Annual Report 2023

1.   Introduction

In this paper, we review a recently reported modified Chen system and examines its singular orbits, giving some new insights into it and extending the existing results ^[1]. It not only contains the classic Chen system as a special case, but also gives rise to multi-wings attractors with the higher largest Lyapunov exponent. To further understand its nature, we will consider the existence of homoclinic and heteroclinic orbits, which involve some real world applications ^{[2,3,4,5,6,7,8,9]}, i.e., heart tissue, neurons, cell signalling, chemistry, biomathematics and mechanics, etc. Particularly, when planetary scientists design space missions, the heteroclinic connections between period orbits need to be studied in planar restricted circular three body problem ^[10,11,12].

For the sake of the following discussion, the concepts of homoclinic and heteroclinic orbits are introduced, i.e., a heteroclinic (resp., homoclinic) orbit is a type of orbit doubly asymptotic to two different equilibrium points or closed orbits (resp., the same equilibrium point or closed orbit) ^[13,14]. In order to uncover homoclinic and heteroclinic orbits, Shilnikov et al. developed a powerful tool, a combination of contraction map and boundary problem, and categorized chaos of three-dimensional quadratic autonomous differential systems, i.e., chaos of the Shilnikov homoclinic or heteroclinic orbit type, or the hybrid type with both Shilnikov homoclinic and heteroclinic orbits, etc. ^[14].

With a combination of definitions of the $\alpha$-limit and $\omega$-limit set and the Lyapunov function, Li et al. revisited the Chen system and proved the existence of a pair of heteroclinic orbits of it ^[15]. Inspired by that example, other researchers began to consider heteroclinic orbits of other Lorenz-like systems ^{[16,17,18,19,20,21,22,23,24,25]} one after another. When studying the Tricomi problem on homoclinic and heteroclinic orbits ^[26], Leonov formulated the fishing principle and applied it to Lorenz-type systems ^[27]. Through tracing the stable and unstable manifolds of the Shimizu-Morioka model, Tigan and Turaev ^[28] detected a pair of homoclinic orbits to the origin. Wiggins, Feng and Hu^[13,29] also utilized the Melnikov method to study homoclinic and heteroclinic orbits and, thus, determined chaos in the sense of Smale horseshoes, such as its existence, stability, bifurcation, mutual position between the stable and the unstable manifold, and so on.

More importantly, some bifurcations of homoclinic or heteroclinic orbits may shed light on the problem of the nonlinear relationship between equilibria and the number of multi-wings/scrolls ^[1,30,31]. Recently, Gilardi-Velázquez et al. ^[32] collided two heteroclinic orbits to create a new square chaotic attractor, providing a mechanism to establish bistability in a new class of piecewise linear (PWL) dynamical systems. By rupturing heteroclinic-like orbits of a class of PWL systems, Escalante-González and Campos ^[33] revealed hidden attractors coexisting with self-excited ones.

So far as is known, scholars seldom consider homoclinic and heteroclinic orbits of the modified Chen system ^[1]. To achieve this target, we reinvestigate it and present the following main contribution:

1) Utilizing a combination of definitions of the $\alpha$-limit and $\omega$-limit set and the Lyapunov function to prove that there may exist many pairs of heteroclinic orbits to (1) $E_{0}$ and $E_{\pm}$ or (2) $E_{+}$ or (3) $E_{-}$.

2) Applying a Hamiltonian function to prove that there may exist multitudinous potential homoclinic orbits to $E_{0}$ and $E_{\pm}$, and heteroclinic orbits to $E_{\pm}$ on the invariant algebraic surface $z = \frac{x^{2}}{2a}$.

This paper is organized as follows. The new modified Chen system is introduced and the main results are reported in Section 2. In Section 3, one derives the existence of multitudinous potential homoclinic and heteroclinic orbits. At last, the conclusions are given in Section 4.

2.   New modified Chen system and the main results

In this section, we consider the modified Chen system ^[1]:

where $a, c, k, k_{1}, b\in \mathbb{R}$. Apparently, for $k = 0$, system (2.1) reduces to the Chen system. Of particular interest is that the new added term $k\sin{(k_{1}z)}$ guarantees the creation of multi-wings attractors with higher largest Lyapunov exponent coexisting with multiple equilibria, as illustrated in [1, Figures 2–4, p.2]. Next, we mainly focus on its multiple homoclinic and heteroclinic orbits.

Figure 1.  Nine solutions of $21 = z(1-0.5\sin{z})$ with $z > 0$ when $(a, c, k, k_{1}) = (35, 28, 0.5, 1)$.

