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A pair of centrosymmetric heteroclinic orbits coined
Advances in Continuous and Discrete Models volume 2024, Article number: 14 (2024)
Abstract
Although the axissymmetric heteroclinic orbits of Lorenzlike systems have been intensively studied in the past decades, scholars seem to pay scant attention to the centrosymmetric ones. To achieve this target, the present paper introduces a new subquadratic centrosymmetric threedimensional Lorenzlike system: \(\dot{x}=a(y  x)\), \(\dot{y}=cx  \sqrt[3]{x^{2}}z\), \(\dot{z}= bz + \sqrt[3]{x^{2}}y\), and proves the existence of a pair of centrosymmetric to \(E_{0}\) and \(E_{\pm}\) combining the definitions of αlimit and ωlimit set, Lyapunov functions. The effectiveness and correctness of the theoretical conclusions are verified via a few numerical examples. Not only does the study provide new ideas for finding heteroclinic orbits, but also it poses an interesting question that the existence of heteroclinic orbits may depend on the degrees of the considered models.
1 Introduction
Since the introduction of the Lorenz system and attractor, the study of chaos entered a new era [1]. Utilizing theoretical tools, numerical and circuit simulation, etc., researchers and engineers performed a systematical analysis for numerous chaotic systems [2–7]. Not only did they reveal distinctive properties, such as extremely sensitive dependence on initial conditions, deterministicness, unpredictability, existence of at least one positive Lyapunov exponent, boundedness, hidden attractors, conservative chaotic flow, multistability, and so on, but also they explained the forming mechanism of strange attractors to some degree, i.e., the bifurcation of singular orbits (including homoclinic and heteroclinic orbits, singularly degenerate heteroclinic cycles, etc.) and invariant algebraic surfaces, the loss of global stability, etc. [2, 4–9].
Recently, Zhang et al. posed the extension of the second part of the celebrated Hilbert’s 16th problem [9, 10], i.e., the degree of polynomials in the studied models determines the number and mutual disposition of attractors and repellers (if they exist). Inspired by this, Wang et al. introduced two new subquadratic Lorenzlike systems of degree \(\frac{4}{3}\) and \(\frac{6}{5}\), and they found a multitude of twowing hidden attractors in a broader range of parameters [11, 12]. Moreover, by aid of Lyapunov functions and the definitions of both αlimit set and ωlimit set [7, 13–22], Wang et al. proved the existence of a pair of heteroclinic orbits when \(\frac{b}{a}\geq \frac{4}{3}\) and \(\frac{b}{a}\geq \frac{6}{5}\), respectively. Another two Lorenzlike analogues of degree \(\frac{4}{3}\) also exhibit two pairs of heteroclinic orbits when \(\frac{b}{a}\geq \frac{4}{3}\) [13, 14]. Likewise, there exist heteroclinic orbits in the cubic and quadratic Lorenzlike systems when \(\frac{b}{a}\geq 3\) and \(\frac{b}{a}\geq 2\) [15–21]. On these grounds, the degree of some Lorenzlike systems may have some relativity with heteroclinic orbits. In addition, all of the aforementioned heteroclinic orbits are axissymmetric or single ones. However, the scenario of centrosymmetric heteroclinic orbits of it is not considered at all to the best of our knowledge. Therefore, it is an urgent task to conduct the study.
The newly introduced Lorenzlike system has to satisfy at least three principles:
(1) This model has to be a centrosymmetric analogue.
(2) The degree of it should guarantee the generation of a pair of nontrivial centrosymmetric equilibria w.r.t. the origin.
(3) The method of combination of Lyapunov functions, the definitions of both αlimit set and ωlimit set should be applicable to it when proving the existence of heteroclinic orbits.
Based on the above three tips and trialanderror, we try to search for a new subquadratic centrosymmetric Lorenzlike system with the targeted heteroclinic orbits.
