Electronic Journal of Differential Equations, Vol. 2024 (2024), No. 57, pp. 1–23. ISSN: 1072-6691. URL: https://ejde.math.txstate.edu, https://ejde.math.unt.edu DOI: 10.58997/ejde.2024.57 GLOBAL DYNAMICS OF A SPECIAL CLASS OF PLANAR SECTOR-WISE LINEAR SYSTEMS QIAN-QIAN HAN, SONG-MEI HUAN Abstract. In this article, we study the global dynamics of a special class of planar sector-wise linear differential systems with two subsystems being the same except for their equilibriums. Taking the position of one equilibrium as the bifurcation parameter, we provide a complete analysis about sliding cy- cle bifurcation and sliding homoclinic bifurcation. Moreover, we obtain the existence of all important separatrix orbits and their dependence on the bifur- cation parameter, including sliding heteroclinic orbit, heteroclinic cycle, limit cycle, sliding homoclinic cycle and sliding cycle. 1. Introduction Piecewise smooth dynamical systems arise in applications, see [1, 2, 3, 4, 6, 9, 10, 13, 32, 33, 34, 35, 45, 46, 47, 51, 52]. As the simplest piecewise smooth dynamical system, planar piecewise linear systems (hereafter referred as PWLSs) with two pieces separated by a straight line has been deeply studied in recent years [12, 14, 15, 16, 17, 19, 23, 29, 30, 36, 42], mainly due to their use in analyzing the dynamics of complex systems locally, the explicit solvability of each linear subsystem and their own applications in modeling real systems. The discontinuity of vector field appeared in piecewise smooth systems induces many dynamics that are more complicated and a lot of DIBs having new bifurcation mechanisms, see [18, 36, 50, 53]. Even in planar PWLSs, we can see some kind of sliding cycles, for example, sliding homoclinic cycles [21, 22] and even double homoclinic loops, which can be found in some engineering models such as Valve oscillators [37], Coulomb friction [51] and so on. In recent years, the main research focus about planar PWLSs with a straight line separation has been transformed from the study of the existence and bifurcation of crossing limit cycles (see [5, 11, 12, 21, 23, 25, 26, 27, 38, 39, 40, 41, 42, 43, 47]) into the study of some critical sliding orbits and their bifurcations. For example, ap- pearance of crossing-sliding bifurcation and a double tangency, pseudo-heteroclinic bifurcations, pseudo-homoclinic bifurcation, pseudo-Hopf bifurcation and so on ([8, 20, 36]). 2020 Mathematics Subject Classification. 34C23, 34C60, 34C37. Key words and phrases. Sector-wise linear systems; sliding cycle bifurcation; sliding homoclinic bifurcation; heteroclinic cycle; non-regular point. ©2024. This work is licensed under a CC BY 4.0 license. Submitted April 2, 2024. Published September 30, 2024. 1 2 Q.-Q. HAN, S.-M. HUAN EJDE-2024/57 In addition to this, the perturbation to discontinuity boundary as an important common phenomenon should also be taken into consideration, since which can lead to many new interesting phenomena and affect both the number and type of crossing limit cycles, see [7, 24, 28, 29, 44, 46, 54, 55, 56]. Therefore, in this article, we study a special family of planar PWLSs with two zones under two considerations, i.e., the discontinuity boundary is given by two rays starting from the same point (which are called planar sector-wise linear systems in [30]) and investigate the global qualitative dynamics of such systems. Particularly, we have obtained explicit dependence on system parameters of the existence, sta- bility and number of all kinds of special sliding points for these planar sector-wise linear systems in [31]. Notice that the discontinuity boundary of planar sector-wise linear systems can be written as Σπ/2 (see [28, Theorem 1]), i.e., Σπ/2 = {(x, y) : x ≥ 0 and y = 0} ∪ {(x, y) : x = 0 and y ≥ 0}. So, here we consider the planar PWLSs ẋ = { F+(x) = (F+ 1 (x), F+ 2 (x))T = A · (x− x+ e ), if x ∈ Rπ/2, F−(x) = (F− 1 (x), F− 2 (x))T = A · (x− x− e ), if x ∈ R3π/2, (1.1) where x = (x, y)T ∈ R2, x± e = (x± e , y ± e ) T ∈ R2, x+ e > 0, y+e = −y−e > 0, A = [aij ] are 2× 2 real invertible matrices, and Rπ/2 = {(x, y) : x > 0 and y > 0}, R3π/2 = {(x, y) : xy = 0, or x < 0 ory < 0}. We will investigate the global dynamics of system (1.1) as the value of x− e changes when both A have two different real eigenvalues and a12a21 ̸= 0. For convenience, we refer to the systems ẋ = A · (x − x+ e ),x ∈ Rπ/2 and ẋ = A · (x− x− e ),x ∈ R3π/2 as the ⊕-system and the ⊖-system, respectively. The remainder of this article is organized as follows. Some preliminaries are given in Section 2. The main results can be found in Section 3. The proofs of the main results and some examples can be found in Section 4 and section 5. Our conclusions are provided in Section 6. 