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Figure 2.  When $(a, c, k, k_{1}) = (35, 28, 0.5, 1)$ and $(x_{0}^{1, 2}, y_{0}^{1, 2}, z_{0}^{1}) = (\pm 0.13, \pm1.3, 1.6)\times 10^{-7}$, (a) $b = 70$, system (2.1) has a pair of heteroclinic orbits to $E_{\pm}^{3}$ and $E_{0}$, which coexist with $E_{\pm}^{1, 2, 4, 5, 6, 7, 8, 9}$; (b) $b = 75$, system (2.1) has a pair of heteroclinic orbits to $E_{\pm}^{1}$ and $E_{0}$, which coexist with $E_{\pm}^{2, 3, 4, 5, 6, 7, 8, 9}$.

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Figure 3.  When $(a, c, k, k_{1}, b) = (35, 28, 0.5, 1, 70)$, (a) $(x_{0}^{3, 4}, y_{0}^{3, 4}, z_{0}^{2}) = (\pm 36.0705, \pm36.0705, 18.5868)$ (resp., $(\pm 36.0705, \pm36.0705, 18.587)$), a pair of heteroclinic orbits to $E_{\pm}^{2}$ and $E_{\pm}^{1}$ (resp., $E_{\pm}^{3}$), (b) $(x_{0}^{5, 6}, y_{0}^{5, 6}, z_{0}^{3}) = (\pm 42.2435, \pm42.2435, 25.494)$ (resp., $(\pm 42.2435, \pm42.2435, 25.492)$), a pair of heteroclinic orbits to $E_{\pm}^{4}$ and $E_{\pm}^{5}$ (resp., $E_{\pm}^{1}$), (c) $(x_{0}^{7, 8}, y_{0}^{7, 8}, z_{0}^{4}) = (\pm 47.4648, \pm47.4648, 32.1845)$ (resp., $(\pm 47.4648, \pm47.4648, 32.1843)$), a pair of heteroclinic orbits to $E_{\pm}^{6}$ and $E_{\pm}^{7}$ (resp., $E_{\pm}^{5}$), (d) $(x_{0}^{9, 10}, y_{0}^{9, 10}, z_{0}^{5}) = (\pm 52.1592, \pm52.1592, 38.8656)$ (resp., $(\pm 52.1592, \pm52.1592, 38.8654)$), a pair of heteroclinic orbits to $E_{\pm}^{8}$ and $E_{\pm}^{9}$ (resp., $E_{\pm}^{3}$). These figures suggest that system (2.1) has multitudinous potential heteroclinic orbits to $E_{+}$ or $E_{-}$.

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Figure 4.  When $(a, c, k, k_{1}, b) = (35, 28, 0.5, 1, 75)$, (a) $(x_{0}^{11, 12}, y_{0}^{11, 12}, z_{0}^{2}) = (\pm 37.3365, \pm37.3365, 18.5868)$ (resp., $(\pm 37.3365, \pm37.3365, 18.587)$), a pair of heteroclinic orbits to $E_{\pm}^{2}$ and $E_{\pm}^{1}$ (resp., $E_{\pm}^{3}$), (b) $(x_{0}^{13, 14}, y_{0}^{13, 14}, z_{0}^{3}) = (\pm 43.7261, \pm43.7261, 25.494)$ (resp., $(\pm 43.7261, \pm43.7261, 25.492)$), a pair of heteroclinic orbits to $E_{\pm}^{4}$ and $E_{\pm}^{5}$ (resp., $E_{\pm}^{1}$), (c) $(x_{0}^{15, 16}, y_{0}^{15, 16}, z_{0}^{4}) = (\pm 49.1307, \pm49.1307, 32.1845)$ (resp., $(\pm 49.1307, \pm49.1307, 32.1843)$), a pair of heteroclinic orbits to $E_{\pm}^{6}$ and $E_{\pm}^{7}$ (resp., $E_{\pm}^{1}$), (d) $(x_{0}^{17, 18}, y_{0}^{17, 18}, z_{0}^{5}) = (\pm 53.9899, \pm53.9899, 38.8656)$ (resp., $(\pm 53.9899, \pm53.9899, 38.8654)$), a pair of heteroclinic orbits to $E_{\pm}^{8}$ and $E_{\pm}^{9}$ (resp., $E_{\pm}^{3}$). These figures suggest that system (2.1) has multitudinous potential heteroclinic orbits to $E_{+}$ or $E_{-}$.

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First of all, let us present equilibria of system (2.1):

Proposition 2.1. 1) If $b = 0$, then $E_{z} = \{(0, 0, z)|z\in \mathbb{R}\}$ is non-isolated equilibria.

2) When $\forall\sigma \neq 0$, $2c-a = \sigma[1-k\sin{(k_{1}\sigma)}]$, $b\sigma > 0$, and $a \neq 0$. System (2.1) may have many pairs of equilibria: $E_{\pm} = \{(\pm\sqrt{b\sigma}, \pm\sqrt{b\sigma}, \sigma)|b\sigma > 0, 2c-a = \sigma[1-k\sin{(k_{1}\sigma)}]\}$, except for $E_{0} = (0, 0, 0)$.