To our knowledge, little seems to be known about the Lorenzlike system yet with cross products \(\sqrt[3]{x^{2}}z\) and \(\sqrt[3]{x^{2}}y\). Innovations of this paper are:
(1) Proposing a new threedimensional subquadratic centrosymmetric Lorenzlike system.
(2) Proving the existence of a pair of centrosymmetric heteroclinic orbits.
(3) Confirming the correlation between the degree and heteroclinic orbits to some extent.
As a result, it is theoretically and practically important to analyze such a Lorenzlike system, motivating the followup research of this work.
2 New subquadratic centrosymmetric Lorenzlike system and basic dynamics
Combining the axissymmetric quadratic/subquadratic Lorenz system family [11–14] and trialanderror, we firstly introduce the new subquadratic centrosymmetric analogue as follows:
which is invariant under transformation \((x,y,z)\rightarrow (x,y,z)\). Next, aiming at revealing the heteroclinic orbits, we present some basic dynamics of system (2.1) in the following theorems, i.e., the distribution of equilibrium points, stability, Hopf bifurcation, inequality, etc.
Theorem 2.1
(1) When \(b = 0\), system (2.1) has nonisolated equilibria \(E_{z} = \{(0, 0, z)z\in \mathbb{R}\}\).
(2) When \(b \neq 0\) and \(bc < 0\), system (2.1) has a single equilibrium point \(E_{0}=(0, 0, 0)\).
(3) When \(bc > 0\), system (2.1) has two equilibria \(E_{\pm} = \pm (\sqrt[4]{(bc)^{3}}, \sqrt[4]{(bc)^{3}}, c\sqrt[4]{bc})\) except for \(E_{0}\).
Theorem 2.2
When \(abc \neq 0\), the local dynamical behaviors of \(E_{0}\) are totally summarized in Table 1. While \(a \neq 0\) and \(b = 0\), all of \(E_{z}\) are unstable.
Set \(W = \{(a, c, b)a \neq 0, bc > 0\}\), \(W_{1}=\{(a, c, b)\in W: a+b > 0, ab+bc\frac{ac}{3} > 0, \frac{4abc}{3}> 0\}\), \(\Delta =ab(a+b)c[\frac{(ab)(3b+a)}{3}]\), \(W_{2}=W\backslash W_{1}\), \(W_{1}^{1}=\{(a, c, b)\in W_{1}: \Delta < 0\}\), \(W_{1}^{2}=\{(a, c, b)\in W_{1}: \Delta = 0\}\), and \(W_{1}^{3}=\{(a, c, b)\in W_{1}: \Delta > 0\}\).
Theorem 2.3
(1) When \((a, c, b)\in W_{1}^{1}\) (resp. \(W_{1}^{3}\)), \(E_{\pm}\) are unstable (resp. asymptotically stable).
(2) When \((a, c, b)\in W_{1}^{2}\), Hopf bifurcation happens at \(E_{\pm}\).
Theorem 2.4
If \(5a > 3b > 0\) and \(t\rightarrow \infty \), then the inequality \(z \geq \frac{3}{5a}\sqrt[3]{x^{5}}\) holds.
The rest content is arranged as follows. The proofs of Theorems 2.2–2.4 are outlined in Sect. 3. Section 4 studies the existence of centrosymmetric heteroclinic orbits. Lastly, some conclusions are drawn, and the correlation between power of the polynomials and dynamics is also discussed.
3 Basic dynamics and proofs of Theorems 2.2–2.4
In this section, proofs of Theorems 2.2–2.4 are sketched as follows.
Proof of Theorem 2.2
Based on linear analysis, Theorem 2.2 easily follows, and the proof is omitted here. □
Proof of Theorem 2.3
The characteristic equation of matrix associated with the vector field of system (2.1) at \(E_{\pm}\) is calculated as follows:
On the basis of Eq. (3.1) and Routh–Hurwitz criterion, \(E_{\pm}\) are unstable (resp. asymptotically stable) when \((a, c, b)\in W_{1}^{1}\) (resp. \(W_{1}^{3}\)).
