations, which show excellent agreement with the analytical findings.

For future work, we will extend our analysis to explore heteroclinic orbits in a more general class of memristive Lorenz-type systems given by

$$\left\{\begin{array}{cc}{}_{t_0}{}^C\mathcal{D}_t^{\alpha }x\hfill & =y,\hfill \\ {}_{t_0}{}^C\mathcal{D}_t^{\alpha }y\hfill & =ax-cy-dxz-e{x}^{2}y-i{z}^{2}y-k{x}^2+h,\hfill \\ {}_{t_0}{}^C\mathcal{D}_t^{\alpha }z\hfill & =-bz+fxy+g{x}^2,\hfill \end{array}\right.$$

(7)

where $a,c,d,e,i,k,h,b,f,g\in \mathbb{R},0<\alpha \le 1$. In fact, motivated by the results [13,17] and the work of this paper, we manage to construct the Lyapunov function $V={\displaystyle \frac{1}{2}}\{y^2+{\displaystyle \frac{d}{b(bf-2g)}}(-bz+g{x}^2{)}^2+{\displaystyle \frac{2bk+3gd{l}_1}{6b{l}_1}}(-l_1^2+x^2{)}^2+{\displaystyle \frac{h}{l_1}}(-l_1+x{)}^2-{\displaystyle \frac{k}{3l_1}}(-l_{1}x+x^2{)}^2\}$, and the corresponding fractional derivative ${}_{t_0}{}^C\mathcal{D}_t^{\alpha }V=-(c+e{x}^2+i{z}^2)y^2-{\displaystyle \frac{d}{bf-2g}}(-bz+g{x}^2{)}^2$, when $b>0$, $c>0$, $i>0$, $d>0$, $bf-2g>0$, $h{l}_1>0$, $k{l}_1<0$, $l_1(2bk+3gd{l}_1)>0$, $gd{l}_1^3+bk{l}_1^2-ab{l}_1-bh=0$. Furthermore, we will investigate other fundamental dynamical properties of these systems, including homoclinic orbits, integrability, and potential engineering applications of the derived results.