We arrive at heteroclinic orbits in the case of non-Hamiltonian modified Chen system (2.1):

Proposition 2.2. If $2c > a > c > 0$, $b \geq 2a > 0$, $\forall\sigma > 0$, $2c-a = \sigma[1-k\sin{(k_{1}\sigma)}]$, then

(i) there are no homoclinic orbits to $E_{+}$ or $E_{-}$, or heteroclinic orbits to $E_{+}$ and $E_{-}$ in system (2.1);

(ii) multitudinous potential heteroclinic orbits to (1) $E_{0}$ and $E_{\pm}$, or (2) $E_{+}$ or (3) $E_{-}$ exist in system (2.1).

Finally, we give homoclinic and heteroclinic orbits on the invariant algebraic surface $z = \frac{x^{2}}{2a}$ in the following statement.

Proposition 2.3. Suppose $c = a$, $b-2a = 0$, $\triangle = x^{2}-\frac{x^{4}}{4a^{2}}+\frac{2}{a}[\frac{ak}{k_{1}^{2}}\sin{(\frac{k_{1}x^{2}}{2a})}-\frac{kx^{2}}{2k_{1}}\cos{(\frac{k_{1}x^{2}}{2a})}]$, $2a^{2} = x_{*}^{2}(1-k\sin{(\frac{k_{1}x_{*}^{2}}{2a})})$ and $\Gamma_{*} = \frac{-ax_{*}^{2}}{2}+\frac{x_{*}^{4}}{8a}-[\frac{ak}{k_{1}^{2}}\sin{(\frac{k_{1}x_{*}^{2}}{2a})}-\frac{kx_{*}^{2}}{2k_{1}}\cos{(\frac{k_{1}x_{*}^{2}}{2a})}]$, then the following two assertions hold.

1) If $\triangle \geq 0$, then system (2.1) has a pair of homoclinic orbits to $E_{0}$: $y = x\pm \sqrt{\triangle}$.

2) If $\triangle+\Gamma_{*} \geq 0$, then system (2.1) has a multitude of heteroclinic or homoclinic orbits to $E_{\pm}$: $y = x\pm \sqrt{\triangle+\Gamma_{*}}$, $|x| < |x_{*}|$.

3.   Homoclinic and heteroclinic orbit and proofs of Proposition 2.2 and 2.3

In this section, we first study the existence of heteroclinic orbits when $2c > a > c > 0$, $b \geq 2a > 0$, $\forall\sigma > 0$, and $2c-a = \sigma[1-k\sin{(k_{1}\sigma)}]$, as ^{[15,16,17,18,19,20,21,22,23,24,25]}.

To facilitate derivation, denote by $\phi_{t}(q_{0}) = (x(t; x_{0}), y(t; y_{0}), z(t; z_{0}))$ a solution of system (2.1) with the initial value $q_{0} = (x_{0}, y_{0}, z_{0})\in\mathbb{R}^{3}$. Let $\gamma^{-}$ (resp., $\gamma^{+}$) be the negative (resp., positive) branch of the unstable manifold $W^{u}(E_{0})$ corresponding to $-x_{+} < 0$ (resp. $x_{+} > 0$) as $t\rightarrow -\infty$, i.e., $\gamma^{\mp} = \{\phi_{t}^{\mp}(q_{0})|\phi_{t}^{\mp}(q_{0}) = (\mp x_{+}(t; x_{0}), \mp y_{+}(t; y_{0}), z_{+}(t; z_{0}))\in W_{\mp}^{u}(E_{0}), t\in\mathbb{R}\}$.

Put the first Lyapunov function

with the derivative of it along trajectories of system (2.1):

for the case of $b > 2a > 0$ and $\sin{(k_{1}z)} = \sin{(k_{1}\sigma)}$, and the second one

with the corresponding derivative:

for the case of $b = 2a > 0$, $z = \frac{x^{2}}{2a}$ and $\sin{(\frac{k_{1}x^{2}}{2a})} = \sin{(k_{1}\sigma)}$.

Here, one only considers the case of heteroclinic orbits to $E_{0}$ and $E_{\pm}$. The other case of heteroclinic orbits to $E_{+}$ or $E_{-}$ is similar to the ones in ^[24,25] and is omitted.

For proving Proposition 2.2, one introduces the following result.

Proposition 3.1. When $2c > a > c > 0$, $b \geq 2a > 0$, $\forall\sigma > 0$, and $2c-a = \sigma[1-k\sin{(k_{1}\sigma)}]$, the following two assertions are right.

(i) If $\exists$ $t_{1, 2}$, such that $t_{1} < t_{2}$ and $V_{1, 2}(\phi_{t_{1}}(q_{0})) = V_{1, 2}(\phi_{t_{2}}(q_{0}))$, then $q_{0}$ is only one of the equilibria.