While \((a, c, b)\in W_{1}^{2}\), Eq. (3.1) has the negative real root \(\lambda _{1} = (a+b) < 0\) and a pair of conjugate purely imaginary roots \(\lambda _{2, 3}=\pm \omega i=\pm 2ab\sqrt{\frac{1}{(ab)(3b+a)}} i\). Furthermore, the transversal condition \(\frac{d\operatorname{\mathit{Re}}(\lambda _{2})}{dc} _{c=c^{*}}= \frac{(ab)(a+3b)}{6[\omega ^{2}+(a+b)^{2}]}\neq 0 \) holds, where \(c_{*} = \frac{3ab(a+b)}{(ab)(3b+a)}\). Therefore, Hopf bifurcation occurs at \(E_{\pm}\), as shown in Fig. 1. The proof is completed.
□
Remark 3.1
When \((a,c,b) = (5,2.8125,1)\), the eigenvalues of \(E_{\pm}\) are \(\lambda _{1}=6\), \(\lambda _{2,3}=\pm 1.7678i\).
Proof of Theorem 2.4
Set \(Q(x, z) = z  \frac{3}{5a}\sqrt[3]{x^{5}}\) and compute the derivative of it along any one orbit of system (2.1): \(\frac{dQ(x, z)}{dt} _{\text{(2.1)}} = bz + \frac{5a}{3} \sqrt[3]{x^{5}}\), i.e., \(\dot{Q} + bQ = (b  \frac{5a}{3})\sqrt[3]{x^{5}}\).
Based on the comparison principle, if \(b  \frac{5a}{3} < 0\), then \(\dot{Q} + bQ \geq 0\) leads to
Namely, for \(b  \frac{5a}{3} < 0\), we arrive at the inequality \(\lim_{t\rightarrow \infty}Q(t) = \lim_{t\rightarrow \infty} [z  \frac{3}{5a}\sqrt[3]{x^{5}}] \geq 0\). The proof is finished. □
For discussion purposes, the following denotations have to be introduced:
(1) \(p(t; q_{0}) = (x(t; x_{0}), y(t; y_{0}), z(t; z_{0}))\): each solution of system (2.1) through the initial value \(q_{0} = (x_{0}, y_{0}, z_{0})\).
(2) \(\gamma ^{\pm} =\{p_{\pm}(t; q_{0})p_{\pm}(t; q_{0})=\pm (x_{+}(t; x_{0}), y_{+}(t; y_{0}), z_{+}(t; z_{0}))\in W_{\pm}^{u}, t\in \mathbb{R}\}\): the branch of the unstable manifold \(W^{u}(E_{0})\) corresponding to \(x_{+} > 0\) and \(x_{+} < 0\) when \(t\rightarrow \infty \).
4 Existence of heteroclinic orbit
Combining the concepts of both αlimit and ωlimit set and Lyapunov functions, we prove the existence of centrosymmetric heteroclinic orbits of system (2.1) and arrive at the following result.
Theorem 4.1
If \(c > 0\) and \(3b\geq 5a > 0\), then there exist no homoclinic orbits but a pair of centrosymmetric heteroclinic orbits: \(\gamma ^{\pm}\) to \(E_{0}\) and \(E_{\pm}\).
Next, we prove Theorem 4.1 in two steps: (1) \(3b5a > 0\), (2) \(3b5a = 0\).
4.1 The case \(3b5a > 0\)
In this subsection, we first construct the following Lyapunov function:
and derive the following statements.
Lemma 4.2
When \(c > 0\) and \(3b5a > 0\), we arrive at the following statements:

1.
If \(\exists t_{1,2}\), \(t_{1} < t_{2}\) and \(V_{1}(p(t_{1}; q_{0})) = V_{1}(p(t_{2}; q_{0}))\), then \(q_{0}\) is one of the equilibrium points.

2.
