Electronic Journal of Differential Equations, Vol. 2025 (2025), No. 42, pp. 1–19. ISSN: 1072-6691. URL: https://ejde.math.txstate.edu, https://ejde.math.unt.edu DOI: 10.58997/ejde.2025.42 CLASSIFICATION OF BOUNDARY-EQUILIBRIA FOR TWO-DIMENSIONAL CONTINUOUS PIECEWISE LINEAR SYSTEMS WITH TWO INTERSECTING SWITCHING LINES XIN YANG, JUELIANG ZHOU Abstract. In this article, we classify the boundary-equilibria of two-dimensional continuous piecewise linear systems with two intersecting switching lines. We present local phase portraits and indices of boundary-equilibria at the intersection point of two switching lines with more abundant dynamics. 1. Introduction The study of equilibria is very important in the researches and applications of dynamical sys- tems, serving as the foundation for local dynamic analysis [11, 23]. The topological structure of orbits near equilibria are rather complicated which can demonstrate local dynamics intuitively. Nonsmooth models induced by mechanics, electromagnetism and biology, are represented by mathematical formalisms, such as piecewise systems, switching systems, impulsive ordinary differ- ential equations [2, 4, 5, 17, 20]. Planar continuous piecewise linear systems are powerful tools to explain a series of natural phenomena, which is in fact part of reasons why they have been attract- ing the attention of an increasing number of scholars in recent decades [1, 6, 7, 8, 9, 10, 12, 13, 14, 15, 16, 19, 20, 21, 22]. Since continuous piecewise linear systems are Lipschitz continuous but not C1, methods of eigenvalues, continuous dependence of eigenvalues on the parameter, center manifold reduction and Taylor series expansion do not hold for these systems. Therefore, a deep study of continuous piecewise linear systems has important practical and theoretical significance. The classification and local phase portraits of boundary-equilibria are crucial for studying con- tinuous piecewise linear systems. Some results on boundary-equilibria of continuous piecewise linear systems can refer to [3, 7, 8, 10, 18]. Chen and his co-workers [7] investigated the clas- sification on boundary-equilibria and local phase portraits near boundary-equilibria of the two dimensional continuous piecewise linear system ẋ = { b1x+ a2y, if x ≤ 0, a1x+ a2y, if x > 0, ẏ = { b2x+ a4y, if x ≤ 0, a3x+ a4y, if x > 0, (1.1) which contains just one switching line x = 0 in the local regionD1 ⊂ R2, where (a1, a2, a3, a4, b1, b2) ∈ R6, (x, y) ∈ D1, a4b1 − a2b2 ̸= 0 and a1a4 − a2a3 ̸= 0. Since system (1.1) is a piecewise linear system, a1 = b1 and a3 = b2 cannot hold simultaneously. Moreover, following [7], we further study local phase portraits near O(0, 0) for two dimensional continuous piecewise linear systems with two intersecting switching lines x = 0 and y = 0. Without loss of generality, when equilibria lie at the intersection of two switching lines, any continuous piecewise linear systems in the local region 2020 Mathematics Subject Classification. 34A36, 34C05. Key words and phrases. Piecewise linear system; switching line; boundary-equilibrium; phase portrait; stability. ©2025. This work is licensed under a CC BY 4.0 license. Submitted November 22, 2024. Published April 21, 2025. 1 2 X. YANG, J. ZHOU EJDE-2025/42 D ⊂ R2 with two intersecting switching lines can be transformed to ẋ =  a1x+ a2y, if x > 0, y > 0, b1x+ a2y, if x ≤ 0, y > 0, b1x+ c1y, if x ≤ 0, y ≤ 0, a1x+ c1y, if x > 0, y ≤ 0, ẏ =  a3x+ a4y, if x > 0, y > 0, b2x+ a4y, if x ≤ 0, y > 0, b2x+ c2y, if x ≤ 0, y ≤ 0, a3x+ c2y, if x > 0, y ≤ 0, (1.2) where κ := (a1, a2, a3, a4, b1, b2, c1, c2) ∈ R8, a1a4 − a2a3 ̸= 0, b1a4 − a2b2 ̸= 0, b1c2 − b2c1 ̸= 0, a1c2 − c1a3 ̸= 0 and (x, y) ∈ D. Since system (1.2) is piecewise linear, a1 = b1 and a3 = b2 (resp. a2 = c1 and a4 = c2) cannot hold simultaneously. In addition, system (1.2) is equivalent to ẋ = ã1x+ ã2y + b̃1|x|+ b̃2|y| := f(x, y), ẏ = ã3x+ ã4y + b̃3|x|+ b̃4|y| := g(x, y), (1.3) where ã1 = (a1+ b1)/2, ã2 = (a2+c1)/2, ã3 = (a3+ b2)/2, ã4 = (a4+c2)/2, b̃1 = (a1− b1)/2, b̃2 = (a2 − c1)/2, b̃3 = (a3 − b2)/2 and b̃4 = (a4 − c2)/2. With the scaling (x, y, t) → (x/k1, y/k2, t/k3), system (1.3) can be changed into ẋ = ã1x/k3 + ã2k1y/(k2k3) + b̃1|x|/k3 + b̃2k1|y|/(k2k3), ẏ = ã3k2x/(k1k3) + ã4y/k3 + b̃3k2|x|/(k1k3) + b̃4|y|/k3, (1.4) where k1 > 0, k2 > 0 and k3 > 0 are constants. Then Jacobian matrices at O(0, 0) in four quadrants of system (1.4) are J̃ ′ i = ( ã1/k3 + b̃1 sgn(x)/k3 ã2k1/(k2k3) + b̃2k1 sgn(y)/(k2k3) ã3k2/(k1k3) + b̃3k2 sgn(x)/(k1k3) ã4/k3 + b̃4 sgn(y)/k3 ) (i = 1, 2, 3, 4). The rest of this paper is organized as follows. In Section 2, we present some notations, definitions and main results. In Section 3, we prove our main results. A brief conclusion is given in Section 4. 2. Main results In this section, we start with topological types of equilibria for the two dimensional linear system ẋ = ax+ by, ẏ = cx+ dy. (2.1) The Jacobian matrix at O(0, 0) of system (2.1) is J = ( a b c d ) , where (a, b, c, d) ∈ R4. It is easy to check that T := trJ = a+ d, D := det J = ad− bc and ∆ := T 2 − 4D = (a− d)2 + 4bc. For simplicity, we define the region D ⊂ R2 and separate D in four subregions Q1 := {(x, y) ∈ R2 : x ≥ 0, y ≥ 0}, Q2 := {(x, y) ∈ R2 : x < 0, y > 0}, Q3 := {(x, y) ∈ R2 : x ≤ 0, y ≤ 0, x2 + y2 > 0}, Q4 := {(x, y) ∈ R2 : x > 0, y < 0}. Then the Jacobian matrices at O(0, 0) of system (1.2) in Qi (i = 1, 2, 3, 4) are J1 = ( a1 a2 a3 a4 ) , J2 = ( b1 a2 b2 a4 ) , J3 = ( b1 c1 b2 c2 ) , J4 = ( a1 c1 a3 c2 ) . Let T1 := trJ1 = a1 + a4, D1 := detJ1 = a1a4 − a2a3, ∆1 := T 2 1 − 4D1, T2 := trJ2 = b1 + a4, D2 := detJ2 = b1a4 − a2b2, ∆2 := T 2 2 − 4D2, T3 := trJ3 = b1 + c2, D3 := detJ3 = b1c2 − c1b2, ∆3 := T 2 3 − 4D3, T4 := trJ4 = a1 + c2, D4 := detJ4 = a1c2 − c1a3 and ∆4 := T 2 4 − 4D4. To study the classification on boundary-equilibria of system (1.2), we shall introduce some notation. We say that O(0, 0) is a saddle in Q1 (resp. Q2, Q3, Q4) when D1 < 0 (resp. D2 < 0, D3 < 0, D4 < 0), denoted by S1 (resp. S2, S3, S4). O(0, 0) is a node in Q1 (resp. Q2, Q3, Q4) EJDE-2025/42 CLASSIFICATION OF BOUNDARY-EQUILIBRIA 3 when D1 > 0 and ∆1 ≥ 0 (resp. D2 > 0 and ∆2 ≥ 0, D3 > 0 and ∆3 ≥ 0, D4 > 0 and ∆4 ≥ 0), denoted by N1 (resp. N2, N3, N4). O(0, 0) is a center/focus in Q1 (resp. Q2, Q3, Q4) when ∆1 < 0 (resp. ∆2 < 0, ∆3 < 0, ∆4 < 0), denoted by M1 (resp. M2, M3, M4). Definition 2.1. We call O(0, 0) of system (1.2) the type of S1234 (resp. N1234, M1234) if O is a saddle (resp. node, center/focus) in Qi (i = 1, 2, 3, 4). For the type of N1234, we only consider four cases N+ 1234, N + 123N − 4 , N+ 13N − 24 and N+ 12N − 34, since the rest cases are topologically equivalent to these four cases. Here for N+ 123N − 4 , N+ 123 means that O is an unstable node in Q1, Q2 and Q3, and N− 4 means that O is a stable node in Q4. All others are understood in a similar way. Definition 2.2. We call O(0, 0) of system (1.2) the type of S-N if O is a saddle in some but not all Qi (i = 1, 2, 3, 4), and is a node in other Qj (j = 1, 2, 3, 4 and j ̸= i). We only consider eight cases S123N + 4 , S12N + 34, S12N + 3 N− 4 , S13N + 24, S13N + 2 N− 4 , S1N + 234, S1N + 23N − 4 and S1N + 24N − 3 , since the rest cases are topologically equivalent to these eight cases. Here for S12N + 3 N− 4 , S12 means that O is a saddle in Q1 and Q2; N + 3 means that O is an unstable node in Q3; and N− 4 means that O is a stable node in Q4. All others are understood in a similar way. Definition 2.3. We call O(0, 0) of system (1.2) the type of S-M if O is a saddle in some but not all Qi (i = 1, 2, 3, 4), and is a center/focus in other Qj (j = 1, 2, 3, 4 and j ̸= i). We only consider four cases S123M4, S12M34, S13M24 and S1M234, since the rest cases are topologically equivalent