2. Preliminaries Let Σ : H(x, y) = 0 be the discontinuity boundary of the planar PWLSs with two zones ẋ = { F+(x) = (F+ 1 (x), F+ 2 (x))T , if x ∈ Σ+ = {(x, y) : H(x, y) > 0}, F−(x) = (F− 1 (x), F− 2 (x))T , if x ∈ Σ− = {(x, y) : H(x, y) < 0}. (2.1) It is often considered that the curve H(x, y) = 0 is smooth, although it maybe piecewise smooth. Precisely, the points on Σ can be divided into two kinds: regular points of the discontinuity boundary (at which the curve have tangent lines) and non-regular points of the discontinuity boundary. In this article, the discontinuity boundary we considered in system (1.1) has a unique non-regular point (0, 0). So, based on the existent definitions about various peculiar sliding points (see [21, 36, 50, 53]), we first introduce some kinds of regular points of Σ for system (2.1), and then study the non-regular point of Σ for system (1.1) separately. EJDE-2024/57 PLANAR SECTOR-WISE LINEAR SYSTEMS 3 Definition 2.1. A regular point p ∈ Σ is called a crossing point if ⟨F+(p), Hx(p)⟩ · ⟨F−(p), Hx(p)⟩ > 0, where Hx(p) is the transpose of the gradient of the function H(x, y) at p. Denoted by Σc the set of all crossing points. Then Σc is the crossing set of system (2.1). Definition 2.2. A regular point p ∈ Σ is called a sliding point if ⟨F+(p), Hx(p)⟩ · ⟨F−(p), Hx(p)⟩ < 0. We denoted by Σs the set of all sliding points. Then Σs is the sliding set of system (2.1). Moreover, Σs is attractive (repulsive) if ⟨F+(p), Hx(p)⟩ < 0(> 0), ⟨F−(p), Hx(p)⟩ > 0(< 0), p ∈ Σs. In addition to this, the sliding vector field Fs on Σs is often defined by using the Filippov convex method ([13, 35]) to be Fs(p) = ⟨F+(p), Hx(p)⟩F−(p)− ⟨F−(p), Hx(p)⟩F+(p) ⟨F+(p)− F−(p), Hx(p)⟩ , p ∈ Σs, (2.2) and a point p ∈ Σ satisfying Fs(p) = 0 is called a pseudo-equilibrium. When p ∈ Σs is a pseudo-equilibrium, p may show properties of a node, a saddle or a focus according to the stability of p inside Σs and the attractivity of Σs, which induces the following definitions. Definition 2.3. Let p ∈ Σs be an isolated pseudo-equilibrium. Then p will be an unstable (a stable) pseudo-node of system (2.1) if Σs is repulsive (attractive) and p is unstable (stable) inside Σs, a pseudo-saddle if one of the followings is true: (i) Σs is attractive and p is unstable inside Σs; (ii) Σs is repulsive and p is stable inside Σs. Moreover, p will be a pseudo-saddle-node if p is half-stable inside Σs. Definition 2.4. The point p ∈ Σ̄ is called a sliding boundary point if ⟨F+(p), Hx(p)⟩ · ⟨F−(p), Hx(p)⟩ = 0. The set of all sliding boundary points, denoted by Σsb, is called the sliding boundary set of system (1.1). Definition 2.5. Let p ∈ Σsb, we call p a boundary equilibrium of system (1.1) if F+(p) = 0 or F−(p) = 0. More specifically, p is called a double boundary equilibrium when F+(p) = F−(p) = 0, and a unilateral boundary equilibrium induced by the ⊖−system (⊕−system) if F−(p) = 0, F+(p) ̸= 0, (F+(p) = 0, F−(p) ̸= 0). Let p0 = (0, 0) and Fs(p),p ∈ Σs be the unique non-regular point of the discon- tinuity boundary Σ and the sliding vector field of system (1.1), respectively. Notice that we did not discuss the definition of the sliding field Fs at p0. In fact, we even do not care about whether or not this point is a sliding point, which we think is not necessary at least now. However, there is no doubt that the dynamics around p0 play an important role in studying the global dynamics. Therefore, we need to consider Fs around this special point. 4 Q.-Q. HAN, S.-M. HUAN EJDE-2024/57 Suppose that the sliding field Fs is well defined on Iyb ∪ Ixa with Iyb = {(0, y) : y ∈ (0, b)}, Ixa = {(x, 0) : x ∈ (0, a)}. Then we call p0 a non-regular boundary saddle of system (1.1) if there exist a > 0, b > 0 such that ⟨Fs(p1), (0, 1)⟩ · ⟨Fs(p2), (1, 0)⟩ < 0, ∀p1 ∈ Iyb , ∀p2 ∈ Ixa , and a non-regular boundary source (sink) if there exist a > 0 and b > 0 such that ⟨Fs(p1), (0, 1)⟩ · ⟨Fs(p2), (1, 0)⟩ > 0, ⟨Fs(p1), (0, 1)⟩ > 0 (< 0), for all p1 ∈ Iyb and all p2 ∈ Ixa . In addition to this, according to [36, 48, 49], we obtain that an orbit which has a common segment with the sliding set is called a sliding orbit, and a closed orbit with sliding motion is called a sliding cycle. Furthermore, a sliding-zero cycle is a closed orbit intersecting Σ at an end point of the sliding region. A sliding heteroclinic orbit (or cycle) is a hereroclinic one with sliding motion. A sliding homoclinic orbit (or cycle) is a homoclinic one with sliding motion. Moreover, for a piecewise smooth dynamical system with a parameter µ, a slid- ing cycle bifurcation happens at µ = µ0 means that there is a sliding-zero cycle when µ = µ0, while for 0 < |µ − µ0| ≪ 1 the sliding-zero cycle disappears, and instead there appear a limit cycle and a sliding cycle when µ crosses µ0 in oppo- site directions, respectively. And a sliding homoclinic bifurcation of the system at µ = µ0 means the system has a sliding homoclinic cycle when µ = µ0, while for 0 < |µ − µ0| ≪ 1 the sliding homoclinic cycle disappears, instead there appears a sliding cycle when µ varies in one direction and neither sliding homoclinic orbits nor sliding cycles when µ varies in another direction. Finally, to give our main results, we introduce some notation for the planar linear system ẋ = A · (x− x± e ), (2.3) where det(A) < 0 (i.e., A has two different real eigenvalues having opposite signs: λ1 > 0 > λ2) and [tr(A)]2 > 4 det(A). Then λ1 = tr(A) + √ [tr(A)]2 − 4 det(A) 2 , λ2 = tr(A)− √ [tr(A)]2 − 4 det(A) 2 . Moreover, denote the invariant manifolds of x± e by l±1 , l ± 2 . Let k ± 1 , k ± 2 be the slopes of l±1 , l ± 2 , y± m1 = (0, y±m1), y± m2 = (0, y±m2) and x± m1 = (x± m1, 0), x± m2 = (x± m2, 0) be the intersections of l±1 and l±2 with the y-axis and the x-axis, respectively. In addition to this, let y± t = (0, y±t ) and x± t = (x± t , 0) be the contact points at which the trajectories of system (2.3) being tangent to the y-axis and the x-axis, respectively. Then by simple calculations, it follows that k±1 = λ1 − a11 a12 , k±2 = λ2 − a11 a12 , (2.4) x± t = x± e + a22 a21 y±e , y±t = y±e + a11 a12 x± e , (2.5) EJDE-2024/57 PLANAR SECTOR-WISE LINEAR SYSTEMS 5 x± m1 = x± e − a12 λ1 − a11 y±e = x± t − λ1 a21 y±e , x± m2 = x± e − a12 λ2 − a11 y±e = x± t − λ2 a21 y±e , y±m1 = y±e − λ1 − a11 a12 x± e = y±t − λ1 a12 x± e , y±m2 = y±e − λ2 − a11 a12 x± e = y±t − λ2 a12 x± e . (2.6) Remark 2.6. It should be noted that, by system (1.1) with λ1 > 0 > λ2, we obtain F1(0, y ± m1) = −λ1x ± e , F1(0, y ± m2) = −λ2x ± e , F2(x ± m1, 0) = −λ1y ± e , F2(x ± m2, 0) = −λ2y ± e , which means l±1 and l±2 are the unstable and stable invariant manifolds of x± e , respectively. 3. Main results In this paper, we give the qualitative analysis of system (1.1) under the following set of conditions (H): det(A) = a11a22 − a12a21 < 0, a12a21 ̸= 0, a21 > 0, a22 > 0, x+ e > 0, y+e = −y−e > 0, (3.1) for which, we have the following considerations. First, in this article we want to investigate the dynamics of system (1.1) related to sliding homoclinic/heteroclinic orbits, so we consider the case det(A) < 0. The eigenvalues of A will be denoted by λ1 > 0 > λ2. Moreover, since the discontinuity boundary of system (1.1) is given by the positive x-axis and the positive y-axis, we use a21a12 ̸= 0 to guarantee that the section return maps can be well defined. Notice that the first condition in (3.1) implies a11a22 ̸= 0. Moreover, the signs of a12 and a21 will be changed at the same time under the transformation (x, y, t) → (−x + 2x+ e , y, t) with the signs of a11 and a22 unchanged. Similarly, the signs of a11 and a22 will be changed simultaneously under the transformation (x, y, t) → (−x + 2x+ e , y,−t) with the signs of a12 and a21 unchanged. Therefore, when we need to analyze the section return maps defined on the discontinuity boundary, we only need to discuss the case when a21 > 0, a22 > 0. More precisely, we only consider the following two cases: Ω1 = {A : a21 > 0, a22 > 0, a12a11 > 0}, Ω2 = {A : a21 > 0, a22 > 0, a12a11 < 0}. Finally, because the global dynamics of piecewise smooth systems is so complex that we put x+ e > 0, y+e = −y−e > 0 and only left x− e as a bifurcation parameter. To state our main results about the global dynamics of system (1.1) under (3.1) by means of providing the existence of all important separatrix in the state space, we first give the following propositions to clarify the existence of sliding set and pseudo-equilibrium points, and the properties of the unique non-regular p0 = (0, 0). 6 Q.-Q. HAN, S.-M. HUAN EJDE-2024/57 Proposition 3.1. Suppose that condition (3.1) holds and A ∈ Ω1 ∪ Ω2. Let µ1 := a22 a21 y+e , µ2 := a12 a11 y+e . Then the sliding set Σs of system (1.1) is Σs = Σx s ∪ Σy s with Σx s = {(x, 0) : x > 0, x ∈ (min{x− t , x + t },max{x− t , x + t })}, =  {(x, 0) : 0 ≤ x < x+ t }, if x− e < µ1, {(x, 0) : x− t < x < x+ t }, if µ1 ≤ x− e < x+ e + 2µ1, ∅, if x− e = x+ e + 2µ1, {(x, 0) : x+ t < x < x− t }, if x− e > x+ e + 2µ1, (3.2) and Σy s = {(0, y) : y > 0, y ∈ (min{y−t , y+t },max{y−t , y+t })}. (3.3) About Σy s , we have the following statements: (a1) When A ∈ Ω1, we have Σy s =  {(0, y) : 0 ≤ y < y+t }, if x− e < µ2, {(0, y) : y−t < y < y+t }, if µ2 ≤ x− e < x+ e + 2µ2, ∅, if x− e = x+ e + 2µ2, {(0, y) : y+t < y < y−t }, if x− e > x+ e + 2µ2. (3.4) (a2) When A ∈ Ω2 with x+ e + µ2 > 0, we have Σy s = { {(0, y) : 0 ≤ y < y−t }, if x− e < µ2, ∅, if x− e ≥ µ2. (3.5) (a3) When A ∈ Ω2 with x+ e + µ2 ≤ 0, we have Σy s =  {(0, y) : y+t < y < y−t }, if x− e < x+ e + 2µ2, ∅, if x− e = x+ e + 2µ2, {(0, y) : y−t < y < y+t }, if x+ e + 2µ2 < x− e ≤ µ2, {(0, y) : 0 ≤ y < y+t }, if x− e > µ2. (3.6) Proof. Note that the discontinuity boundary of system (1.1) is composed of the origin, the positive x-axis and the positive y-axis. So according to Definition 2.2, when x = (x, 0) is on the positive x-axis, we have ⟨F+(p), Hx(p)⟩ · ⟨F−(p), Hx(p)⟩ = F+ 2 (x, 0)F− 2 (x, 0) = a221(x− x+ t )(x− x− t ), and when x = (0, y) is on the positive y-axis, we have ⟨F+(p), Hx(p)⟩ · ⟨F−(p), Hx(p)⟩ = F+ 1 (0, y)F− 1 (0, y) = a212(y − y+t )(y − y−t ). Moreover, when A ∈ Ω1 ∪ Ω2, we have x+ t > 0 and 0 < µ1 < 2µ1 + x+ e under the condition (3.1). And by easy computations, it follows that x− e = µ1 ⇔ x− t = 0, x− e = x+ e + 2µ1 ⇔ x− t = x+ t . (3.7) It is obvious that x− t increases with respect to x− e by (2.5). Then (3.2) and (3.3) can be obtained directly. In addition, by simple calculations, we obtain x− e = µ2 ⇔ y−t = 0, x− e = x+ e + 2µ2 ⇔ y−t = y+t . (3.8) EJDE-2024/57 PLANAR SECTOR-WISE LINEAR SYSTEMS 7 On the one hand, when A ∈ Ω1, by (2.5), we have y+t > 0, and y−t is a linear monotone increasing function with respect to x− e . Then (3.4) can be obtained easily. On the other hand, when A ∈ Ω2, by (2.5), it follows that y+t { < 0, if x+ e + µ2 > 0, ≥ 0, if x+ e + µ2 ≤ 0. (3.9) And y−t is a linear monotone decreasing function with respect to x− e . Then (3.5) and (3.6) can be proved directly. □ Based on the results about sliding sets given in Proposition 3.1, we can now write out the sliding vector field of system (1.1) defined by using the Filippov convex method as follows: Fs(p) =  1 F+ 2 (p)−F− 2 (p) ( F+ 2 (p)F− 1 (p)− F− 2 (p)F+ 1 (p) 0 ) , if p ∈ Σx s ; 1 F+ 1 (p)−F− 1 (p) ( 0 F+ 1 (p)F− 2 (p)− F− 1 (p)F+ 2 (p) ) , if p ∈ Σy s . (3.10) Then the existence, number and stability of pseudo-equilibrium points (i.e., the equilibrium points of Fs) of system (1.1) are provided in the next proposition. Proposition 3.2. Suppose that condition (3.1) holds and A ∈ Ω1 ∪ Ω2. There exists at most one pseudo-equilibrium point in system (1.1). More precisely, about the pseudo-equilibrium point in Σx s , we have: (a1) If x− e ≥ −x+ e and x− e ̸= x+ e + 2µ1, there exists a unique pseudo-equilibrium point (x∗, 0), which is a stable (an unstable) pseudo-node when x− e < x+ e + 2µ1 (x− e > x+ e + 2µ1); (a2) If x− e < −x+ e or x− e = x+ e + 2µ1, there exists no pseudo-equilibrium point in Σx s . About the pseudo-equilibrium point in Σy s , we have: (b1) If x− e ≤ −x+ e and x− e ̸= x+ e + 2µ2, there exists a unique pseudo-equilibrium point (0, y∗), which is a stable (an unstable) pseudo-node when a11[x − e − (x+ e + 2µ2)] < 0 (a11[x − e − (x+ e + 2µ2)] > 0); (b2) If x− e > −x+ e or x− e = x+ e + 2µ2, there exists no pseudo-equilibrium point in Σy s . Proof. By the definition of a pseudo-equilibrium given in Definition 2.2 and (3.10), we know that pex = (x∗, 0) is a pseudo-equilibrium of system (1.1) if and only if pex ∈ Σx s and F+ 2 (x∗, 0)F− 1 (x∗, 0)− F− 2 (x∗, 0)F+ 1 (x∗, 0) = −2 det(A) · y+e · (x∗ − x− e + x+ e 2 ) = 0. (3.11) Similarly, pey = (0, y∗) is a pseudo-equilibrium of system (1.1) if and only if pey ∈ Σy s and F+ 1 (0, y∗)F− 2 (0, y∗)− F− 1 (0, y∗)F+ 2 (0, y∗) = −det(A) [ (x+ e − x− e )y ∗ + (x+ e + x− e )y + e ] = 0. (3.12) 8 Q.-Q. HAN, S.-M. HUAN EJDE-2024/57 Then we obtain x∗ = x− e +x+ e 2 and y∗ = x− e +x+ e x− e −x+ e y+e directly from (3.11) and (3.12). In addition to this, we have x∗ = x+ t + x− t 2 = x+ m1 + x− m1 2 = x+ m2 + x− m2 2 , which means that pex is the midpoint of x± e , x ± t , x ± m1 and x± m2 at the same time. And lim x− e →−∞ y∗ = y+e , lim x− e →(x+ e )− y∗ = −∞, lim x− e →(x+ e )+ y∗ = +∞, lim x− e →+∞ y∗ = y+e , this implies that y∗ decreases with respect to x− e , which is illustrated in Figure 1. -x e + y e + x e - y* x e + Figure 1. The changing of y∗ with respect to x− e On the one hand, it is easy to see that x∗ ≥ 0 when x− e ≥ −x+ e , and x∗ = 0 if and only if x− e = −x+ e . Since x− e = −x+ e < 0 < µ1, which means that pex = (0, 0) ∈ Σx s is a pseudo-equilibrium point by (3.2). Furthermore, to prove whether pex = (x∗, 0) ∈ Σx s as x∗ > 0, we need the following calculations: x∗ − x− t = 1 2 (x+ e + 2µ1 − x− e ), x∗ − x+ t = 1 2 (x− e − 2µ1 − x+ e ), which implies that x− t < x∗ < x+ t , if x− e < x+ e + 2µ1; x+ t < x∗ < x− t , if x− e > x+ e + 2µ1. Then by (3.2), we obtain pex = (x∗, 0) ∈ Σx s , that is pex = (x∗, 0) is a pseudo- equilibrium point of system (1.1) only as x− e > −x+ e and x− e ̸= x+ e + 2µ1. On the other hand, according to Figure 1, it is easy to see that y∗ > 0 when x− e ∈ (−∞,−x+ e ) ∪ (x+ e ,+∞), and y∗ = 0 if and only if x− e = −x+ e . Firstly, when x− e = −x+ e , simple computation shows that x− e − µ2 = −x+ e − µ2 = −a12 a11 y+t . Then we obtain that pey = (0, 0) ∈ Σy s as long as x+ e +µ2 ̸= 0 by (3.4)-(3.6). Furthermore, EJDE-2024/57 PLANAR SECTOR-WISE LINEAR SYSTEMS 9 to prove whether pey = (0, y∗) ∈ Σy s as x− e ∈ (−∞,−x+ e ) ∪ (x+ e ,+∞), we need the following statements: y∗ − y−t = x− e x− e − x+ e a11 a12 (x+ e + 2µ2 − x− e ), y∗ − y+t = x+ e x− e − x+ e a11 a12 (x+ e + 2µ2 − x− e ), x+ e + 2µ2 > 0, if A ∈ Ω1, x+ e + 2µ2 < 0, if A ∈ Ω2, which implies that as A ∈ Ω1 and x− e < x+ e (A ∈ Ω2 and x− e > x+ e ), we have x− e < x+ e + 2µ2 (x− e > x+ e + 2µ2), then we obtain that y−t < y∗ < y+t (y∗ > max{y−t , y+t }). As A ∈ Ω1 and x− e > x+ e (A ∈ Ω2 and x− e < −x+ e ), we obtain that y∗ < min{y−t , y+t }(y−t < y∗ < y+t ), if x− e > x+ e + 2µ2; y∗ > max{y−t , y+t }(y+t < y∗ < y−t ), if x− e < x+ e + 2µ2. Then by (3.4)-(3.6), it follows that pey = (0, y∗) ∈ Σy s , that is pey = (0, y∗) is a pseudo-equilibrium point of system (1.1) only as x− e < −x+ e and x− e ̸= x+ e + 2µ2. Finally, we discuss the stability of pex and pey. Let f(x) = 2 det(A) · y+e · (x− x− e +x+ e 2 ) and g(x) = det(A) · [(x+ e − x− e )y + (x+ e + x− e )y + e ]. Then x∗ and y∗ are the isolated root of the equation f(x) = 0 and g(x) = 0, respectively. We first discuss the stability of pex. By Definition 2.3, on the one hand, Σx s = {(x, 0) : F+ 2 (x, 0) · F− 2 (x, 0) < 0}. So the sliding set Σx s is escaping if F− 2 (x, 0) < 0 and attracting if F− 2 (x, 0) > 0. On the other hand, the sliding field (3.10) at (x, 0) ∈ Σx s is Fsx(x, 0) = ( F 1 sx(x, 0) 0 ) , F 1 sx(x, 0) = f(x) F− 2 (x, 0)− F+ 2 (x, 0) . (3.13) Then from (3.13) and the fact that F+ 2 (x, 0) · F− 2 (x, 0) < 0, we have sign(F 1 sx(x, 0)) = sign(F− 2 (x, 0)f(x)). (3.14) And it is easy to see f ′(x∗) = 2 det(A) · y+e < 0. So we must have f(x) > 0 on (x∗ − ε, x∗) and f(x) < 0 on (x∗, x∗ + ε) for some small ε > 0. From (3.14), if the sliding set is attracting (escaping), i.e., F− 2 (x, 0) > 0(< 0), pex will be stable (unstable). That is pex will always be a pseudo-node of system (1.1) with A ∈ Ω1∪Ω2, which is a stable (an unstable) pseudo-node when x− e < x+ e +2µ1(x − e > x+ e +2µ1). Similarly, we can also get the stability of pey. The proof is complete. □ Next, we give the properties of the non-regular point p0. Proposition 3.3. Suppose that condition (3.1) holds and A ∈ Ω1 ∪ Ω2. We have the following statements about the non-regular point p0. (a) When A ∈ Ω1, about the non-regular point p0, we have (a1) As x− e = µ1 or x− e = µ2, p0 is a boundary equilibrium point; (a2) As min{µ1, µ2} < x− e < max{µ1, µ2}, p0 is a sliding boundary point; (a3) As x− e > max{µ1, µ2}, p0 is a crossing point; 10 Q.-Q. HAN, S.