(ii) If $\lim\phi_{t}(q_{0})_{t\rightarrow -\infty} = E_{0}$, and $\exists$ $t\in\mathbb{R}$, $x(t; x_{0}) < 0$, then $V_{1, 2}(E_{0}) > V_{1, 2}(\phi_{t}(q_{0}))$ and $x(t; x_{0}) < 0$ for all $t\in\mathbb{R}$. Therefore, $q_{0}\in W_{-}^{u}(E_{0})$.

Proof. (ⅰ) If $2c > a > c > 0$, $b \geq 2a > 0$, $\forall\sigma > 0$, and $2c-a = \sigma[1-k\sin{(k_{1}\sigma)}]$, then one arrives at $\frac{dV_{1, 2}(\phi_{t}(q_{0}))}{dt}\big|_{(2.1)} \leq 0$ from Eqs (3.1) and (3.2). On the basis of the hypothesis (ⅰ), one obtains $\frac{dV_{1, 2}(\phi_{t}(q_{0}))}{dt}\big|_{(2.1)} = 0$, $\forall t\in(t_{1}, t_{2})$, and derives that $q_{0}$ is one stationary point, i.e.,

namely, $\dot{x}(t; x_{0}) = a(y - x) = 0$ implies $x(t) = x_{0}$ and $\dot{y}(t, y_{0}) = 0$, $y(t) = y_{0} = 0$, $\forall t \in \mathbb{R}$.

Since $z = \frac{x^{2}}{2a}$ is an invariant algebraic surface of system (2.1) with cofactor $-2a$ for $b = 2a$ ^[34], $\phi_{t}(q_{0})\in\left\{y - x = 0\right\}$ results in Eq (3.3).

(ⅱ) Due to $\lim\phi_{t}(q_{0})_{t\rightarrow -\infty} = E_{0}$ and $x(t; x_{0}) < 0$, $\exists$ $t\in\mathbb{R}$, $q_{0}$ cannot be a fixed point. Otherwise, $0 < V_{1, 2}(E_{0}) \leq V_{1, 2}(\phi_{t_{0}}(q_{0}))$, $\exists t_{0}\in \mathbb{R}$. Therefore, one arrives at $V_{1, 2}(\phi_{t_{1}}(q_{0})) \leq V_{1, 2}(\phi_{t_{0}}(q_{0}))$, $\exists t_{1}\in \mathbb{R}$. Based on $\frac{dV_{1, 2}(\phi_{t}(q_{0}))}{dt}\big|_{(2.1)} \leq 0$, $V_{1, 2}(\phi_{t_{1}}(q_{0})) = V_{1, 2}(\phi_{t_{0}}(q_{0}))$ and assertion (ⅰ), $q_{0}$ is one stationary point. The condition $\lim_{t\rightarrow -\infty}\phi_{t}(q_{0}) = E_{0}$ leads to $q_{0} = E_{0}$ and $x(t; x_{0}) = 0$, $\forall t \in \mathbb{R}$. A contradiction happens. The fact $V_{1, 2}(E_{0}) > V_{1, 2}(\phi_{t}(q_{0}))$ holds, $\forall t \in \mathbb{R}$.

Next, let us show $x(t, x_{0}) < 0$, $\forall t\in \mathbb{R}$. If not, $\exists$ $t^{'} \in \mathbb{R}$, $x(t^{'}, x_{0}) \geq 0$. Based on the hypothesis (ⅱ), one obtains $x(t^{''}, x_{0}) < 0$, $\exists$ $t^{''} \in \mathbb{R}$. As a result, $x(\tau, x_{0}) = 0$, $\exists$ $\tau\in \mathbb{R}$. According to $V_{1, 2}(E_{0}) > V_{1, 2}(\phi_{t}(q_{0}))$, $\forall t \in \mathbb{R}$, one gets $\phi_{\tau}(q_{0})\in \{(x, y, z): V_{1, 2}(E_{0}) > V_{1, 2}(x, y, z)\} \cap \{x = 0\}$, which is contradiction. Actually, both $\{(x, y, z): V_{1}(E_{0}) > V_{1}(x, y, z)\} \cap \{x = 0\} = \{(x, y, z): \frac{1}{2}[\frac{2ab(b-2a)\sigma}{2c-a}y^{2} + 2ab^{2}z^{2}+(b-2a)b^{2}\sigma^{2}] < \frac{(b-2a)b^{2}\sigma^{2}}{2}\}$ and $\{(x, y, z): V_{2}(E_{0}) > V_{2}(x, y, z)\} \cap \{x = 0\} = \{(x, y, z): 2a^{2}\sigma y^{2}+2(2c-a)a^{2}\sigma^{2} < 2(2c-a)a^{2}\sigma^{2}\}$ are all empty. So, $\forall t \in \mathbb{R}$, $x(t, q_{0}) < 0$. The proof is completed.

From Proposition 3.1, one proves Proposition 2.2 as follows.