to these four cases. Here for S123M4, S123 means that O is a saddle in Q1, Q2 and Q3, and M4 means that O is center/focus in Q4. All others are understood in a similar way. Definition 2.4. We call O(0, 0) of system (1.2) the type of N -M if O is a node in some but not all Qi (i = 1, 2, 3, 4), and is a center/focus in other Qj (j = 1, 2, 3, 4 and j ̸= i). We only consider eight cases M1N + 234, M1N + 23N − 4 , M1N + 24N − 3 , M12N + 34, M12N + 3 N− 4 , M13N + 24, M13N + 2 N− 4 and M123N + 4 , since the rest cases are topologically equivalent to these eight cases. Here for M1N + 234, M1 means that O is a center/focus in Q1, and N+ 234 means that O is an unstable node in Q2, Q3 and Q4. All others are understood in a similar way. Definition 2.5. We call O(0, 0) of system (1.2) the type of S-N -M if O is a saddle in some but not all Qi (i = 1, 2, 3, 4), is a node in some Qj (j = 1, 2, 3, 4 and j ̸= i), and is a center/focus in other Qk (k = 1, 2, 3, 4, k ̸= i and k ̸= j). We only consider eight cases S12N + 3 M4, S13N + 2 M4, S3N + 12M4, S3N + 1 N− 2 M4, S2N + 13M4, S2N + 1 N− 3 M4, S1N + 2 M34 and S1N + 3 M24, since the rest cases are topologically equivalent to these eight cases. Here for S12N + 3 M4, S12 means that O is a saddle in Q1 and Q2, N + 3 means that O is an unstable node in Q3, and M4 means that O is a center/focus in Q4. All others are understood in a similar way. Lemma 2.6 ([23, Section 3.6]). The index of O(0, 0) of system (1.2) is IO := 1+(e−h)/2, where e is the number of elliptic sectors, and h is the number of hyperbolic sectors. Suppose that for any 0 < δ ≪ 1, there exists a neighborhood Sδ(O) of O(0, 0). We now present local phase portraits of system (1.2) in Sδ(O). All definitions similar to I0S1234 in the following theorems are given in Appendix A. Theorem 2.7. When O(0, 0) of system (1.2) is the type of S1234, local phase portraits have four hyperbolic sectors, where two separatrixes are stable manifolds and the other two separatrixes are unstable manifolds, see Figure 1. Theorem 2.8. When O(0, 0) of system (1.2) is the type of N1234 corresponding to four cases N+ 1234, N + 123N − 4 , N+ 13N − 24 and N+ 12N − 34, we have (a) N+ 1234 is a node (IO = 1) if κ ∈ I0 N+ 1234 , and the phase portrait consists of four parabolic sectors, see Figure 2(a). (b) N+ 123N − 4 is a node (IO = 1) if κ ∈ I0 N+ 123N − 4 , and the phase portrait consists of four parabolic sectors, see Figure 2(b). The phase portrait of N+ 123N − 4 consists of two hyperbolic sectors (IO = 0) (resp. one elliptical sector, one hyperbolic sector and two parabolic sectors 4 X. YANG, J. ZHOU EJDE-2025/42 (a) κ ∈ I0S1234 (b) κ ∈ I2S1234 (c) κ ∈ I2S1234 (d) κ ∈ I3S1234 (e) κ ∈ I4S1234 Figure 1. Local phase portraits of S1234 for system (1.2) in Sδ(O) (IO = 1); two elliptical sectors and two parabolic sectors (IO = 2)) if κ ∈ I ′ N+ 123N − 4 (resp. κ ∈ I1 N+ 123N − 4 , κ ∈ I2 N+ 123N − 4 ), see Figure 2(c) (resp. 2(d), 2(e)). (c) N+ 13N − 24 is a node (IO = 1) if κ ∈ I0 N+ 13N − 24 , and the phase portrait consists of four parabolic sectors, see Figure 2(f). N+ 13N − 24 is an unstable (resp. a stable) node when a2a3 < 0 and −a2(a1 − a4) > 0 (resp. < 0). (d) The phase portrait of N+ 12N − 34 consists of two hyperbolic sectors (IO = 0) (resp. one ellip- tical sector, one hyperbolic sector and one parabolic sector (IO = 1), two elliptical sectors and two parabolic sectors (IO = 2)) if κ ∈ I0 N+ 12N − 34 (resp. κ ∈ I1 N+ 12N − 34 , κ ∈ I2 N+ 12N − 34 ), see Figure 2(g) (resp. 2(h), 2(i). (a) κ ∈ I0 N+ 1234 (b) κ ∈ I0 N+ 123N − 4 (c) κ ∈ I ′ N+ 123N − 4 (d) κ ∈ I1 N+ 123N − 4 (e) κ ∈ I2 N+ 123N − 4 (f) κ ∈ I0 N+ 13N − 24 (g) κ ∈ I0 N+ 12N − 34 (h)κ ∈ I1 N+ 12N − 34 (i)κ ∈ I2 N+ 12N − 34 Figure 2. Local phase portraits of N1234 for system (1.2) in Sδ(O) Theorem 2.9. O(0, 0) of system (1.2) is monodromic if and only if (ã2 + b̃2)(ã2 − b̃2) ( ã3 + b̃3 + (−ã1 + ã4 − b̃1 + b̃4)u− (ã2 − b̃2)u 2 ) × ( ã3 + b̃3 + (−ã1 + ã4 − b̃1 − b̃4)u− (ã2 + b̃2)u 2 ) × ( ã3 − b̃3 + (−ã1 + ã4 + b̃1 − b̃4)u− (ã2 + b̃2)u 2 ) × ( ã3 − b̃3 + (−ã1 + ã4 + b̃1 + b̃4)u− (ã2 − b̃2)u 2 ) ̸= 0 (2.2) holds, where u = tan θ. Let w = ∫ 2π 0 f(cos θ, sin θ) cos θ + g(cos θ, sin θ) sin θ g(cos θ, sin θ) cos θ − f(cos θ, sin θ) sin θ dθ. EJDE-2025/42 CLASSIFICATION OF BOUNDARY-EQUILIBRIA 5 When (2.2) holds and a2 < 0 (resp. > 0), O is a center, a stable focus and an unstable focus if and only if w = 0, w < 0 (resp. > 0) and w > 0 (resp. < 0), respectively. Phase portraits of system (1.2) when O is monodromic are shown in Figure 3. (a) w = 0 and a2 < 0 (b) w < 0 and a2 < 0 (c) w > 0 and a2 < 0 (d) w = 0 and a2 > 0 (e) w > 0 and a2 > 0 (f) w < 0 and a2 > 0 Figure 3. Local phase portraits of monodromic for system (1.2) in Sδ(O) Note that when O(0, 0) is the type of M1234, it is monodromic. In addition, when O is a saddle or a node in one of Qi (i = 1, 2, 3, 4), it can also be monodromic, see Figures 5(e) and 6(o). Theorem 2.10. When O(0, 0) of system (1.2) is the type of S-N corresponding to eight types S123N + 4 , S12N + 34, S12N + 3 N− 4 , S13N + 24, S13N + 2 N− 4 , S1N + 234, S1N + 23N − 4 and S1N + 24N − 3 , we have (a) S123N + 4 is a saddle (IO = −1) if κ ∈ I0 S123N + 4 and the phase portrait consists of four hyperbolic sectors, see Figure 4(a). (b) The phase portrait of S12N + 34 (resp. S13N + 24) consists of two hyperbolic sectors and two parabolic sectors (IO = 0) if κ ∈ I0 S12N + 34 (resp. κ ∈ I0 S13N + 24 ), see Figure 4(b) (resp. 4(e)). (c) The phase portrait of S12N + 3 N− 4 consists of two hyperbolic sectors and two parabolic sectors (IO = 0) (resp. three hyperbolic sectors and one elliptical sector (IO = 0)) if κ ∈ I0 S12N + 3 N− 4 (resp. κ ∈ I1 S12N + 3 N− 4 ), see Figure 4(c) (resp. 4(d)). (d) The phase portrait of S13N + 2 N− 4 consists of two hyperbolic sectors and one parabolic sector (IO = 0) (resp. one elliptical sector, one hyperbolic sector and two parabolic sectors (IO = 1)) if κ ∈ I0 S13N + 2 N− 4 (resp. κ ∈ I1 S13N + 2 N− 4 ), see Figure 4(f) (resp. 4(g)). (e) S1N + 234 is a node (IO = 1) if κ ∈ I0 S1N + 234 , see Figure 4(h). (f) S1N + 23N − 4 is a node (IO = 1) if κ ∈ I0 S1N + 23N − 4 and the phase portrait consists of four hyperbolic sectors, see Figure 4(i). The phase portrait of S1N + 23N − 4 consists of one elliptical sector, one hyperbolic sector and two parabolic sectors (IO = 1) if κ ∈ I1 S1N + 23N − 4 , see Figure 4(j). (g) The phase portrait of S1N + 24N − 3 consists of two hyperbolic sectors (IO = 0) (resp. one elliptical sector, one hyperbolic sector and two parabolic sectors (IO = 1), two elliptical sectors and two hyperbolic sectors (IO = 1)) if κ ∈ I0 S1N + 24N − 3 (resp. κ ∈ I1 S1N + 24N − 3 , κ ∈ I2 S1N + 24N − 3 ), see Figure 4(k) (resp. 4(l), 4(m)). And when there are two elliptical sectors in the phase portrait of S1N + 24N − 3 , there must be two hyperbolic sectors. Theorem 2.11. When O(0, 0) of system (1.2) is the type of S-M corresponding to four types S123M4, S12M34, S13M24 and S1M234, we have 6 X. YANG, J. ZHOU EJDE-2025/42 (a) κ ∈ I0 S123N + 4 (b) κ ∈ I0 S12N + 34 (c) κ ∈ I0 S12N + 3 N− 4 (d) κ ∈ I1 S12N + 3 N− 4 (e) κ ∈ I0 S13N + 24 (f) κ ∈ I0 S13N + 2 N− 4 (g) κ ∈ I1 S13N + 2 N− 4 (h) κ ∈ I0 S1N + 234 (i) κ ∈ I0 S1N + 23N − 4 (j) κ ∈ I1 S1N + 23N − 4 (k) κ ∈ I0 S1N + 24N − 3 (l) κ ∈ I1 S1N + 24N − 3 (m) κ ∈ I2 S1N + 24N − 3 Figure 4. Local phase portraits of the type of S-N for system (1.2) in Sδ(O) (a) S123M4 is a saddle (IO = −1) if κ ∈ IS123M4 , see Figure 5(a). (b) The phase portrait of S12M34 (resp. S13M24, S1M234) consists of two hyperbolic sectors (IO = 0) if κ ∈ IS12M34 (resp. κ ∈ IS13M24 , κ ∈ IS1M234), see Figure 5(b) (resp. 5(c), 5(d)). (c) S1M234 is a center or focus (IO = 1) if κ ∈ I ′S1M234 , see Figure 5(e). (a) κ ∈ IS123M4 (b) κ ∈ IS12M34 (c) κ ∈ IS13M24 (d) κ ∈ IS1M234 (e) κ ∈ I ′S1M234 Figure 5. Local phase portraits of the type of S-M for system (1.2) in Sδ(O) Theorem 2.12. When O(0, 0) of system (1.2) is the type of N -M corresponding to eight types M1N + 234, M1N + 23N − 4 , M1N + 24N − 3 , M12N + 34, M12N + 3 N− 4 , M13N + 24, M13N + 2 N− 4 and M123N + 4 , we have (a) M1N + 234 (resp. M12N + 34, M13N + 24) is a node (IO = 1) if κ ∈ I0 M1N + 234 (resp. κ ∈ I0 M12N + 34 , κ ∈ I0 M13N + 24 ), see Figure 6(a) (resp. 6(g), 6(j)). EJDE-2025/42 CLASSIFICATION OF BOUNDARY-EQUILIBRIA 7 (b) The phase portrait of M1N + 23N − 4 consists of two hyperbolic sectors (IO = 0) (resp. one elliptical sector and one hyperbolic sector (IO = 1), two elliptical sectors and two parabolic sectors (IO = 2)) if κ ∈ I0 M1N + 24N − 3 (resp. κ ∈ I1 M1N + 24N − 3 , κ ∈ I2 M1N + 24N − 3 ), see Figure 6(b) (resp. 6(c), 6(d)). (c) The phase portrait of M1N + 24N − 3 (resp. M12N + 3 N− 4 ) consists of one elliptical sector, one hyperbolic sector and two parabolic sectors (IO = 1) if κ ∈ I1 M13N + 24 (resp. κ ∈ I1 M12N + 3 N− 4 ) and consists of four parabolic sectors (IO = 1) (resp. two parabolic sectors (IO = 1)) if κ ∈ I0 M13N + 24 (resp. κ ∈ I0 M12N + 3 N− 4 ), see Figures 6(f) and 6(e) (resp. 6(i) and 6(h). (d) The phase portrait of M13N + 2 N− 4 consists of two parabolic sectors (IO = 1) (resp. one elliptical sector, one hyperbolic sector and two parabolic sectors (IO = 1), two elliptical sectors (IO = 2)) if κ ∈ I0 M13N + 2 N− 4 (resp. κ ∈ I1 M13N + 2 N− 4 , κ ∈ I2 M13N + 2 N− 4 ), see Figure 6(k) (resp. 6(l), 6(m)). (e) M123N + 4 is a node (IO = 1) if κ ∈ I0 M123N + 4 (resp. κ ∈ I ′ M123N + 4 ) and the phase portrait consists of two parabolic sectors (resp. center/focus (IO = 1)), see Figure 6(n) (resp. 6(o)). (a) κ ∈ I0 M1N + 234 (b) κ ∈ I0 M1N + 23N − 4 (c) κ ∈ I1 M1N + 23N − 4 (d) κ ∈ I2 M1N + 23N − 4 (e) κ ∈ I0 M1N + 24N − 3 (f) κ ∈ I1 M1N + 24N − 3 (g) κ ∈ I0 M12N + 34 (h) κ ∈ I0 M12N + 3 N− 4 (i) κ ∈ I1 M12N + 3 N− 4 (j) κ ∈ I0 M13N + 24 (k) κ ∈ I0 M13N + 2 N− 4 (l) κ ∈ I1 M13N + 2 N− 4 (m) κ ∈ I2 M13N + 2 N− 4 (n) κ ∈ I0 M123N + 4 (o) κ ∈ I ′ M123N + 4 Figure 6. Local phase portraits of N -M for system (1.2) in Sδ(O) Theorem 2.13. When O(0, 0) of system (1.2) is the type of S-N -M corresponding to eight types S12N + 3 M4, S13N + 2 M4, S3N + 12M4, S3N + 1 N− 2 M4, S2N + 13M4, S2N + 1 N− 3 M4, S1N + 2 M34 and S1N + 3 M24, we have (a) The phase portrait of S3N + 12M4 (resp. S12N + 3 M4, S13N + 2 M4, S2N + 13M4) consists of two hyperbolic sectors and two parabolic sectors (IO = 0) (resp. two hyperbolic sectors and one parabolic sector (IO = 0)) if κ ∈ I0 S3N + 12M4 (resp. κ ∈ I0 S12N + 3 M4 , κ ∈ I0 S13N + 2 M4 , κ ∈ I0 S2N + 13M4 ), see Figure 7(c) (resp. 7(a), 7(b), 7(f)). 8 X. YANG, J. ZHOU EJDE-2025/42 (b) The phase portrait of S3N + 1 N− 2 M4 (resp. S2N + 1 N− 3 M4) consists of two hyperbolic sectors and two parabolic sectors (IO = 0) (resp. two hyperbolic sectors and one parabolic sector) and consists of one elliptical sector, one hyperbolic sector and two parabolic sectors (IO = 1) if κ ∈ I0 S3N + 1 N− 2 M4 and κ ∈ I1 S3N + 1 N− 2 M4 (resp. κ ∈ I0 S2N + 1 N− 3 M4 and κ ∈ I1 S2N + 1 N− 3 M4 ), see Figures 7(d) and 7(e) (resp. 7(g) and 7(h)). (c) S1N + 2 M34 (resp. S1N + 3 M24) is a node(IO = 1) if κ ∈ I0 S1N + 2 M34 (resp. κ ∈ I0 S1N + 3 M24 ) and the phase portrait consists of two parabolic sectors, see Figure 7(i) (resp. 7(j)). (a) κ ∈ I0 S12N + 3 M4 (b) κ ∈ I0 S13N + 2 M4 (c) κ ∈ I0 S3N + 12M4 (d) κ ∈ I0 S3N + 1 N− 2 M4 (e) κ ∈ I1 S3N + 1 N− 2 M4 (f) κ ∈ I0 S2N + 13M4 (g) κ ∈ I0 S2N + 1 N− 3 M4 (h) κ ∈ I1 S2N + 1 N− 3 M4 (i) κ ∈ I0 S1N + 2 M34 (j) κ ∈ I0 S1N + 3 M24 Figure 7. Local phase portraits of the type of S-N -M for system (1.2) in Sδ(O) For nodes, there are two eigenvalues λ1 and λ2. When |λ1| > |λ2| > 0, we call the characteristic direction corresponding to λ1 (resp. λ2) is the non-principal (resp. principal) direction. From phase portraits of node, we can obtain that orbits are tangent to the manifold whose direction is principal direction at the origin as t → −∞ or +∞. We provide the types of boundary-equilibria O(0, 0) of system (1.2), local phase portraits in Qδ(O), and corresponding indices in Table 1. 