-M. HUAN EJDE-2024/57 (a4) When a12 > 0 (a12 < 0), p0 is a non-regular boundary saddle (a non-regular boundary sink), a stable pseudo-node (a pseudo-saddle-node), a non-regular boundary saddle (a non-regular boundary source) for x− e < −x+ e , x − e = −x+ e and −x+ e < x− e < min{µ1, µ2}, respectively. (b) When A ∈ Ω2, about the non-regular point p0, we have (b1) As x− e = µ1 or x− e = µ2, p0 is a boundary equilibrium point; (b2) As x− e > µ1, p0 is a crossing point; (b3) When x+ e + µ2 > 0 (x+ e + µ2 < 0), p0 is a non-regular boundary sink (a sliding boundary point), a pseudo-saddle-node (a stable pseudo-node), a non-regular boundary source (a non-regular boundary saddle), a sliding boundary point (a non-regular boundary saddle) for x− e < min{−x+ e , µ2}, x− e = −x+ e , min{−x+ e , µ2} < x− e < max{−x+ e , µ2} and max{−x+ e , µ2} < x− e < µ1, respectively. Proof. To discuss the non-regular point p0, we need the following statements: µ2 − µ1 = −det(A) a11a21 y+e , µ2 − (−x+ e ) = a12 a11 y+t , µ1 − (−x+ e ) = x+ t > 0. Then it is easy to see that when A ∈ Ω1, we obtain −x+ e < 0 < µ1 < µ2, for a12 > 0; −x+ e < 0 < µ2 < µ1, for a12 < 0. And when A ∈ Ω2, we obtain −x+ e < µ2 < 0 < µ1, for x+ e + µ2 > 0; (3.15) µ2 < −x+ e < 0 < µ1, for x+ e + µ2 ≤ 0. (3.16) And for x− e = −x+ e , it is easy to see pex = pey = p0. Thus by Proposition 3.1 and Proposition 3.2, the statements in this proposition can be obtained directly. □ Before we give the global qualitative dynamics of system (1.1), we denote µ3 := x+ e + a22 − a11 a21 y+e , µ4 := x+ e + a22(λ2 − a11)− a12a21 a21(λ2 − a11) y+e , µ5 := x+ e − 2a12 λ2 − a11 y+e , µ6 := (λ2 − a11)x + e − 2a12y + e λ1 − a11 , µ7 := −a11x + e + 2a12y + e λ1 − a11 , µ8 := 2a12y + e − (λ2 − a11) a11 , µ9 := − a12 λ1 − a11 y+e . (3.17) And by simple computations, it follows that x− e = µ3 ⇔ x+ m2 = x− m1, x− e = µ4 ⇔ x+ m2 = x− t , x− e = µ5 ⇔ x+ m2 = x− m2, x− e = µ6 ⇔ y−m1 = y+m2, x− e = µ7 ⇔ y−m1 = y+t , x− e = µ8 ⇔ y−t = y+m2, x− e = µ9 ⇔ y−m1 = 0. (3.18) EJDE-2024/57 PLANAR SECTOR-WISE LINEAR SYSTEMS 11 Secondly, to make our analysis of the dynamics more concrete, without loss of generality, we assume that x+ m1 > 0 ⇔ 0 < y+e < λ1 − a11 a12 x+ e . (3.19) Since the discussion with x+ m1 ≤ 0 is similar, so we omit it. Theorem 3.4. Suppose that condition (3.1) holds and A ∈ Ω1 ∪ Ω2. We have max{2µ1 + x+ e , µ3} < µ4 < µ5 and the following statements hold. (a) When x− e = µ3, there exists one heteroclinic cycle; (b) When min{2µ1+x+ e , µ3} < x− e < max{2µ1+x+ e , µ3}, there exists one limit cycle, which is repulsive (attractive) if µ3 < 2µ1 + x+ e (µ3 > 2µ1 + x+ e ); (c) When x− e = µ4 and x− e = µ5, there both exist two sliding heteroclinic orbits. Theorem 3.5. Suppose that condition (3.1) holds and A ∈ Ω1, the following state- ments are true. (a) When a11 > 0, we have µ6 < µ7 < 0 < µ1 < µ3 < 2µ1 + x+ e < µ4 < µ5 and the following statements hold. (a1) When x− e < min{−x+ e , µ6}, there exists one sliding heteroclinic orbit; (a2) When x− e = µ6, there exist one heteroclinic orbit and one sliding hetero- clinic orbit; (a3) When x− e = −x+ e , if x − e < µ6, there exists one sliding heteroclinic orbit; Or there exist two sliding heteroclinic orbits; (a4) When min{−x+ e , µ6} < x− e < µ3, there exist two sliding heteroclinic orbits. (b) When a11 < 0, we have −x+ e < 0 < µ9 < min{2µ1 + x+ e , µ3} < max{2µ1 + x+ e , µ3} < µ4 < µ5 and the following statements. (b1) When x− e < −x+ e , there exists one sliding orbit containing pey, p0 and x+ e ; (b2) When −x+ e ≤ x− e < µ9, there exists one sliding heteroclinic orbit; (b3) When µ9 ≤ x− e < min{2µ1 + x+ e , µ3}, there exist two sliding heteroclinic orbits. Theorem 3.6. Suppose that condition (3.1) holds and A ∈ Ω2, the following state- ments are true. (a) When a11 < 0 and 0 < y+e < −a11 a12 x+ e , we have µ8 < min{µ6, µ2} < 0 < µ1 < max{2µ1 + x+ e , µ3} < µ4 < µ5 and the following statements hold. (a1) When x− e < min{µ8,−x+ e }, there exists one sliding orbit containing x+ e , p0 and pey; (a2) When x− e = µ8, there exists one sliding cycle; (a3) When x− e = µ6, there exists one sliding heteroclinic orbit; (a4) If µ8 < −x+ e , we obtain (a41) When µ8 < x− e < −x+ e , there exists one sliding cycle; (a42) When x− e = −x+ e , there exist one sliding homoclinic cycle and one sliding heteroclinic orbit; 12 Q.-Q. HAN, S.-M. HUAN EJDE-2024/57 (a5) When min{−x+ e , µ6, µ2} < x− e < min{2µ1+x+ e , µ3}, there exist two sliding heteroclinic orbits and at most one sliding cycle. (b) When a11 < 0 and −a11 a12 x+ e < y+e < λ1−a11 a12 x+ e , we have µ8 < µ6 < µ7, µ8 < 2µ2 + x+ e < µ2 < −x+ e < 0 < max{2µ1 + x+ e , µ3} < µ4 < µ5 and the following statements hold. (b1) When x− e = µ8, there exists one sliding heteroclinic orbit; (b2) When x− e = µ6, there exists one heteroclinic orbit; (b3) If 2µ2 + x+ e < µ6 < µ2, when µ8 < x− e < 2µ2 + x+ e , there exists one sliding homoclinic cycle; (b4) If µ6 < 2µ2 + x+ e , when µ8 < x− e < 2µ2 + x+ e , there exist a sliding cycle bifurcation and a sliding homoclinic bifurcation; (b5) When 2µ2 + x+ e < x− e < min{2µ1 + x+ e , µ3}, there exist two sliding hetero- clinic orbits. There is no other separatrix of system (1.1) except for the separatrixes given in the above theorems. 