In recent years, chaotic dynamical systems have become a major topic of interest in the nonlinear science. The understanding of chaotic behavior is crucial for the development of chaotic dynamical systems. It has many potential applications in engineering fields, such as biology, mechanical engineering, private communication, and artificial intelligence.^[1–13]

Since 1963, Lorenz put his research findings,^[14] the first physical evidence of chaos, into the public article, various researchers all over the world sparked off a craze for studying chaotic behaviors. A lot of achievements^[15–21]have been made in the chaotic field. However, interesting issues that how to ensure the existence of heteroclinic cycles and how the heteroclinic cycles lead to chaotic behaviors in the high-dimensional dynamical systems are still here. Fortunately, the famous Shil’nikov theorem^[22,23] overcomes difficulties and gives part of the answers to that issue. The theorem suggests that the existence of heteroclinic cycles means chaos presenting in a small enough neighborhood of this special orbit. In other words, the Shil’nikov theorem gives a criterion to ensure the existence of chaos by heteroclinic cycles in smooth systems. For the discontinuity systems, because of the discontinuity in the switch manifold, the Shil’nikov theorem cannot be used directly. Meanwhile, the theorem is hard to do in practice, so detecting the appearance of heteroclinic cycles and chaos in high-dimensional systems is still a great challenge now.

However, for some discontinuous piecewise systems (DPSs), we can obtain the analytic expressions for stable manifolds and unstable manifolds of subsystem, and then describe the intersection of these manifolds and switch manifolds, whereby finding a heteroclinic cycle. So scholars and experts have paid their attention to these special DPSs, and there have been several theoretical results. For instance, in 2008, Carmona^[24] gave an analytical proof of the existence of a reversible T-point heteroclinic cycle in a continuous piecewise linear version of the widely studied Michelson system. In 2009, Li^[25] proved the existence of the heteroclinic cycle in a kind of piecewise linear chaotic system. In 2011, Bao^[26] provided a new series method for continuous-time autonomous dynamical systems. In 2014, Leonov^[27] formulated and proved a fishing principle for the existence of homoclinic and heteroclinic trajectories. In 2015, based on the heteroclinic Shil’nikov theorem and switching control, Han^[28] constructed a kind of multipiecewise linear chaotic system. In 2016, Wu^[29] considered a design of chaos generator and gave the existence of heteroclinic cycle and chaos in some classes of three-dimensional (3D) piecewise affine systems (PASs). In 2017, Wang^[30] studied a class of 3D PASs and presented succinct sufficient conditions for the existence of three types of heteroclinic cycles by mathematical analysis. Carmona,^[31] taking advantage of the reversibility and some geometrical features of a piecewise linear version of the Michelson system, constructed a global problem that includes homoclinic connections and T-point heteroclinic cycles as particular cases. In 2018, Chen^[32] investigated the existence of heteroclinic cycles in 3D three-zone PASs with two switching planes. Yang and Lu^[33] gave the existence of orbit homoclinic and chaos in four-dimensional (4D) PASs. Wu^[34] also considered a class of 4D PASs where A and B both have two differential pairs of conjugate complex eigenvalues. Furthermore, the conditions for the existence of heteroclinic cycles were presented, and chaotic invariant set in a small neighborhood of the heteroclinic cycles was obtained by numerical simulation. However, to the best of our knowledge, except for several works, little seems to be known about heteroclinic cycle in 4D DPSs, even in the system governed by the simplest 4D equations.

In this paper, the existence of heteroclinic cycles is established in a new class of 4D two-zone DPASs. By analyzing the stable and unstable manifolds of equilibriums of the subsystems, we describe the intersection of these manifolds. Then, the heteroclinic cycles are achieved accurately. Furthermore, under the help of Shil’nikov theory, a chaotic invariant set is also obtained by numerical simulation.

The rest of this paper is organized as follows. Section 2 introduces a new class of 4D two-zone DPASs, and proposes a criterion, proved by rigorous analysis, to detect the appearance of heteroclinic cycles. Section 3 offers two numerical examples, displaying chaotic behavior, to depict the availability of analytic results. Section 4 discusses the results found in this paper.

In this section, a new class of 4D DPASs are introduced first. Then the existence of heteroclinic cycle in this system will be shown in detail.