If \(\lim_{t\rightarrow \infty}p(t; q_{0}) = E_{0}\) and \(x(t_{3}; q_{0}) > 0\), \(\exists t_{3}\in \mathbb{R}\), then \(V_{1}(E_{0}) > V_{1}(p(t; q_{0}))\) and \(x(t; x_{0}) > 0\), \(\forall t\in \mathbb{R}\). As a result, \(q_{0}\in \gamma ^{+}\).
Proof
(1) Taking the derivative of \(V_{1}\) along \(p(t; q_{0})\) results in
and thus leads to
under the condition of (1), \(\forall t\in (t_{1}, t_{2})\).
On the basis of Eq. (4.2) and system (2.1), we obtain the identities \(\dot{x}(t; x_{0})\equiv \dot{y}(t; y_{0})\equiv \dot{z}(t; z_{0}) \equiv 0\), \(\forall t\in (t_{1}, t_{2})\). Therefore, system (2.1) has the stationary point \(q_{0}\).
(2) Let us first show \(V_{1}(E_{0}) > V_{1}(p(t; q_{0}))\), \(\forall t\in \mathbb{R}\). Otherwise, \(V_{1}(E_{0})\leq V_{1}(p(t; q_{0}))\), \(\exists t\in \mathbb{R}\). Then, the first assertion yields that \(q_{0}\) is just one of the equilibria of system (2.1), which contradicts the assumed condition \(\lim_{t\rightarrow  \infty}p(t; q_{0}) = E_{0}\) and \(x(t_{3}; x_{0}) > 0\). Namely, \(V_{1}(E_{0}) > V_{1}(p(t; q_{0}))\), \(\forall t\in \mathbb{R}\).
Next, we prove \(x(t; x_{0}) > 0\), \(\forall t\in \mathbb{R}\). If not, \(x(t_{4}; x_{0}) \leq 0\), \(\exists t_{4}\in \mathbb{R}\). Due to \(x(t_{3}; x_{0}) > 0\), \(t_{3}\in \mathbb{R}\), we obtain \(x(t_{5}; x_{0})=0\), \(\exists t_{5}\in \mathbb{R}\). Since \(V_{1}(E_{0}) > V_{1}(p(t; q_{0}))\), \(\forall t\in \mathbb{R}\), we arrive at
In addition, the fact holds:
A contradiction happens. Consequently, the fact \(x(t; x_{0}) > 0\), \(\forall t\in \mathbb{R}\) holds, and the proof is completed. □
Lemma 4.3
When \(c > 0\), \(3b>5a > 0\), and \(t\rightarrow \infty \), each solution of system (2.1) approaches one of its equilibrium points. In a word, closed orbits are nonexistent in system (2.1).
Proof
From Eq. (4.1), we deduce \(\lim_{t\rightarrow +\infty}V_{1}(p(t; q_{0}))=\Phi (q_{0})\) and \(0\leq V_{1}(p(t; q_{0})) \leq V_{1}(p(0; q_{0})) = V_{1}(q_{0})\), \(\forall t \geq 0 \), implying the boundedness of \(x(t; x_{0})\), \(y(t; y_{0})\) and \(z(t; z_{0})\), \(t\in [0, +\infty )\). Namely, the set \(\{p(t; q_{0})t \geq 0\}\) is bounded.
Let \(\Omega (q_{0})\neq \emptyset \) be the ωlimit set of \(p(t; q_{0})\). For \(\forall q \in \Omega (q_{0})\), i.e., \(\exists \{t_{n}\}\), such that
Next, \(\forall t\in \mathbb{R}\), \(p(t; q) = \lim_{n \to +\infty} p(t; p(t_{n}; q_{0})) = \lim_{n \to +\infty} p(t+t_{n}; q_{0})\) yields \(V_{1}(p(t; q))=V_{1}[\lim_{n \to +\infty}p(t; p(t_{n}; q_{0}))] = \lim_{n \to +\infty}V_{1}(p(t+t_{n}; q_{0}))= \Phi (q_{0})\). Therefore, \(q \in \{E_{}, E_{0}, E_{+}\}\). Since \(\Omega (q_{0})\) is connected, we only obtain \(\Omega (q_{0}) = \{E_{}\}\) or \(\Omega (q_{0}) = \{E_{0}\}\), or \(\Omega (q_{0}) = \{E_{+}\}\), which suggests that \(p(t; q_{0})\) converges to one of the equilibria when \(t\to +\infty \). Therefore, the proof is finished. □
From Lemmas 4.2–4.3, we prove the existence of heteroclinic orbits.