4. Proofs of main results Proof of Theorem 3.4. By (2.5) and (2.6), it is easy to see that x− m1 x− m2 and x− t increase with respect to x− e . And by (3.19) and simple calculation, it follows that for A ∈ Ω1 ∪ Ω2, we have x− m2 < x− t < x− m1, 0 < x+ m1 < x+ t < x+ m2. (4.1) Then µ3 < µ4 < µ5 and 2µ1 + x+ e < µ4 can be obtained directly since (3.7) and (3.18), which means max{µ3, 2µ1 + x+ e } < µ4 < µ5. As x− e = µ3, we know x∓ m1 = x± m2, it is also easy to see x+ m1 < min{x+ t , x − t } < max{x+ t , x − t } < x+ m2 by (3.19). So there is a heteroclinic cycle connecting x+ e with x− e . In addition, by simple calculation, it follows that µ3 − (2µ1 + x+ e ) = − y+e a21 (a11 + a22). Then for A ∈ Ω1 ∪ Ω2, we obtain that µ3 > 2µ1 + x+ e , if a11 + a22 < 0; µ3 < 2µ1 + x+ e , if a11 + a22 > 0. (4.2) For µ3 < x− e < 2µ1 + x+ e , we have x+ m1 < x− m2 < x− t < x+ t < x+ m2 < x− m1. And by Proposition 3.2, pex ∈ Σx s is stable. Then there must exist a repulsive limit cycle L1. To prove this, we need to construct the Poincaré map as follows: P1 : DP1 7→ DP1 , P1(x0, 0) = (x1, 0), whereDP1 = {(x, 0) : x ∈ [x+ t , x + m2]}, and (x1, 0) is the first arriving point atDP1 of the orbit of system (1.1) with the initial point (x0, 0) ∈ DP1 . Since x− t < x+ t < x+ m2, there must exist (x⋆ 1, 0), (x ⋆ 2, 0) ∈ DP1 and x+ t < x⋆ 2 < x⋆ 1 < x+ m2, such that P1(x ⋆ 1, 0) = (x+ m2, 0), P1(x ⋆ 2, 0) = (x+ t , 0). This means there must be (x⋆, 0) ∈ DP1 , such that P1(x ⋆, 0) = (x⋆, 0), EJDE-2024/57 PLANAR SECTOR-WISE LINEAR SYSTEMS 13 which implies the existence of the repulsive limit cycle L1. Similarly, for µ3 < x− e < 2µ1 + x+ e , we can prove the existence of an attractive limit cycle L2. As x− e > max{2µ1+x+ e , µ3}, we have x+ t < x− t and x− m1 > x+ m2. Then according to Proposition 3.1 and Proposition 3.2, it follows that Σx s = {(x, 0) : x+ t < x < x− t } is repulsive and pex is unstable. So there is no other special dynamics expect for two sliding heteroclinic orbits L3 and L4, both of which connect x± e and pex as x− e = µ4 and x− e = µ5, respectively. The proof is complete. □ Proof of Theorem 3.5. (a) In this case, since a11 > 0 and (3.19), we obtain that y+m2 > y+t > y+e > 0 > y+m1, 0 < x+ m1 < x+ e < x+ t < x+ m2. By (2.6), it is obvious that y−m1 decreases with respect to x− e . So we obtain µ6 < µ7 < 0 directly because of (3.18). Moreover, when x− e = µ5, we have x∓ m1 = x± m2, which means x− t > 0. So we obtain that 0 < µ1 < µ3. Since (4.2), it is easy to see µ3 < 2µ1 + x+ e . Then by Theorem 3.4, we obtain that µ6 < µ7 < 0 < µ1 < µ3 < 2µ1 + x+ e < µ4 < µ5. In addition, by simple calculation, it follows that µ6− (−x+ e ) = (a22 − a11)x + e − 2a12y + e λ1 − a11 , µ7− (−x+ e ) = λ2 − a11 − 2a12y + e λ1 − a11 , (4.3) the signs of which are not sure. Then −x+ e < µ6 < µ7 < 0, µ6 < −x+ e < µ7 < 0 or µ6 < µ7 < −x+ e < 0 could happen. Firstly, according to Proposition 3.1, Proposition 3.2 and Remark 2.6, it is obvi- ous to see that there is a sliding heteroclinic orbitM1 from x+ e to pey for x− e ≤ −x+ e , and a sliding heteroclinic orbit M2 from x+ e to pex for x− e > −x+ e . Moreover, a heteroclinic orbit M3 from x− e to x+ e appears only as x− e = µ6 obviously. For µ6 < x− e < µ7, the orbit of system (1.1) starting from (0, y−m1) will arrive at the sliding set Σs. And for µ7 ≤ x− e < µ3, the invariant manifold l−1 of x− e will inter- sects with Σs. So there is a sliding heteroclinic orbit M4 from x− e to the unique pseudo-equilibrium point pex or pey for µ6 < x− e < µ3. Thus we obtain that M1 exists for x− e < min{−x+ e , µ6}. When min{−x+ e , µ6} < x− e < µ3, there exist two sliding heteroclinic orbits M2 and M4. As x− e = µ6, there exists the heteroclinic orbit M3 and the sliding heteroclinic orbit M4. (b) Since a11 < 0, we obtain that x− m2 < x− t < x− m1 < x− e , 0 < x+ e < x+ m1 < x+ t < x+ m2. By (2.6) and (3.18), we know x− m1 = y−m1 = 0 as x− e = µ9, then 0 < µ9 < min{2µ1+ x+ e , µ3} can be obtained directly. And by Theorem 3.4, we have −x+ e < 0 < µ9 < min{2µ1 + x+ e , µ3} < max{2µ1 + x+ e , µ3} < µ4 < µ5. According to Propositions 3.1, 3.2 and 3.3, it is easy to see there exists a sliding orbit connecting x+ e , pey and the non-regular boundary sink p0 for x− e < −x+ e , which will become a sliding heteroclinic orbit from x+ e , pex for −x+ e ≤ x− e < min{2µ1+x+ e , µ3}. Once x− e = µ9, there appears a sliding heteroclinic orbit from x− e to pex, which will persist until x− e = min{2µ1+x+ e , µ3}. The proof is complete. □ Proof of Theorem 3.6. When A ∈ Ω2, it is not difficult to verify that the case of matrix A with a21 > 0, a22 > 0, a12 < 0, a11 > 0 is invalid since det(A) = a11a22 − a12a21 > 0, which contradicts that det(A) < 0 in the condition (3.1). So 14 Q.-Q. HAN, S.-M. HUAN EJDE-2024/57 we just need to give the discussion of system (1.1) with A ∈ Ω2 and a11 < 0 under assumption (3.19) as follows. (a) When a11 < 0 and 0 < y+e < −a11 a12 x+ e , we know that y+t < 0 by (3.9). In addition, we obtain that y+m2 > y+e > 0 > y+t > y+m1, 0 < x+ m1 < x+ e < x+ t < x+ m2. Note that y−m1 > y−t and both of them decrease with respect to x− e by (2.5) and (2.6). Then as x− e = µ8, we must have y−m1 > y−t = y+m2 > 0, which means µ8 < µ6 < 0 and µ8 < µ2 < 0 since (3.8) and (3.18). Moreover, by simple calculation, it follows that µ8 − (−x+ e ) = 2a11y + e − (λ2 − 2a11)x + e a11 , µ2 − µ6 = (λ1 + a11)a12y + e − a11(λ2 − a11)x + e (λ1 − a11)a11 . (4.4) And by (4.3), we know the signs of µ8 − (−x+ e ), µ2 − µ6 and µ6 − (−x+ e ) are not sure. Then by (3.15) and Theorem 3.4, we know µ8 < min{µ6, µ2} < 0 < µ1 < max{2µ1 + x+ e , µ3} < µ4 < µ5. Meanwhile, we obtain that −x+ e < µ8 < min{µ6, µ2}, µ8 < −x+ e < min{µ6, µ2} or µ8 < µ6 < −x+ e < µ2 could take place. By Propositions 3.1, 3.2 and 3.3, and Remark 2.6, for x− e < min{µ8,−x+ e }, there is a sliding orbit J1 containing pey, x + e and p0, which is a non-regular boundary sink. As x− e = µ8, if µ8 < −x+ e , there is a sliding cycle J2 connecting pey, x + e and p0, which is a non-regular boundary sink; If µ8 > −x+ e , there is a sliding cycle J3 connecting pex, x + e and p0, which is a non-regular boundary source. Moreover, as x− e = µ6, there is a heteroclnic orbit J4 from x− e to x+ e . As µ8 < x− e < µ3, the orbits of ⊕-system starting from (0, y−t ) will arrive at Σx s since y+t < 0 and Σx s is attractive before x− e = µ3. Then on the one hand, if µ8 < −x+ e , for µ8 < x− e < −x+ e , J2 will become a sliding cycle J21 connecting pey and the non-regular boundary sink p0. Then J21 will turn into a sliding homoclinic cycle J22 as x− e = −x+ e , which containing the pseudo-saddle-node pex = pey = p0, meanwhile, a sliding heteroclinic orbit J5 from x+ e to pex occurs, which will exist until x− e = min{2µ1+x+ e , µ3}. And J22 will turn into a sliding cycle J23 connecting pex and the non-regular boundary source p0 as long as the first arriving point on Σx s of the orbit of ⊕-system starting from (0, y−t ) is on the right of pex. However, if µ8 > −x+ e , J3 can only becomes J23 before x− e = µ3. On the other hand, if µ6 < −x+ e < µ2, we have as µ6 < x− e < µ2, J4 will firstly become a sliding heteroclinic orbit J41 from x− e to pey for x− e < −x+ e , then J41 will become a sliding heteroclinic orbit J42 from x− e to pex for x− e ≥ −x+ e , which will persist until x− e = min{2µ1 + x+ e , µ3}; If −x+ e < min{µ6, µ2}, we obtain that J4 will only become J42. (b) In this case, we know y+t > 0 by (3.9). In addition, we obtain that y+m2 > y+e > y+t > 0 > y+m1, y−m1 > y−t < y−m2. Then since y−t decreases with respect to x− e , we obtain µ8 < 2µ2+x+ e < µ2 directly by (3.8) and (3.18). Note that µ2 < −x+ e by (3.16). So by Theorem 3.4, we obtain µ8 < 2µ2 + x+ e < µ2 < −x+ e < 0 < max{2µ1 + x+ e , µ3} < µ4 < µ5. EJDE-2024/57 PLANAR SECTOR-WISE LINEAR SYSTEMS 15 By (3.8) and (3.18), it easy to see that µ8 < µ6 < µ7 and 2µ3+x+ e < µ7. However, the signs of µ7 − (−x+ e ) and µ2 − µ6 in this case are not sure by (4.3) and (4.4), respectively. And by easy computation, we obtain that 2µ2 + x+ e − µ6 = λ1 − λ2 λ1 − a11 x+ e + 2a12λ1 a11(λ1 − a11) y+e , which implies 2µ2 + x+ e < µ6 and 2µ2 + x+ e > µ6 both may be true. By Propositions 3.1, 3.2 and 3.3, and Remark 2.6, we see that there is no special separatrixes as x− e < µ8. As x− e = µ8, there exists a sliding heteroclinic orbit from pey to x+ e . In addition, it is obvious that a heteroclinic orbit from x− e to x+ e appears as x− e = µ6. Furthermore, as x− e > max{µ6, µ2}, the orbit of ⊕-system starting from y− m1 will arrive Σs. Thus there will exist a sliding heteroclinic orbit from x− e to pey if x− e < −x+ e , or from x− e to pex if x− e ≥ −x+ e until x− e = min{2µ1+x+ e , µ3}. In addition to this, when x− e < 2µ2 + x+ e , we have y−t > y+t > 0, then let y1 = (0, y1) and y2 = (0, y2) be the first arriving point on the positive y-axis of the forward orbits of ⊖-system starting from y+ t and pey, respectively. And when x− e < µ2, denote by y− = (0, y−) and y+ = (0, y+) the first arriving point on the positive y-axis of the forward and backward orbits of ⊖-system and ⊕-system with initial condition p0, respectively. Then when x− e < 2µ2 + x+ e < µ2, we must have y−m1 > y− > y1 > y2. Since y+ = (0, y+) and y+ m2 = (0, y+m2) will not change with the value of x− e , moreover, y+m2 > y+, so there must be ξ1 < ξ2, ξ1 < µ6, ξ2 < µ7 such that y− = y+m2, if x− e = ξ1 ; y− = y+, if x− e = ξ2 . This mean there exist a sliding homoclinic cycle containing x+ e called H1 and a sliding-zero cycle containing p0 called H2 as x− e = ξ1 and x− e = ξ2, respectively. Apart from this, one the one hand, if µ6 < 2µ2 + x+ e < µ2, there must be ξ11 < ξ12 < ξ1, such that y1 = y+m2, if x− e = ξ11; y2 = y+m2, if x− e = ξ12, which mean there both appears a sliding heteroclinc orbit H3 from pey to x+ e as x− e = ξ11 and x− e = ξ12, that is H3 and H1 will appear in turn before x− e = µ6; If 2µ2 + x+ e < µ6 < µ2, it is easy to see only H1 will appear before x− e = µ6; If 2µ2 + x+ e < µ2 < µ6, neither H3 nor H1 will appear before x− e = µ6 since p0 ∈ Σs. On the other hand, because of the sign of 2µ2 + x+ e − µ6 is not sure, so the size relationship of ξ2 with 2µ2 + x+ e is uncertain. Then there may exist some special separatrixes, which will be discussed as follows: If y− > y+ is true for x− e < 2µ2+x+ e , there must be a sliding cycle H4 containing p0 before y− = y+ since the orbit of ⊕-system starting from y− will arrive at Σx s . If y− > y+ is true for 2µ2 + x+ e < x− e < µ2, H4 must exist, meanwhile, there must exist a repulsive limit cycle H5 before x− e = ξ2. To prove this, we need constructing the Poincaré map as follows: P2 : DP2 7→ DP2 , P2(0, y0) = (0, y1), where DP2 = {(0, y) : y ∈ [y−t , y −]}, and (0, y1) is the first arriving point at DP2 of the forward orbit of system (1.1) starting from (0, y0). Then we must have P2((0, y +)) = (0, y−). And there must be (0, y∗1) and (0, y∗2), which satisfy 16 Q.-Q. HAN, S.-M. HUAN EJDE-2024/57 y+t < y∗2 < y∗1 < y+, such that P2(0, y ∗ 1) = (0, y∗2) since pey is atable. So there must exist y∗1 < y∗p < y+, such that P2(0, y ∗ p) = (0, y∗p). which implies the existence of the repulsive H5. Whether y− = y+ is true for x− e < 2µ2+x+ e or 2µ2+x+ e < x− e < µ2, it is obvious that there must arise a sliding-zero cycle H2. If y− < y+ is true for x− e < 2µ2 + x+ e , we obtain a attractive limit cycle H6 before x− e = 2µ2 + x+ e . To prove this, we need constructing the Poincaré map as follows: P3 : DP3 7→ DP3 , P3(0, y0) = (0, y1), where DP3 = {(0, y) : y ∈ [0, y+t ]}, and (0, y1) is the first arriving point at DP3 of the forward orbit of system (1.1) starting from (0, y0). Thus there must be (0, y⋄1) and (0, y⋄2), 0 < y⋄1 < y⋄2 < y+t , such that P3(0, 0) = (0, y⋄1), P3((0, y + t )) = (0, y⋄2), since