Consider the 4D piecewise affine systems 1 where x = (x₁ x₂ x₃ x₄)^T ∈ ℝ⁴, a, b ∈ ℝ⁴ are constant vectors, and for 4 × 4 invertible matrices P, Q, 2 with α, β, ρ, ω > 0, λ_1,2, γ_1,2 < 0.

Let the right half part the left half part and the switch mainfold

It is clear that the left subsystem of system (1) on Σ⁻ 3 has an equilibrium point denoted by p = (x_1p x_2p x_3p x_4p)^T = – A⁻¹ a. And the right subsystem of system (1) on Σ⁺ 4 has an equilibrium point denoted by q = (x_1q x_2q x_3q x_4q)^T = – B⁻¹ b. Because p ∈ Σ⁻ and q ∈ Σ⁺, so we have 2x_1p + x_3p < 1 and 2x_1q + x_3q > 1.

For convenience, let P = (ξ₁ ξ₂ ξ₃ ξ₄), Q = (η₁ η₂ η₃ η₄) with ξ_i, η_i ∈ ℝ⁴, (i = 1, 2, 3, 4) being the general eigenvectors of matrix A and matrix B, respectively. Assume that ψ₁(t,x₀) is a solution of system (3) with initial value x₀, and ψ₂(t,y₀) is a solution of system (4) with initial value y₀, where 5 6 From systems (3) and (4), we obtain 7 8 Substituting Eq. (5) into Eq. (7) and Eq. (6) into Eq. (8), we have 9 10

According to Eqs. (9) and (10), it is clear that the unstable mainfolds, through p, q and denoted by W^u(p), W^u(q), are two-dimensional invariant mainfolds. The stable invariant mainfolds, through p, q and denoted by W^s(p), W^s(q), also are two dimensional. The concrete forms are as follows: 11 12 13 14 with k₁, k₂, k₃, k₄ and

The following theorem establishes a criterion for the existence of heteroclinic cycle of 4D piecewise affine system (1).

The condition (i) in Theorem 1 is equivalent to the condition (H₁), which can be obtained from simple analysis. In the following we further show condition (H₂) holding.

Let 15 where c′ = (2 0 1 0) and ψ₁ is defined as Eq. (9). According to p₁ ∈ Ł₁, one obtains 16 17 where

If g₁(t) < 1 – 2x_1p – x_3p for arbitrary t > 0, it is clear that the negative orbit of p₁ satisfies {ψ₁(t,p₁) | t < 0} ⊂ Σ⁻. That means we simply need to prove g₁(t) < 1 – 2x_1p – x_3p for all t > 0. In fact, by Eq. (16) and the third inequation in condition (ii) in Theorem 1, we obtain This means g₁(t) < 1 – 2x_1p – x_3p for t ∈ (0,ε) with sufficient small ε. Let with sufficiently small ε, then g₁(t) < 1 – 2x_1p – x_3p for We further show g₁(t) < 1 – 2x_1p – x_3p for

Let t₀ > 0, then t₀ satisfies 18 i.e., either 19 or 20 Moreover, from Eq. (17), we have 21 When equation (19) holds for t₀ ∈ I, we have which means that t₀ is a local minimum point of g₁(t). Similarly, when equation (20) holds for t₀ ∈ I, we obtain which means that t₀ is a local maximum point of g₁(t).

If k₁ (2ξ₂₁ + ξ₂₃) – k₂(2ξ₁₁ + ξ₁₃) ≥ 0, we obtain that the local maximum points t_0k^[35] of g₁(t) satisfy If k₁ (2ξ₂₁ + ξ₂₃) – k₂(2ξ₁₁ + ξ₁₃) < 0, we obtain that the local maximum points t_0k of g₁(t) satisfy Then the local maximum value is given by 22 According to α > 0 and the first inequation in condition (iii) of Theorem 1, it yields g₁(t_0k0) < ··· < g₁(t₀₁) < 1 – 2x_1p – x_3p. Then g₁(t) < 1 – 2x_1p – x_3p on t ∈ I. Therefore (H₂) holds.

By the same method, (H₄) also can be proved if the fourth inequation in condition (ii) and the second inequation in condition (iii) of Theorem 1 hold.

Next (H₃), {ψ₁(t,q₁) | t > 0} ⊂ Σ⁻, will be shown. Let Then {ψ₁(t,q₁) | t > 0} ⊂ Σ⁻ will holdif g₂(t) < 1 – 2x_1p – x_3p for t > 0. Thus g₂(t) < 1 – 2x_1p – x_3p for t > 0 will be shown in following.