Theorem 4.4
If \(c > 0\) and \(3b > 5a > 0\), then

1.
Homoclinic orbits are nonexistent in system (2.1);

2.
System (2.1) has a pair of centrosymmetric heteroclinic orbits: \(\gamma ^{+}\) joining \(E_{+}\) and \(E_{0}\), and \(\gamma ^{}\) joining and \(E_{}\) and \(E_{0}\).
Proof
Let us prove that both homoclinic orbits and heteroclinic orbits to \(E_{\pm}\) are nonexistent in system (2.1) when \(c > 0\) and \(3b > 5a > 0\). Otherwise, let \(p(t) = (x(t), y(t), z(t))\) be a homoclinic (resp. heteroclinic) orbit to \(E_{0}\) or \(E_{+}\), or \(E_{}\) (resp. \(E_{+}\) and \(E_{}\)), i.e., \(\lim_{t \to \infty} p(t) = e^{}\), \(\lim_{t \to + \infty} p(t) = e^{+}\), where \(e^{} = e^{+} \in \{E_{}, E_{0}, E_{+}\}\) or \(\{e^{}, e^{+}\} = \{E_{}, E_{+}\}\).
It follows from Eq. (4.1) that
holds. In either case, we only obtain the relation \(V_{1}(e^{}) = V_{1}(e^{+})\), which thus results in \(V_{1}(p(t))\equiv V_{1}(e^{+})\). In virtue of the first assertion of Lemma 4.2, \(p(t)\) is just one of the fixed points. As a result, there exist neither homoclinic orbits to \(E_{0}\) or \(E_{+}\), or \(E_{}\), nor heteroclinic orbits to \(E_{}\) and \(E_{+}\).
Next, we show that \(\gamma ^{+}\) is a heteroclinic orbit to \(E_{0}\) and \(E_{+}\), i.e., \(\lim_{t \to +\infty}p(t) = E_{+}\). On the basis of the definition of \(\gamma ^{+}\) and the second assertion of Lemma 4.2, we arrive at \(x_{+}(t) > 0\), \(\forall t\in \mathbb{R}\), which also yields \(\lim_{t \to +\infty}p(t) \neq E_{}\). Meanwhile, the definition of \(\gamma ^{+}\) leads to \(\lim_{t \to +\infty}p(t) \neq E_{0}\). Thus, \(\lim_{t\to +\infty}p_{+}(t)= E_{+}\) holds.
At last, we prove that, if there exists a heteroclinic orbit to \(E_{0}\) and \(E_{+}\) in system (2.1), then it is nothing but \(\gamma ^{+}\).
Define the \(p_{1}(t) = (x_{1}(t), y_{1}(t), z_{1}(t))\) to be any one solution of system (2.1) such that
where \(\{e_{1}^{}, e_{1}^{+}\} = \{E_{0}, E_{+}\}\). Similar to Eq. (4.3), we arrive at \(V_{1}(e_{1}^{})\geq V_{1}(p_{1}(t))\geq V_{1}(e_{1}^{+})\), \(\forall t\in \mathbb{R}\), based on Eq. (4.2). Since \(V_{1}(E_{0}) > V_{1}(E_{+})\), we deduce that \(e_{1}^{} = E_{0}\) and \(e_{1}^{+} = E_{+}\), i.e.,
which results in \(p_{1}(t)\in \gamma ^{+}\) from the second assertion of Lemma 4.2. Since system (2.1) is centrosymmetrical w.r.t. the origin \(E_{0}\), there is a unique heteroclinic orbit \(\gamma ^{}\) centrosymmetrical to \(\gamma ^{+}\). The proof is completed. □
4.2 The case \(3b5a = 0\)
This subsection first introduces the second Lyapunov function
and the following statements.