3. Proofs of main results It is clear that separatrixes of system (1.2) are orbits along the characteristic direction. We consider the eigenvectors of J̃i = ( ã1 + sgn(x)b̃1ã2 + sgn(y)b̃2 ã3 + sgn(x)b̃3ã4 + sgn(y)b̃4 ) , (3.1) where J̃i (i = 1, 2, 3, 4) are Jacobian matrices of system (1.3) in Qi (i = 1, 2, 3, 4). Take the characteristic direction to be ⟨x1, y1⟩T . From (3.1) it follows that( ã1 + sgn(x)b̃1 − λ ) x1 + ( ã2 + sgn(y)b̃2 ) y1 = 0, (3.2)( ã3 + sgn(x)b̃3 ) x1 + ( ã4 + sgn(y)b̃4 − λ ) y1 = 0, (3.3) where λ is the corresponding eigenvalue of ⟨x1, y1⟩T . By (3.2) and (3.3), we have( −ã3 − sgn(x)b̃3 ) x2 1 + ( ã2 + sgn(y)b̃2 ) y21 + ( ã1 + sgn(x)b̃1 − ã4 − sgn(y)b̃4 ) x1y1 = 0. (3.4) EJDE-2025/42 CLASSIFICATION OF BOUNDARY-EQUILIBRIA 9 Table 1. Type of O(0, 0) of system (1.2) Type of O Value of IO Values of (e, h) Local phase portraits in Qδ(O) S1234 -1 (0,4) Figure 1 N1234 0 (0,2) Figures 2(c, g) 1 (0,0) Figures 2(a-b, f) (1,1) Figures 2(d, h) 2 (2,0) Figures 2(e, i) monodromy 1 (0,0) Figures 3 S-N -1 (0,4) Figure 4(a) 0 (0,2) Figures 4(b-c, e-f, k) (1,3) Figure 4(d) 1 (1,1) Figures 4(g, j, l) (0,0) Figures 4(h-i) (2,2) Figure 4(m) S-M -1 (0,4) Figure 5(a) 0 (0,2) Figures 5(b-d) 1 (0,0) Figure 5(e) N -M 0 (0,2) Figure 6(b) 1 (0,0) Figures 6(a, e, g-h, j-k, n-o) (1,1) Figures 6(c, f, i, l) 2 (2,0) Figures 6(d, m) S-N -M 0 (0,2) Figures 7(a-d, f-g) 1 (0,0) Figures 7(i-j) (1,1) Figures 7(e, h) For simplicity, we use ⟨x, y⟩T to replace with ⟨x1, y1⟩T . In Qi (i = 1, 2, 3, 4), (3.4) can be rewritten as −a3x 2 + (a1 − a4)xy + a2y 2 = 0, (3.5) −b2x 2 + (b1 − a4)xy + a2y 2 = 0, (3.6) −b2x 2 + (b1 − c2)xy + c1y 2 = 0, (3.7) −a3x 2 + (a1 − c2)xy + c1y 2 = 0, (3.8) respectively. Then, the number of separatrixes is the number of solutions of (3.5)-(3.8) in Qi (i = 1, 2, 3, 4). Lemma 3.1. The number and the distribution of solutions of (3.5) in Q1 are given in Table 2. Proof. We consider three cases: a2 ̸= 0, a2 = 0 and a3 ̸= 0, and a2 = 0 and a3 = 0. Case 1. We consider (3.5) with a2 ̸= 0. Take x = 0 into (3.5), then (x, y) = (0, 0) which contradicts x2 + y2 > 0. This implies that (3.5) has no solutions when a2 ̸= 0 and x = 0, i.e., the y-axis. Therefore, when a2 ̸= 0 and x ̸= 0, equation (3.5) can be rewritten as a2 (y x )2 + (a1 − a4) y x − a3 = 0. (3.9) Since (x, y) is the characteristic direction of a separatrix, we define the slope of the separatrix as k = y/x. Clearly, (3.9) has two distinct solutions (resp. solution of multiplicity, no solu- tions) determined by ∆1 := (a1 − a4) 2 + 4a2a3 > 0 (resp. = 0, < 0). When O in Q1 is a saddle (resp. node) corresponding to ∆1 > 0 (resp. ∆1 ≥ 0), it implies that (3.5) has two (resp. at most two) solutions defined by (x1, y1) and (x2, y2). In addition, we let k11 = y1/x1 and k21 = y2/x2. Then, k11 + k21 = −(a1 − a4)/a2 and k11k 2 1 = −a3/a2, where k11 = (−(a1 − a4) +√ (a1 − a4)2 + 4a2a3)/(2a2) and k21 = (−(a1−a4)− √ (a1 − a4)2 + 4a2a3)/(2a2). Then we get the characteristic directions D1 1 and D2 1 corresponding to slopes of separatrixes k11 and k21 in Q1, where 10 X. YANG, J. ZHOU EJDE-2025/42 Table 2. The number and the distribution of solutions of (3.5) in Q1 Number of solutions Distribution of solutions Conditions two solutions ∆1 > 0 two solutions inQ1 except the x-axis (a) −a3/a2 > 0, −(a1 − a4)/a2 > 0 one solution in Q1 except the x-axis, the other one on the x-axis (b) −a3/a2 = 0, −(a1 − a4)/a2 > 0 one solution in Q1 except the x-axis, the other one outside Q1 (c) −a3/a2 < 0 one solution on the