Proof of Proposition 2.2:. (ⅰ) When $2c > a > c > 0$, $b \geq 2a > 0$, $\forall\sigma > 0$, and $2c-a = \sigma[1-k\sin{(k_{1}\sigma)}]$, there are neither homoclinic orbits to $E_{+}$ or $E_{-}$, nor heteroclinic orbits to $E_{+}$ and $E_{-}$ in system (2.1). Otherwise, suppose $p(t) = (x, y, z)$ is a homoclinic orbit to $E_{+}^{'}$ or $E_{-}^{'}$, or heteroclinic orbit to $E_{\pm}^{'}$, where $\forall E_{+}^{'}\in E_{+}$ and $\forall E_{-}^{'}\in E_{-}$, i.e., $\lim\limits_{t \to -\infty} p(t) = e_{-}, \quad \lim\limits_{t \to + \infty} p(t) = e_{+},$ where $e_{-}$ and $e_{+}$ satisfy either $e_{-} = e_{+} \in \{E_{-}^{'}, E_{0}, E_{+}^{'}\}$ or $\{e_{-}, e_{+}\} = \{E_{-}^{'}, E_{+}^{'}\}$.

Based on $\frac{dV_{1, 2}(\phi_{t}(q_{0}))}{dt}\big|_{(2.1)} \leq 0$, one arrives at

In any case, the relation $V_{1, 2}(e_{-}) = V_{1, 2}(e_{+})$ all hold, which also leads to $V_{1, 2}(p(t))\equiv V_{1, 2}(e_{-})$. According to the assertion (ⅰ) of Proposition 3.1, $p(t)$ is just one stationary point. In a word, homoclinic orbits to $E_{+}$ or $E_{-}$, or heteroclinic orbits to $E_{+}$ and $E_{-}$ do not exist in system (2.1).

(ⅱ) Next, one proves that $p(t)$ is a heteroclinic orbit to $E_{0}$ and $E_{-}^{'}$, i.e., $\lim\limits_{t \to +\infty}p(t) = E_{-}^{'}$. On the basis of the definition of $p(t)$ and the second assertion of Proposition 3.1, $V_{1, 2}(E_{0}) > V_{1, 2}(p(t))$ holds, which also yields $\lim\limits_{t \to +\infty}p(t) \neq E_{0}$, i.e., $\lim\limits_{t \to +\infty}p(t) = E_{-}^{'}$.

At last, let us show that, if system (2.1) has a heteroclinic orbit to $E_{0}$ and $E_{-}^{'}$, then this orbit is just $p(t)$.

Suppose $p^{*}_{-}(t) = (x^{*}(t), y^{*}(t), z^{*}(t))$ is a solution of system (2.1) such that

where $e_{1}^{-}$ and $e_{1}^{+}$ satisfy $\{e_{1}^{-}, e_{1}^{+}\} = \{E_{0}, E_{-}^{'}\}.$ Since $\frac{dV_{1, 2}(\phi_{t}(q_{0}))}{dt}\big|_{(2.1)} \leq 0$, $\forall t\in \mathbb{R}$, $V_{1, 2}(e_{1}^{-})\geq V_{1, 2}(p^{*}_{-}(t))\geq V_{1, 2}(e_{1}^{+})$ holds. Due to $V_{1, 2}(E_{0}) > V_{1, 2}(E_{-}^{'})$, one gets $e_{1}^{-} = E_{0}$ and $e_{1}^{+} = E_{-}^{'}$, i.e.,

which yields $p^{*}_{-}(t)\in \gamma^{-}$ based on the assertion (ⅱ) of Proposition 3.1. Since system (2.1) is symmetrical w.r.t. the $z$–axis, there exists a unique heteroclinic orbit $p^{*}_{+}(t)\in \gamma^{+}$, i.e., $\lim\limits_{t\to -\infty} p^{*}_{+}(t) = E_{0}, \lim\limits_{t\to +\infty} p^{*}_{+}(t) = E_{+}^{'}$.

Due to the term of $k\sin{(k_{1}z)}$, $E_{\pm}$ may include many isolated stationary points $E_{\pm}^{'}$ and, thus, may generate many heteroclinic orbits to $E_{0}$ and $E_{\pm}$. The proof is over.

Set $(a, c, k, k_{1}) = (35, 28, 0.5, 1)$, $b = 70, 75$. The nontrivial equilibria of system (2.1) satisfies $2c-a = z(1-k\sin{(k_{1}z)})$, i.e., $21 = z(1-0.5\sin{(z)})$ with $z > 0$. At this time, system (2.1) has nine pairs of nontrivial equilibria: $E_{\pm}^{i} = (\pm \sqrt{bz_{i}}, \pm \sqrt{bz_{i}}, z_{i})$, where $z_{1} = 16.319$, $z_{2} = 18.5869$, $z_{3} = 21.9081$, $z_{4} = 25.493$, $z_{5} = 27.7653$, $z_{6} = 32.1844$, $z_{7} = 33.7036$, $z_{8} = 38.8655$, and $z_{9} = 39.6184$, as depicted in Figure 1 and Tables 1 and 2.

