Existence of homoclinic orbits and heteroclinic cycle in a class of three-dimensional piecewise linear systems with three switching manifolds

In this article, we construct a kind of three-dimensional piecewise linear (PWL) system with three switching manifolds and obtain four theorems with regard to the existence of a homoclinic orbit and a heteroclinic cycle in this class of PWL system. The first theorem studies the existence of a heteroclinic cycle connecting two saddle-foci. The existence of a homoclinic orbit connecting one saddle-focus is investigated in the second theorem


I. INTRODUCTION
In the past 50 years, chaos has been a hot research field in nonlinear science.It is well known that Lorenz discovered the first chaotic attractor in the literature, 1 which indicates that it is unrealistic to predict weather conditions for a long time afterward and leads the trend of researching chaos by numerical simulations.Since then, more and more scholars have begun to devote themselves to the study of chaos.3][4][5][6] However, most of the literature have no strict mathematical proof for the existence of chaos, which can only be verified by computers.One of the main reasons for this is the difficulty in proving the existence of chaos.
With the efforts of many scholars, some achievements have been made in the mathematical proof of chaos in smooth systems, such as the famous Smale Horseshoe 7 and Shilnikov's theorem. 8One of the key issues is the existence of the homoclinic orbit or the heteroclinic cycle, which has been given in some studies.For example, the perturbation is used to study the homoclinic orbit and the heteroclinic cycle of systems. 9,10Leonov showed the fishing method is a good way to prove the existence of the homoclinic orbit and the heteroclinic cycle. 11

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In the past few years, with the development of the study of smooth systems, some limitations of smooth systems in the mathematical modeling of more complex phenomena in the real world have gradually been realized.Therefore, researchers are paying more and more attention to non-smooth dynamic systems, especially the simpler PWL systems.Compared with smooth systems, PWL systems can more accurately characterize complex phenomena in the real world, such as collisions in mechanical systems 12,13 and switching in the circuit. 14lthough smooth systems are similar to PWL systems in some aspects, basic concepts and definitions cannot be copied directly.Therefore, the study of the basic definitions has become one of the hotspots for PWL systems.Some corresponding results were reported in Refs.15 and 16.On the other hand, some unique complex bifurcations caused by discontinuities in PWL systems, such as boundary collision bifurcations and sliding bifurcations, have also attracted the attention of many scholars.In recent years, relevant research results of this research direction have been reflected in Refs.17 and 18. Inspired by Shil'nikov's theorem in smooth systems, the existence of the homoclinic orbit and the heteroclinic cycle is crucial to PWL systems.0][21][22][23][24][25][26] Some researchers have also studied the existence of homoclinic orbits, heteroclinic cycles, and chaos in higher-dimensional PWL systems, and obtained some results [27][28][29][30][31] by constructing a Poincaré map and proving the existence of topological horseshoes.
However, there are few studies on the existence of homoclinic orbits or heteroclinic cycles in PWL systems with multiple manifolds.In Ref. 32, Chen et al. studied the existence of heteroclinic cycles in several kinds of 3D three-region PWL systems with two switching planes.In Ref. 33, Lu et al. proposed a new 3D threeregion PWL system with two discontinuous boundaries.For three different situations, (i) one saddle and two foci, (ii) two saddles and one focus, and (iii) three saddles, some criteria for the existence of heteroclinic cycles are provided.In addition, sufficient conditions for the existence of chaos are obtained.In Ref. 34, Lu et al. studied the coexistence problems of homoclinic orbit connected with one saddle point and heteroclinic cycle connected with two saddle points for a new class of 3D three-region piecewise affine systems (PASs).Recently, Lu et al. further proposed some criteria to locate the coexistence of homoclinic cycles and heteroclinic cycles in a class of 3D PASs and gave a mathematical proof of chaos by analyzing the constructed Poincaré map. 35s far as we know, few people have studied the existence of homoclinic orbits and heteroclinic cycles in PWL systems with three or more switching manifolds.The purpose of this paper is to explore the existence of homoclinic orbits and heteroclinic cycles in a class of 3D PWL system with three switching manifolds.The main idea is to obtain the stable and unstable manifolds, as well as the intersections of the stable manifolds and unstable manifolds with switching manifolds, respectively, and then obtain the corresponding theorems by basic mathematical analyses.
This article is organized as follows.In Sec.II, a novel PWL system with four regions is introduced.Next, the existence theorems of homoclinic orbits and heteroclinic cycles are given in Sec.III.In order to verify the correctness of these theorems, a concrete example and its numerical simulation are given in Sec.IV.Finally, Sec.V discusses the research content of this paper and these problems are worthy of further research in the future.

II. THE PIECEWISE LINEAR SYSTEM WITH THREE SWITCHING MANIFOLDS
This section provides a new class of 3D PWL system with four regions and gets some basic dynamic properties.