Lemma 4.5
For \(c > 0\) and \(3b=5a > 0\), the following assertions hold:
(i) If \(\lim_{t \rightarrow \infty}p(t; q_{0})\) is bounded, then \(z(t; z_{0}) = \frac{3}{5a}\sqrt[3]{x^{5}(t; x_{0})}\);
(ii) If \(z(t; z_{0}) = \frac{3}{5a}\sqrt[3]{x^{5}(t; x_{0})}\), then \(\frac{dV_{2}(p(t; q_{0}))}{dt} _{\textit{(2.1)}} =  \frac{25a^{3}}{9}(yx)^{2} \leq 0\);
(iii) If \(z(t; z_{0}) = \frac{3}{5a}\sqrt[3]{x^{5}(t; x_{0})}\) and \(V_{2}(p(t_{1}; q_{0})) = V_{2}(p(t_{2}; q_{0}))\), \(\exists t_{1,2}\), \(t_{1} < t_{2}\), then \(q_{0}\) is one of the equilibrium points;
(iv) If \(\lim_{t \rightarrow \infty}p(t; q_{0}) = E_{0}\) and \(x(t_{3}; q_{0}) > 0\), \(\exists t_{3}\in \mathbb{R}\), then \(V_{2}(E_{0}) > V_{2}(p(t; q_{0}))\) and \(x(t; q_{0}) > 0\), \(\forall t\in \mathbb{R}\). In a word, \(q_{0} \in \gamma ^{+}\).
Proof
(i) From Proof of Theorem 2.4, we arrive at \(\frac{dQ(p(t; q_{0}))}{dt} _{\text{(2.1)}} = \frac{5a}{3}Q(p(t; q_{0}))\), i.e.,
Because \(\lim_{\tau \rightarrow \infty}p(\tau ; q_{0})\) is bounded, Eq. (4.4) suggests \(Q(p(t; q_{0})) \equiv 0\), i.e., \(z(t; z_{0}) \equiv \frac{3}{5a}\sqrt[3]{x^{5}(t; x_{0})}\).
(ii) The result easily follows from the first assertion \(z(t; z_{0}) \equiv \frac{3}{5a}\sqrt[3]{x^{5}(t; x_{0})}\) and system (2.1).
(iii) The second assertion yields \(\frac{dV_{2}(p(t, q_{0}))}{dt} _{\text{(2.1)}} = 0\), \(\forall t \in (t_{1}, t_{2})\), i.e.,
Combining \(\dot{x}=a(y  x)\), Eq. (4.5), and \(z(t; z_{0}) \equiv \frac{3}{5a}\sqrt[3]{x^{5}(t; x_{0})}\), we derive
Hence, \(q_{0}\) is just one of the equilibria.
(iv) Let us prove \(V_{2}(E_{0}) > V_{2}(p(t; q_{0}))\), \(\forall t \in \mathbb{R}\). Otherwise, \(V_{2}(E_{0}) \leq V_{2}(p(t_{0}; q_{0}))\), \(\exists t_{0} \in \mathbb{R}\). On the other hand, assertions (i)–(iii) yield that \(q_{0}\) is one of the equilibria, which contradicts \(\lim_{t \rightarrow  \infty}p(t; q_{0}) = 0\) and \(x(t_{3}; x_{0}) > 0\). Thus, we arrive at \(V_{2}(E_{0}) > V_{2}(p(t; q_{0}))\), \(\forall t \in \mathbb{R}\).