Table 1.  The dynamics of $E_{\pm}^{i}$ with $(a, c, k, k_{1}, b) = (35, 28, 0.5, 1, 70)$, $i = 1, 2, \cdots, 9$.

$E_{\pm}$	Classification	Eigenvalues
$E_{\pm}^{1} = (\pm 33.7984, \pm 33.7984, 16.319)$	Stable focus	$-3.5\pm95.3515i, -70$
$E_{\pm}^{2} = (\pm 36.0705, \pm 36.0705, 18.5869)$	Saddle	$97.589, -104.589, -70$
$E_{\pm}^{3} = (\pm 39.1608, \pm 39.1608, 21.9081)$	Stable focus	$-3.5\pm 134.9i, -70$
$E_{\pm}^{4} = (\pm 42.2435, \pm 42.2435, 25.493)$	Saddle	$137.312, -144.312, -70$
$E_{\pm}^{5} = (\pm 44.086, \pm 44.086, 27.7653)$	Stable focus	$-3.5\pm158.18i, -70$
$E_{\pm}^{6} = (\pm 47.4648, \pm 47.4648, 32.1844)$	Saddle	$153.3704, -160.3704, -70$
$E_{\pm}^{7} = (\pm 48.5721, \pm 48.5721, 33.7036)$	Stable focus	$-3.5\pm166.07i, -70$
$E_{\pm}^{8} = (\pm 52.1592, \pm 52.1592, 38.8655)$	Saddle	$135.592, -142.592, -70$
$E_{\pm}^{9} = (\pm 52.662, \pm 52.662, 39.6184)$	Stable focus	$-3.5\pm142.18i, -70$

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Table 2.  The dynamics of $E_{\pm}^{i}$ with $(a, c, k, k_{1}, b) = (35, 28, 0.5, 1, 75)$, $i = 1, 2, \cdots, 9$.

$E_{\pm}$	Classification	Eigenvalues
$E_{\pm}^{1} = (\pm 34.9846, \pm 34.9846, 16.319)$	Stable focus	$-3.5\pm98.7041i, -75$
$E_{\pm}^{2} = (\pm 37.3365, \pm 37.3365, 18.5869)$	Saddle	$101.1325, -108.1325, -75$
$E_{\pm}^{3} = (\pm 40.5353, \pm 40.5353, 21.9081)$	Stable focus	$-3.5\pm 139.64i, -75$
$E_{\pm}^{4} = (\pm 43.7261, \pm 43.7261, 25.493)$	Saddle	$142.2534, -149.2534, -75$
$E_{\pm}^{5} = (\pm 45.6333, \pm 45.6333, 27.7653)$	Stable focus	$-3.5\pm163.73i, -75$
$E_{\pm}^{6} = (\pm 49.1307, \pm 49.1307, 32.1844)$	Saddle	$158.8772, -165.8772, -75$
$E_{\pm}^{7} = (\pm 50.2769, \pm 50.2769, 33.7036)$	Stable focus	$-3.5\pm171.9i, -75$
$E_{\pm}^{8} = (\pm 53.9899, \pm 53.9899, 38.8655)$	Saddle	$140.4712, -147.4712, -75$
$E_{\pm}^{9} = (\pm 54.5104, \pm 54.5104, 39.6184)$	Stable focus	$-3.5\pm147.19i, -75$

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DownLoad: CSV

In contrast to the Lorenz-like systems ^{[19,20,24,25]}, Figures 2–4 show multitudinous heteroclinic orbits to (1) $E_{0}$ and $E_{\pm}$, (2) $E_{+}$, (3) $E_{-}$.

Remark 1. Motivated by ^[16], a pair of homoclinic orbits to $E_{0}$ may exist in the scenario of non-Hamiltonian modified Chen system (2.1) when $(a, c, k, k_{1}, b) = (1, 0.8, 0.5, 1, 1.088966)$, as shown in Figure 5.

Figure 5.  When $(a, c, k, k_{1}, b) = (1, 0.8, 0.5, 1, 1.088966)$ and $(x_{0}^{1, 2}, y_{0}^{1, 2}, z_{0}^{1}) = (\pm 0.13, \pm1.3, 1.6)\times 10^{-7}$, a pair of homoclinic orbits to $E_{0}$ may exist in system (2.1).

DownLoad: Full-Size Img PowerPoint

Remark 2. When $a = c$, $V_{2}$ is a first integral. Proposition 2.2 includes the result on heteroclinic orbits given in ^[15,16] as a special case when $k = 0$.

Finally, for $b = 2a$, let us consider homoclinic and heteroclinic orbits on the invariant algebraic surface $z = \frac{x^{2}}{2a}$ ^[34] of system (2.1), i.e., the ones of following system:

When $c = a$, system (3.5) reduces into a Hamiltonian system with the corresponding Hamiltonian function

Obviously, the equilibrium points of system (3.5) are $S_{0} = (0, 0)$, $S_{\pm} = (\pm x, \pm x)$ for $2a^{2} = x^{2}(1-k\sin{(\frac{k_{1}x^{2}}{2a})})$. Further, the proof of Proposition 2.3 easily follows and is omitted here.