Chaos
ARTICLE scitation.org/journal/cha and Denote ψ A 1 (t, x 0 ), ψ A 2 (t, y 0 ), ψ A 3 (t, z 0 ) and ψ A 4 (t, w 0 ) as the solutions of subsystem (2) with the initial values x 0 , y 0 , z 0 , and w 0 , respectively.Therefore, we can get and Denote W u (E s ) and W s (E s ) are the unstable and stable manifolds of E s (s = 2, 3), respectively.From formulas (8) and ( 9), we can see W u (E s ) is two-dimensional and W s (E s ) is one-dimensional.Therefore, we obtain the following formulas: Suppose , and where , respectively.For points q 1 and p 3 , according to Eqs. ( 3) and ( 6), we know that there exist σ i and τ i (i = 1, 2, 3) such that Choosing points p 1 ∈ L 1 and q 3 ∈ L 2 , there exist σ i and τ i (i = 1, 2, 3) similarly such that

III. MAIN RESULTS
In this section, the main results of this paper are given.That is, the conditions for the existence of heteroclinic cycle, homoclinic orbits in the system (1) and related proofs.
First, we will ensure that system (1) has a heteroclinic cycle 1 .The general idea is given here.As shown in Fig. 1, to prove that Theorem 3.1 holds, only the following conditions are satisfied: Theorem 3.1.Suppose that there exist constants k i , k i , i = 1, 2, 3 and points p 2 , q 2 such that the following conditions (i)-(ii) hold: (i) (ii)
Proof.According to the first equation of ( 11), {ψ A 2 (t, p 2 )| t > 0} is a straight line and ψ A 2 (t, p 2 ) → E 2 , when t → +∞.From the previous assumption E 2 ∈ S 2 , we can see that Similarly, to prove formula ( 16), from the second equation of (11), {ψ A 3 (t, q 2 )|t > 0} is a straight line and ψ A 3 (t, q 2 ) → E 3 , when t → +∞.Due to E 3 ∈ S 3 , then Next, we have to prove that {ψ A 2 (t, q 2 )|t < 0} ⊂ S 2 , which is equivalent to Denote a function f . From representation ( 8) and the second equation of condition (i) of Theorem 3.1, then f 1 (t) has the following form: where In order to prove {ψ A 2 (−t, q 2 )|t > 0} ⊂ S 2 , it needs to be verified that From Eqs. ( 8) and ( 19), we can get the following formulas: According to the first inequality in condition (ii) of Theorem 3.1, we have
So, we can get local maximum points π ] by calculation.Thus, the corresponding maximum values are According to α A 2 > 0 and the second inequality in condition (ii) of Theorem 3.1, then where T 1 = t 00 .Similarly, to prove the maximum values of f 1 (t), the minimum point of f 1 (t) is and the minimum value of f 1 (t) is Due to the third inequality of condition (ii) of Theorem 3.1.Then, has been proven.
According to the inequalities of Theorem 3.1, one gets which mean that 1 transversally intersects 2,3 .In summary, the proof of Theorem 3.1 is completed.
Next, the theorem that system (1) has the homoclinic orbit 2 is given.
Before formally proving Theorem 3.2, the general idea of the proof is given here.As shown in Fig. 2, to prove Theorem 3.2 is holds, only the following conditions are satisfied:

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(ii) there exists a constant T 0 > 0 such that where then, system (1) has a homoclinic orbit 2 connecting equilibrium E 2 .Moreover, the homoclinic orbit 2 transversally intersects switching manifold 1,2 at points p 1 and q 1 , as shown in Fig. 2.
Due to the inequalities of Theorem 3.2, there exist which means that the homoclinic orbit 2 transversally intersects switching manifold 1,2 .
Hence, the proof of Theorem 3.2 is completed.Remark 3.1.In Sec.II, regarding the position of equilibrium E 1 , it is assumed that E 1 is located in S 1 for convenience.In fact, even if E 1 is the virtual equilibrium point, where E 1 lies in S 2 , S 3 , or S 4 , we can get something similar to Theorem 3.2.
Then, the theorem that system (1) has the homoclinic 3 is also given below.
Before formally proving Theorem 3.3, the general idea of the proof is given here.As shown in Fig. 3, to prove that Theorem 3.3 holds, only the following conditions are satisfied: Theorem 3.3.Suppose that there exist constants l i , i = 1, 2, 3 and points p 3 , q 3 such that the following conditions (i)-(iii) hold: (i) where then, system (1) has a homoclinic orbit 3 connecting equilibrium E 3 .Moreover, homoclinic orbit 3 intersects switching manifold 3,4 at points p 3 and q 3 transversally, as shown in Fig. 3.
Proof.Similar to the proof of Theorem 3.2, if conditions of Theorem 3.3 are satisfied, we can get and there exists a constant T 4 > 0 such that and Due to formulas (29), ( 30), (31), and (32), system (1) has a homoclinic orbit 3 connecting equilibrium E 3 , and the homoclinic orbit 3 intersects switching manifold 3,4 at points p 3 and q 3 .
According to the inequalities of Theorem 3.3, we can obtain which indicate that the homoclinic orbit 3 transversally intersects switching manifold 3,4 at points p 3 and q 3 .So, the proof of Theorem 3.3 is completed.Remark 3.2.Same as Remark 3.1, in Sec.II, regarding the position of equilibrium E 4 , it is assumed that E 4 is located in S 4 for convenience.In fact, even if E 4 is the virtual equilibrium point, where E 4 lies in S 1 , S 2 , or S 3 , we can get something similar to Theorem 3.3.
At last, the theorem of coexistence of heteroclinic cycle 1 , homoclinic orbits 2 and 3 of system (1) is given.
Proof.Combining with Theorems 3.1-3.3,we can easily get the proof of Theorem 3.4.For the sake of simplicity, we will not repeat them in detail.
Thus, the proof of Theorem 3.4 is completed.Remark 3.3.Same as Remarks 3.1 and 3.2, in Sec.II, regarding the position of equilibrium E 1 and E 4 , it is assumed that E 1 is located in S 1 and E 4 is located in S 1 for convenience.In fact, even if E 1 and E 4 are the virtual equilibrium, where E 1 lies in S 2 , S 3 , or S 4 and E 4 lies in S 1 , S 2 , or S 3 , we can get something similar to Theorem 3.4.

IV. NUMERICAL SIMULATIONS FOR THEORETICAL RESULTS AND CHAOS
In this section, the correctness of theorems is verified by some numerical simulations of a specific case that conforms to theorems.
Considering the 3D PWL system with four regions, The equilibria of the four subsystems are There are invertible matrices

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There exist  and Therefore, this instance satisfies Theorem 3.1, the heteroclinic cycle 1 of this PWL system exists and 1 transversally intersects the switching manifold 2,3 at points p 2 , q 2 , as shown in Fig. 5.The case satisfies Theorem 3.2 too.So, the homoclinic orbit 2 of the FIG. 8. Phase diagram of the coexistence of the heteroclinic cycle 1 , the homoclinic orbit 2 , and the homoclinic orbit 3 in system (33) satisfying Theorem 3.4.Among them, the blue line is heteroclinic cycle, and the red and green lines are homoclinic orbits.

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scitation.org/journal/chasystem exists and 2 transversally intersects the switching manifold 1,2 at points p 1 and q 1 , as shown in Fig. 6.The example also satisfies Theorem 3.3, the homoclinic orbit 3 of the PWL systems exists, and 3 transversally intersects the switching manifold 3,4 at points p 3 and q 3 , as shown in Fig. 7.It is clear that Theorem 3.4 holds for the system, then the heteroclinic cycle 1 and the homoclinic orbits 2 and 3 coexist, as shown in Fig. 8. Furthermore, the chaotic invariant set is shown in Fig. 9.The largest Lyapunov exponent with the Wolf's algorithm is 1.039 for t ∈ [0, 967 000] with a RK4 with a step of 0.001.
A subset of the basin of attraction on the plane {X ∈ R 3 |z = 0} has been numerically found and it is shown in Fig. 10.The blue dots are initial conditions that belong to the basin of attraction while the yellow dots do not.

V. CONCLUSION
This paper introduces a new class of 3D PWL systems with four regions.The analysis on the existence of homoclinic orbits or heteroclinic cycles is presented with four subsystems.We establish sufficient conditions for the coexistence of homoclinic orbits and heteroclinic cycle of 3D PWL systems by rigorous proof.A numerical example with homoclinic orbits, heteroclinic cycle, and chaos is given to illustrate the validity of the presented method and the obtained theoretical results.In addition, the basin of attraction for the chaotic attractor is given.However, for higher-dimensional PWL systems with more switching manifolds, the existence of homoclinic orbit, heteroclinic cycle, and chaos needs further study.

FIG. 4 .
FIG. 4.Schematic diagram of the co-existence of the heteroclinic cycle 1 , the homoclinic orbit 2 , and the homoclinic orbit 3 satisfying Theorem 3.4.The blue line represents the heteroclinic cycle 1 , and the red and green lines represent the homoclinic orbit 2 and the homoclinic orbit 3 , respectively.

FIG. 9 .
FIG. 9. Chaos in system (33) satisfying Theorem 3.4 when the initial point is (3, −2.1, −2) near q 3 : (a) phase diagram in the x-y-z space and (b) projection of (a) on the x-y plane.

FIG. 10 .
FIG. 10.The blue dots on the plane {X ∈ R 3 : z = 0} have been evaluated and belong to the basin of attraction.