Next, one shows \(x(t; x_{0}) > 0\), \(\forall t \in \mathbb{R}\). Otherwise, \(\exists t_{4} \in \mathbb{R}\) such that \(x(t_{4}; x_{0}) \leq 0\). Because of \(x(t_{3}; x_{0}) > 0\), \(\exists t_{5} \in \mathbb{R}\) such that \(x(t_{5}; x_{0}) = 0\). Since \(V_{2}(E_{0}) > V_{2}(p(t; q_{0}))\), \(\forall t \in \mathbb{R}\), we arrive at
Instead, \(\{(x, y, z)V_{2}(x, y, z) < V_{2}(E_{0})\}\cap \{(x, y, z)x = 0\} = \{(0, y, z)\frac{1}{2}[\frac{25a^{2}}{9}y^{2}+ \frac{50a^{2}c^{2}}{27}\sqrt{\frac{5ac}{3}}]<\frac{25a^{2}c^{2}}{27} \sqrt{\frac{5ac}{3}} \} = \emptyset \). A contradiction occurs. Consequently, \(x(t; x_{0}) > 0\) is true, \(\forall t \in \mathbb{R}\). □
Lemma 4.6
Consider \(c > 0\) and \(3b=5a > 0\). If any one negative orbit with initial point \(q_{0}\) is bounded, then \(\lim_{t\rightarrow \infty}p(t, q_{0})\) converges to one of the equilibria of system (2.1). Therefore, closed orbits are nonexistent in system (2.1).
Proof
It follows from assertions (i)–(ii) that \(\lim_{t \rightarrow  \infty}V_{2}(p(t; q_{0}))=\Psi (q_{0})\) exists. Suppose \(q \in \alpha (q_{0})\), i.e., \(\exists \{t_{n}\}\) such that
\(\forall t \in \mathbb{R}\), the fact
results in
According to Lemma 4.5, we arrive at \(q \in \{E_{}, E_{0}, E_{+}\}\). Thus,
Since \(\alpha (q_{0})\) is connected, we obtain \(\alpha (q_{0}) = \{E_{}\}\) or \(\alpha (q_{0}) = \{E_{0}\}\), or \(\alpha (q_{0}) = \{E_{+}\}\), which yields that \(\lim_{n \rightarrow +\infty}p(t; q_{0})\) converges to one of the equilibria. The proof is over. □
Theorem 4.7
If \(c > 0\) and \(3b=5a > 0\), then
(i) system (2.1) has no homoclinic orbits;
(ii) system (2.1) has only a pair of centrosymmetric heteroclinic orbits: \(\gamma ^{+}\) joining \(E_{0}\) and \(E_{+}\), and \(\gamma ^{}\) joining \(E_{0}\) and \(E_{}\).
Proof
(i) Let us prove that neither homoclinic orbits nor heteroclinic orbits joining \(E_{}\) and \(E_{+}\) exist in system (2.1) when \(c > 0\) and \(3b=5a > 0\). If not, suppose that \(p(t) = (x(t), y(t), z(t))\) is a homoclinic or heteroclinic orbit to \(E_{}\) and \(E_{+}\), i.e.,
where \(e^{}\) and \(e^{+}\) satisfy either
It follows from Lemma 4.5 and \(V_{2}(e^{}) = V_{2}(e^{+})\) that \(p(t)\) is just one of the stationary points.
As a result, both homoclinic orbits and heteroclinic orbits to \(E_{\pm}\) are nonexistent in system (2.1).
(ii) Let us prove that if system (2.1) has a heteroclinic orbit to \(E_{0}\) and \(E_{+}\), then it is just \(\gamma ^{+}\).
Assume \(p_{1}(t) = (x_{1}(t), y_{1}(t), z_{1}(t))\) is any one solution of system (2.1) such that
where \(\{e_{1}^{}, e_{1}^{+}\} = \{E_{0}, E_{+}\}\). For \(\forall t \in \mathbb{R}\), assertions (i)–(ii) of Lemma 4.5 suggest
Since \(V_{2}(E_{0}) > V_{2}(E_{+})\), we arrive at \(e_{1}^{} = E_{0}\) and \(e_{1}^{+} = E_{+}\), i.e.,
yielding \(p_{1}(t) \in \gamma ^{+}\) based on the fourth assertion of Lemma 4.5.