Set $(a, c, k, k_{1}) = (\pm35, \pm35, 0.5, 1)$, except for $S_{0}$, Figures 6 and 8 show another fifteen pairs of nontrivial equilibria $S_{\pm}^{i}$ of system (3.5), i.e., the $E_{\pm}^{i}$ of system (2.1), $i = 1, 2, \cdots, 15$, as shown in Tables 3 and 4. Obviously, $S_{\pm}^{i}$ are included in $E_{\pm}^{i}$ and are not listed here, $i = 1, 2, \cdots, 15$. Further, system (3.5) (resp., (2.1)) has eight pairs of homoclinic orbits to $S_{\pm}^{i}$ and $S_{0}$ (resp., $E_{\pm}^{i}$ and $E_{0}$), $i = 2, 4, 6, 8, 10, 12, 14$, as depicted in Figures 7 and 9. Being different from the ones ^[25,35], the heteroclinic orbits to $S_{\pm}^{i}$ (resp., $E_{\pm}^{i}$) are not observed in numerical simulation.

Figure 6.  Fifteen pairs of nontrivial equilibria $S_{\pm}^{i}$ when $(a, c, k, k_{1}) = (35, 35, 0.5, 1)$, $i = 1, 2, \cdots, 15$.

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Figure 7.  When $(a, c, k, k_{1}, b) = (35, 35, 0.5, 1, 70)$, (a) (b) eight pairs of homoclinic orbits to $S_{\pm}^{i}$ and $S_{0}$ of system (3.5), (c) eight pairs of homoclinic orbits to $E_{\pm}^{i}$ and $E_{0}$ of system (2.1), $i = 2, 4, 6, 8, 10, 12, 14$.

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Figure 8.  Fifteen pairs of nontrivial equilibria $S_{\pm}^{i}$ when $(a, c, k, k_{1}) = (-35, -35, 0.5, 1)$, $i = 1, 2, \cdots, 15$.

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Table 3.  The dynamics of $E_{\pm}^{i}$ with $(a, c, k, k_{1}, b) = (35, 35, 0.5, 1, 70)$, $i = 1, 2, \cdots, 15$.

$E_{\pm}$	Classification	Eigenvalues
$E_{\pm}^{1} = (\pm 40.4862, \pm 40.4862, 23.4162)$	Andronov-Hopf bifurcation	$\pm72.37i, -70$
$E_{\pm}^{2} = (\pm 40.9545, \pm 40.9545, 23.9610)$	Saddle	$\pm73.1982, -70$
$E_{\pm}^{3} = (\pm 44.8424, \pm 44.8424, 28.7263)$	Andronov-Hopf bifurcation	$\pm 168.62i, -70$
$E_{\pm}^{4} = (\pm 46.7089, \pm 46.7089, 31.1674)$	Saddle	$\pm174.6572, -70$
$E_{\pm}^{5} = (\pm 49.2008, \pm 49.2008, 34.5817)$	Andronov-Hopf bifurcation	$\pm210.46i, -70$
$E_{\pm}^{6} = (\pm 51.4735,$ $\pm 51.4735, 37.8503)$	Saddle	$\pm217.0728, -70$
$E_{\pm}^{7} = (\pm 53.286, \pm 53.286, 40.5628)$	Andronov-Hopf bifurcation	$\pm240.47i, -70$
$E_{\pm}^{8} = (\pm 55.7622, \pm 55.7622, 44.4203)$	Saddle	$\pm245.134, -70$
$E_{\pm}^{9} = (\pm 57.1156, \pm 57.1156, 46.6027)$	Andronov-Hopf bifurcation	$\pm261.48i, -70$
$E_{\pm}^{10} = (\pm 59.7153, \pm 59.7153, 50.9417)$	Saddle	$\pm261.5179, -70$
$E_{\pm}^{11} = (\pm 60.7207, \pm 60.7207, 52.6715)$	Andronov-Hopf bifurcation	$\pm272.84i, -70$
$E_{\pm}^{12} = (\pm 63.4129, \pm 63.4129, 57.4457)$	Saddle	$\pm 263.8527, -70$
$E_{\pm}^{13} = (\pm 64.1281, \pm 64.1281, 58.7488)$	Andronov-Hopf bifurcation	$\pm 271.2i, -70$
$E_{\pm}^{14} = (\pm 66.9144, \pm 66.9144, 63.9648)$	Saddle	$\pm241.3816, -70$
$E_{\pm}^{15} = (\pm 67.3528, \pm 67.3528, 64.8057)$	Andronov-Hopf bifurcation	$\pm245.16i, -70$

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Table 4.  The dynamics of $E_{\pm}^{i}$ with $(a, c, k, k_{1}, b) = (-35, -35, 0.5, 1, -70)$, $i = 1, 2, \cdots, 15$.