According to Eq. (9) and q₁ ∈ Ł₂, we have 23 where

From expression (23) and the first in equation in condition (ii) of Theorem 1, we have

Since function h(t) is monotonous, h(t) = 0 has only one solution at most for t > 0.

If h(t) = 0 has no solution for t > 0, owing to h(0) < 0, we have h(t) < 0 for t > 0, which implies for t > 0. Hence g₂(t) < 1 – 2x_1p – x_3p for t > 0 as shown in Fig. 1(a). Alternatively, h(t) = 0 has unique solution owing to h(0) < 0, which implies for and for Combining with g₂(t) → 0 for t → + ∞, we always have g₂(t) < 1 – 2x_1p – x_3p for t > 0 as shown in Fig. 1(b). Thus condition (H₃) is proved. Similarly, througth the second inequation in condition (ii), we also can prove the condition (H₅).

Fig. 1.

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	Fig. 1. (color online) The plot of g2(t) for t > 0 when (a) h(t) = 0 has no solution, (b) h(t) = 0 has a unique solution.

According to the inequalities in conditions (ii) and (iii) of Theorem 1, we have which imply condition (H₆). Therefore, the proof of Theorem 1 is completed.

In this section, two numerical tests are given to illustrate the effectiveness of our main results.

By simple calculations, we obtain the equilibrium points of subsystem of subsystem There exist invariant matrices We can respectively convert matrices A and B to Jordan matrices

Choose k₁ = 0, k₂ = 2, k₃ = 11/9, k₄ = 1/9 and we have Therefore, it is easy to see that It means system (24) satisfies the conditions (i)–(iii) in Theorem 1. So the system (24) exists a heteroclinic cycle Γ connecting equilibrium p and q, which intersects Σ transversally at points p₁ and q₁.

Fig. 2.

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	Fig. 2. (color online) The 3D projections of Γ for system (24) in (a) x1–x2–x3 space, (b) x1–x2–x4 space, (c) x1–x3–x4 space, (d) x2–x3–x4 space.

Fig. 3.

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	Fig. 3. (color online) The 3D projections of chaotic invariant set for system (24) in (a) x1–x2–x3 space, (b) x1–x2–x4 space, (c) x1–x3–x4 space, (d) x2–x3–x4 space.

Moreover, the eigenvalues of the matrices A and B satisfy . Thus, by Remark 1, the system (24) has a chaotic invariant set in an adequately small neighborhood of Γ. The 3D projections of Γ and chaotic invariant set of system (24) are shown in Figs. 2 and 3, respectively.

Similarly, according to Theorem 1, choose k₁ = 0, k₂ = 2, k₃ = –2, k₄ = 7 and we have Therefore, it is easy to see that It means that system (25) satisfies the conditions (i)–(iii) in Theorem 1. From Theorem 1, we can also see that system (25) exists a heteroclinic cycle Γ connecting equilibrium points p and q, which intersects Σ transversally at points p₁ and q₁.

Fig. 4.

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	Fig. 4. (color online) The 3D projections of Γ for system (25) in (a) x1–x2–x3 space, (b) x1–x2–x4 space, (c) x1–x3–x4 space, (d) x2–x3–x4 space.

Fig. 5.

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	Fig. 5. (color online) The 3D projections of chaotic invariant set for system (25) in (a) x1–x2–x3 space, (b) x1–x2–x4 space, (c) x1–x3–x4 space, (d) x2–x3–x4 space.

In addition, the eigenvalues of the matrices A and B satisfy . Thus, by Remark 1, the system (25) has a chaotic invariant set in an adequately small neighborhood of Γ. The 3D projections of Γ and chaotic invariant set of system (25) are shown in Figs. 4 and 5, respectively.

We have constructed a new class of 4D two-zone DPASs. The criterion, ensuring that 4D two-zone DPASs can possess heteroclinic cycles to spiral saddle-foci, is given here. From Shil’nikov type theory, it is clear that such 4D piecewise affine systems display chaotic behaviors. The chaotic invariant sets, in a neighborhood of heteroclinic cycles, are also worked out by numerical method. This makes it a good method to analyze theheteroclinic cycles and chaotic invariant sets of 4D piecewise affine systems. Two numerical examples are used to demonstrate the efficiency of our theoretical results.