At last, let us prove that \(\gamma ^{+}\) is a heteroclinic orbit to \(E_{0}\) and \(E_{+}\), i.e., \(\lim_{t \rightarrow +\infty} p_{+}(t) = E_{+}\). From Lemma 4.5, we arrive at
The second equation of Eq. (4.8) suggests that \(\lim_{t \rightarrow \infty}V_{2}(p_{+}(t))=v\) exists. Again, Eq. (4.8) indicates the boundedness of \(x_{+}(t)\), \(y_{+}(t)\) and \(z_{+}(t)\), \(\forall t \in [0, +\infty )\), i.e., the set \(\{p_{+}(t)t \geq 0\}\) is bounded. Define Ω to be the ωlimit set of solution \(p_{+}(t)\). Suppose \(q \in \Omega \), i.e., \(\exists \{t_{n}\}\) such that \(\lim_{n \rightarrow +\infty} t_{n} = +\infty \) and \(\lim_{n \rightarrow +\infty} p_{+}(t_{n}) = q\). Therefore, \(\forall t \in \mathbb{R}\), the relation
and together with assertions (i)–(iii) of Lemma 4.5 results in \(q \in \{E_{}, E_{0}, E_{+}\}\). Consequently, \(\Omega \subseteq \{E_{}, E_{0}, E_{+}\}\). Because Ω is connected, we obtain \(\Omega = E_{}\) or \(\Omega = E_{0}\), or \(\Omega = E_{+}\). On the basis of assertion (ii) of Lemma 4.5 and the fourth equality of Eq. (4.8), we arrive at \(\Omega \neq E_{0}\) and \(\Omega \neq E_{}\). Thus, \(\Omega = E_{+}\), i.e., \(\lim_{n \rightarrow +\infty} p_{+}(t) = E_{+}\). Due to the central symmetry of system (2.1), there is a unique heteroclinic orbit \(\gamma ^{}\) to \(E_{}\) and \(E_{0}\), as illustrated in Fig. 2. The proof is completed.
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5 Conclusions
As an effective method to study the existence of heteroclinic orbits, the combination of Lyapunov functions, the definitions of ωlimit set and αlimit set has been widely applied in many axissymmetric Lorenzlike systems. Whether or not it is applicable to the centrosymmetric ones. In this effort, based on the extension of the second part of the celebrated Hilbert’s 16th problem and a trial and error process, this paper reports another new subquadratic threedimensional Lorenzlike system and proves the existence of a pair of centrosymmetric heteroclinic orbits by aid of the aforementioned method. Moreover, for \((a,c) = (3, 12)\) and \(b=5, 8\), Fig. 2 validates the correctness of theoretical results.
In future work, some interesting issues deserve consideration. First, whether or not strange attractors and pseudo singularly degenerate heteroclinic cycles exist. Second, the existence of some other global dynamics, i.e., homoclinic orbits, boundedness, and so on. Finally, the relationship between the degree and heteroclinic orbits, and real world applications.
Data availability
There is no data because the results obtained in this paper can be reproduced based on the information given in this paper.
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Acknowledgements
This work is supported in part by the National Natural Science Foundation of China under Grant 12001489, in part by Zhejiang Public Welfare Technology Application Research Project of China Grant LGN21F020003, in part by the Natural Science Foundation of Zhejiang Guangsha Vocational and Technical University of Construction under Grant 2022KYQDKGY, in part by 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.
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Haijun Wang: conceptualization, software, writing—original draft, Investigation. Jun Pan: supervision, visualization, validation, writing—review & editing. Guiyao Ke: software, methodology, investigation, visualization. Feiyu Hu: software, validation.
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Wang, H., Pan, J., Ke, G. et al. A pair of centrosymmetric heteroclinic orbits coined. Adv Cont Discr Mod 2024, 14 (2024). https://doi.org/10.1186/s13662024038094
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DOI: https://doi.org/10.1186/s13662024038094