y− < y+ and 0 < y+t < y−t , respectively. So there must be (0, y∗p), such that y⋄1 < y⋄p < y⋄2 P3(0, y ⋄ p) = (0, y⋄p). which implies the existence of the attractive H6. If y − < y+ is true for 2µ2 +x+ e < x− e < µ2, there must exist a sliding heteroclinic orbit H71 from x+ e to pey until x− e = −x+ e since Σy s is attractive. Then H71 turns into a sliding heteroclinic orbit H72 from x+ e to pex until x− e = min{2µ1 + x+ e , µ3}. Thus according to the above analysis, we obtain H4, H2 and H6 will appears in turn if y− < y+ is true for x− e < 2µ2+x+ e , which implies that there is a sliding cycle bifurcation as x− e = ξ2. In addition, H1 and H4 will appear in turn if y− > y+ is true for x− e < 2µ2+x+ e , which implies that there is a sliding homoclinic bifurcation as x− e = ξ1. The proof is complete. □ 5. Examples In this section, three examples are provided to illustrate some separatrixes in the theorems in Section 3. It should be noted that the red and the blue curves in the following figures represent the orbits of the ⊕-system and the ⊖-system, respectively. And the arrows represents the directions of the orbits on the forward time. Example 5.1. Let A = ( −4 2 3 1 ) , X− e = ( 5 −3 ) , X+ e = ( 2 3 ) . (5.1) By simple calculations, we have x+ t = 3, x+ m1 = 1, x+ m2 = 8, x− t = 4, x− m1 = 6, x− m2 = −1, 2µ1 + x+ e = 4 < µ3 = 7. Through numerical simulation, there exists an attractive limit cycle illustrated in Figure 2, which supported the conclusion about L2 in Theorem 3.4. EJDE-2024/57 PLANAR SECTOR-WISE LINEAR SYSTEMS 17 Figure 2. Existence of the attractive limit cycle of equation (5.1) Example 5.2. Let A = ( −1 2 3 4 ) , X− e = ( 9 −3 ) , X+ e = ( 2 3 ) . (5.2) By simple calculations, we have x+ t = 6, x+ m1 = 1, x+ m2 = 8, x− t = 5, x− m1 = 10, x− m2 = 3, 2µ1 + x+ e = 10 > µ3 = 7. Through numerical simulation, there exists a repulsive limit cycle illustrated in Figure 3, which supported the conclusion about L1 in Theorem 3.4. Figure 3. Existence of the repulsive limit cycle of equation (5.2) Example 5.3. Let A = ( −4 2 3 1 ) , X− e = ( µ −10 ) , X+ e = ( 4 10 ) . (5.3) 18 Q.-Q. HAN, S.-M. HUAN EJDE-2024/57 For −9 < µ < −6, we will show there exist a sliding homoclinic bifurcation and a sliding cycle bifurcation. By easy calculations, it follows that x+ t = 22/3, y+t = 2, y+m1 = −2, y+m2 = 12. In addition, we obtain the following statements: (1) As µ = −8.4, we have x− t = −176/15, y−t = 6.8, y−m1 = 15.2, y−m2 = −14.2. By numerical simulation, we have y− = y+m2, and there exists a sliding homoclinic cycle illustrated in Figure 4, which is in agreement with the orbit H1 in Theorem 3.6. Figure 4. Existence of the sliding homoclinic cycle of equation (5.3) with µ = −8.4 (2) As µ = −7, we have x− t = −31/3, y−t = 4, y−m1 = 11, y−m2 = −27/2. By numerical simulation, we have y− < y+m2, and there exists a sliding cycle illus- trated in Figure 5, which is in agreement with the orbit H4 in Theorem 3.6. (3) As µ = −6.2, we have x− t = −143/15, y−t = 2.4, y−m1 = 8.6, y−m2 = −13.1. By numerical simulation, we have y− = y+, and there exists a sliding-zero cycle illustrated in Figures 6 and 7, which is in agreement with the orbit H2 in Theorem 3.6. (4) As µ = −6.15, we have x− t = −123/20, y−t = 2.3, y−m1 = 8.45, y−m2 = −523/40. By numerical simulation, we have y− < y+, and there exists the attractive limit cycle illustrated in Figures 8 and 9, which is in agreement with the orbit H6 in Theorem 3.6. It follows that there is a sliding homoclinic bifurcation at µ = −8.4 and a sliding cycle bifurcation at µ = −6.2. EJDE-2024/57 PLANAR SECTOR-WISE LINEAR SYSTEMS 19 Figure 5. Existence of the sliding cycle of equation (5.3) with µ = −7 Figure 6. Existence of the sliding-zero cycle of equation (5.3) with µ = −6.2 6. Conclusions In this article, taking the perturbations to the separation boundary into consid- eration, we studied a special class of planar sector-wise linear systems, which are separated by two rays starting from the same point. More precisely, the two subsys- tems of this planar sector-wise linear systems are the same except for the positions of x± e . We mainly discussed the global qualitative dynamics of the system above with x± e being saddles, since the analysis of dynamics with x± e being sink or source points is similar. Because the global dynamics of piecewise smooth systems is so complex we put x+ e > 0, y+e = −y−e > 0 and only left x− e as a bifurcation param- eter. Based on our results of the existence of sliding set and pseudo-equilibrium points, and the properties of the unique non-regular point p0 = (0, 0), we obtained that there exist sliding cycle bifurcation and sliding homoclinic bifurcation in the above system. Moreover, we also got the existence of all important separatrixes 20 Q.-Q. HAN, S.-M. HUAN EJDE-2024/57 Figure 7. Expansion of the sliding-zero cycle in Figure 6 Figure 8. 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Chaos, 31 (2021): 2150136-1–17. Qian-Qian Han (corresponding author) School of Mathematics and Statistics, North China University of Water Resources and Electric Power, Zhengzhou, Henan 450046, China Email address: hanqianqian@ncwu.edu.cn Song-Mei Huan School of Mathematics and Statistics, Huazhong University of Science and Technol- ogy, Hubei Key Laboratory of Engineering Modeling and Scientific Computing, Wuhan, Hubei 430074, China Email address: smhuan@hust.edu.cn 1. Introduction 2. Preliminaries 3. Main results 4. Proofs of main results 5. Examples 6. Conclusions Acknowledgments References