$E_{\pm}$	Classification	Eigenvalues
$E_{\pm}^{1} = (\pm 42.5879, \pm 42.5879, -25.9104)$	Shilnikov-Hopf bifurcation	$\pm138.54i, 70$
$E_{\pm}^{2} = (\pm 44.0511, \pm 44.0511, -27.7214)$	Saddle	$\pm142.9652, 70$
$E_{\pm}^{3} = (\pm 47.0547, \pm 47.0547, - 31.6306)$	Shilnikov-Hopf bifurcation	$\pm 191.48i, 70$
$E_{\pm}^{4} = (\pm 49.1642, \pm 49.1642, - 34.5303)$	Saddle	$\pm198.1574, 70$
$E_{\pm}^{5} = (\pm 51.2773, \pm 51.2773, - 37.5623)$	Shilnikov-Hopf bifurcation	$\pm226.65i, 70$
$E_{\pm}^{6} = (\pm 53.6664, \pm 53.6664, -41.144)$	Saddle	$\pm232.5816, 70$
$E_{\pm}^{7} = (\pm 55.2308, \pm 55.2308, -43.5777)$	Shilnikov-Hopf bifurcation	$\pm252.09i, 70$
$E_{\pm}^{8} = (\pm 57.7749, \pm 57.7749, -47.6848)$	Saddle	$\pm254.8295, 70$
$E_{\pm}^{9} = (\pm 58.9443, \pm 58.9443, -49.6347)$	Shilnikov-Hopf bifurcation	$\pm268.49i, 70$
$E_{\pm}^{10} = (\pm 61.5922, \pm 61.5922, -54.1943)$	Saddle	$\pm264.7765, 70$
$E_{\pm}^{11} = (\pm 62.4478, \pm 62.4478, -55.7104)$	Shilnikov-Hopf bifurcation	$\pm2.7403i, 70$
$E_{\pm}^{12} = (\pm 65.1845, \pm 65.1845, -60.7003)$	Saddle	$\pm 257.1704, 70$
$E_{\pm}^{13} = (\pm 65.7631, \pm 65.7631, -61.7826)$	Shilnikov-Hopf bifurcation	$\pm262.71i, 70$
$E_{\pm}^{14} = (\pm 68.6152, \pm 68.6152, -67.2578)$	Saddle	$\pm205.6317, 70$
$E_{\pm}^{15} = (\pm 68.8911, \pm 68.8911, -67.7998)$	Shilnikov-Hopf bifurcation	$\pm207.64i, 70$

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Figure 9.  When $(a, c, k, k_{1}, b) = (-35, -35, 0.5, 1, -70)$, (a) (b) eight pairs of homoclinic orbits to $S_{\pm}^{i}$ and $S_{0}$ of system (3.5), (c) eight pairs of homoclinic orbits to $E_{\pm}^{i}$ and $E_{0}$ of system (2.1), $i = 2, 4, 6, 8, 10, 12, 14$.

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4.   Conclusions

In the present work, the existence of homoclinic orbits and heteroclinic orbits of both cases of non-Hamiltonian and Hamiltonian modified Chen system was proved based on definitions of both $\alpha$-limit set and $\omega$-limit set, the theory of the Lyapunov function, and the Hamiltonian function. In the former case, there existed multitudinous potential heteroclinic orbits to (1) $E_{0}$ and $E_{\pm}$, (2) $E_{+}$, or (3) $E_{-}$ in that modified Chen system. In the latter case, there was a multitude of potential homoclinic orbits to $E_{0}$ and $E_{\pm}$ or heteroclinic orbits to $E_{\pm}$ on the invariant algebraic surface $z = \frac{x^{2}}{2a}$. Moreover, when $(a, c, k, k_{1}) = (35, 28, 0.5, 1)$, $b = 70, 75$ (resp., $(a, c, k, k_{1}, b) = (\pm35, \pm35, 0.5, 1, \pm70)$), numerical simulations illustrated nine pairs of heteroclinic orbits (resp., eight pairs of homoclinic orbits). What follows is to study the multi-wing Lü system ^[1], Lorenz system, and other hyperchaotic Lorenz-like systems.

Use of AI tools declaration

The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

Acknowledgments

This work is supported in part by the National Natural Science Foundation of China under Grant 12001489, the Zhejiang Public Welfare Technology Application Research Project of China under Grant LGN21F020003, the Natural Science Foundation of Zhejiang Guangsha Vocational and Technical University of Construction under Grant 2022KYQD-KGY, and the Natural Science Foundation of Taizhou University under Grant T20210906033. Meanwhile, the authors would like to express their sincere thanks to the anonymous editors and reviewers for their conscientious reading and numerous constructive comments, which improved the manuscript substantially.

Conflict of interest

The authors declare there is no conflicts of interest.