x-axis, the other one outside Q1 (d) −a3/a2 = 0, −(a1 − a4)/a2 < 0 two solutions outside Q1 (e) −a3/a2 > 0, −(a1 − a4)/a2 < 0 one solution in Q1 except the y-axis, the other one on the y-axis (f) −a2/a3 = 0, (a1 − a4)/a3 > 0 one solution on the y-axis, the other one outside Q1 (g) −a2/a3 = 0, (a1 − a4)/a3 < 0 one solution on the x-axis, the other one on the y-axis (h) a2 = 0, a3 = 0 one solution ∆1 = 0 the solution in Q1 except the x-axis (i) −a3/a2 > 0, −(a1 − a4)/a2 > 0 the solution on the x-axis (j) −a3/a2 = 0, −(a1 − a4)/a2 = 0 the solution outside Q1 (k) −a3/a2 > 0, −(a1 − a4)/a2 < 0 the solution on the y-axis (l) −a2/a3 = 0, (a1 − a4)/a3 = 0 D1 1 = ( −2a2, a1 − a4 − √ (a1 − a4)2 + 4a2a3 ) and D2 1 = ( −2a2, a1 − a4 + √ (a1 − a4)2 + 4a2a3 ) . When the separatrix with the characteristic direction D1 1 is located in Q1 except the x-axis (resp. on the x-axis, outside Q1), we obtain H1 1 := −2a2 ( a1 − a4 − √ (a1 − a4)2 + 4a2a3 ) > (resp. =, <) 0. When the separatrix with the characteristic direction D2 1 is located in Q1 except the x-axis (resp. on the x-axis, outside Q1), we obtain H2 1 := −2a2 ( a1 − a4 + √ (a1 − a4)2 + 4a2a3 ) > (resp. =, <) 0, and the eigenvalues corresponding to D1 1 and D2 1 are λ1 1 = a1 + a4 + √ (a1 − a4)2 + 4a2a3 2 and λ2 1 = a1 + a4 − √ (a1 − a4)2 + 4a2a3 2 . Next, we consider the distribution of solutions of (3.5) based on k11 and k21. When equation (3.5) has two distinct solutions, there are five subcases: two solutions are located in Q1 except the x-axis, i.e., k11k 2 1 > 0 and k11 + k21 > 0, one of which is located in Q1 except the x-axis, the other one is located on the x-axis, i.e., k11k 2 1 = 0 and k11 + k21 > 0, one of which is located in Q1 except the x-axis, the other one is located outside Q1, i.e., k 1 1k 2 1 < 0, one of which is located on the x-axis, the other one is located outside Q1, i.e., k 1 1k 2 1 = 0 and k11 + k21 < 0, and two solutions are located outside Q1, i.e., k 1 1k 2 1 > 0 and k11 + k21 < 0. These five subcases correspond to the following five conditions (a) in Table 2, (b) in Table 2, (c) in Table 2, (d) in Table 2, (e) in Table 2, respectively. When equation (3.5) has one repeating solution, there are three subcases: the solution is located EJDE-2025/42 CLASSIFICATION OF BOUNDARY-EQUILIBRIA 11 in Q1 except the x-axis, i.e., k11 = k21 > 0, the solution is located on the x-axis, i.e., k11 = k21 = 0, and the solution is located outside Q1, i.e., k 1 1 = k21 < 0. These three subcases correspond to the following three conditions (i) in Table 2, (j) in Table 2, (k) in Table 2, respectively. Case 2. We consider (3.5) with a2 = 0 and a3 ̸= 0. Similar to the proof of Case 1, equation (3.5) has no solutions when a3 ̸= 0 and y = 0, i.e., x-axis. Then (3.5) can be rewritten as −a3 (x y )2 + (a1 − a4) x y + a2 = 0. (3.10) Since (x, y) is the characteristic direction of a separatrix, we define the reciprocal of the slope of the separatrix as k̄ = x/y. Clearly, (3.10) has two distinct solutions (resp. solution of multiplic- ity, no solutions) as determined by ∆1 := (a1 − a4) 2 + 4a2a3 > 0 (resp. = 0, < 0). When O in Q1 is a saddle (resp. node) corresponding to ∆1 > 0 (resp. ∆1 ≥ 0), it implies that (3.5) has two (resp. at most two) solutions defined as (x1, y1) and (x2, y2). In addition, we let k̄11 = x1/y1 and k̄21 = x2/y2. Then, k̄11 + k̄21 = (a1 − a4)/a3 and k̄11 k̄ 2 1 = −a2/a3, where k̄11 = (−(a1−a4)+ √ (a1 − a4)2 + 4a2a3)/(−2a3) and k̄21 = (−(a1−a4)− √ (a1 − a4)2 + 4a2a3)/(−2a3). Then, we get characteristic directions D̄1 1 and D̄2 1 corresponding to reciprocals of slopes of sepa- ratrixes k̄11 and k̄21 in Q1, where D̄1 1 = ( a1 − a4 − √ (a1 − a4)2 + 4a2a3, 2a3 ) and D̄2 1 = ( a1 − a4 +√ (a1 − a4)2 + 4a2a3, 2a3 ) . When the separatrix with the characteristic direction D̄1 1 is located in Q1 except the y-axis (resp. on the y-axis, outside Q1), we obtain H̄1 1 := 2a3 ( a1 − a4 − √ (a1 − a4)2 + 4a2a3 ) > (resp. =, <) 0. When the separatrix with the characteristic direction D̄2 1 is located in Q1 except the y-axis (resp. on the y-axis, outside Q1), we obtain H̄2 1 := 2a3 ( a1 − a4 + √ (a1 − a4)2 + 4a2a3 ) > (resp. =, <) 0, and the eigenvalues corresponding to D̄1 1 and D̄2 1 are λ̄1 1 = a1 + a4 + √ (a1 − a4)2 + 4a2a3 2 and λ̄2 1 = a1 + a4 − √ (a1 − a4)2 + 4a2a3 2 are eigenvalues corresponding to D̄2 1 and D̄1 1. We further study the distribution of solutions of (3.5) based on k̄11 and k̄21. When equation (3.5) has two distinct solutions, there are two subcases: one of which is located in Q1 except the y-axis, the other one is located on the y-axis, i.e., k̄11 k̄ 2 1 = 0 and k̄11 + k̄21 > 0, one of which is located on the y-axis, the other one is located outside Q1, i.e., k̄ 1 1 k̄ 2 1 = 0 and k̄11 + k̄21 < 0. These two subcases correspond to the following two conditions (f) in Table 2 and (g) in Table 2, respectively. Since a2 = 0, k̄11 k̄ 2 1 = −a2/a3 = 0, implying that (3.5) has solution of multiplicity located on the y-axis, i.e., k̄11 = k̄21 = 0 corresponding to conditions (h) in Table 2. Case 3. When a2 = 0 and a3 = 0, equation (3.5) has two distinct solutions in Q1: one is located on the x-axis, and the other one is located on the y-axis corresponding to separatrixes with the characteristic directions (1, 0) and (0, 1), respectively. □ As discussed in Lemma 3.1, we can obtain the following results on the number and distribution of solutions of (3.6)-(3.8) in Q2-Q4, respectively. Lemma 3.2. The number and the distribution of solutions of (3.6) in Q2 are given in Table 3. The number and the distribution of solutions of (3.7) in Q3 are given in Table 4. The number and the distribution of solutions of (3.8) in Q4 are given in Table 5. Proof of Theorem 2.7. When ∆i > 0 (i = 1, 2, 3, 4), by a straightforward argument we can deduce that equations (3.5)-(3.8) have four solutions in Qi (i = 1, 2, 3, 4) which indicate that system (1.2) has four separatrixes in Qi (i = 1, 2, 3, 4), where two of separatrixes are stable manifolds and the other two are unstable manifolds. Thus the local phase portrait has exactly four hyperbolic sectors. It is easy to see that four separatrixes can be located on the axis when κ ∈ I0S1234 (resp. three separatrixes can be located on the axis when κ ∈ I1S1234 , two separatrixes can be located on 12 X. YANG, J. ZHOU EJDE-2025/42 Table 3. The number and the distribution of solutions of (3.6) in Q2 Number of solutions Distribution of solutions Conditions two solutions ∆2 > 0 two solutions in Q2 (a) −b2/a2 > 0, −(b1 − a4)/a2 < 0 one solution in Q2, the other one on the x-axis (b) −b2/a2 = 0, −(b1 − a4)/a2 < 0 one solution in Q2, the other one outside Q2 except the x-axis (c) −b2/a2 < 0 two solutions outside Q2 and one of which on the x-axis (d) −b2/a2 = 0, −(b1 − a4)/a2 > 0 two solutions outside Q2 except the x-axis (e) −b2/a2 > 0, −(b1 − a4)/a2 > 0 one solution in Q2, the other one on the y-axis (f) −a2/b2 = 0, (b1 − a4)/b2 < 0 two solutions outside Q2 and one of which on the y-axis (g) −a2/b2 = 0, (b1 − a4)/b2 > 0 one solution on the x-axis and the other one on the y-axis (h) a2 = 0, b2 = 0 one solution ∆2 = 0 the solution in Q2 (i) −b2/a2 > 0, −(b1 − a4)/a2 < 0 the solution on the x-axis (j) −b2/a2 = 0, −(b1 − a4)/a2 = 0 the solution outside Q2 except the x-axis (k) −b2/a2 > 0, −(b1 − a4)/a2 > 0 the solution on the y-axis (l) −a2/b2 = 0, (b1 − a4)/b2 = 0 the axis when κ ∈ I2S1234 , one separatrix can be located on the axis when κ ∈ I3S1234 , and none of four separatrixes can be located on the axis when κ ∈ I4S1234 ), as shown in Figure 1(a) (resp. 1(b), 1(c), 1(d), 1(e)). □ Proof of Theorem 2.8. (a) For N+ 1234, the phase portrait can only have unstable manifolds. To prove that N+ 1234 is not monodromic, we proceed by way of contradiction. If N+ 1234 is monodromic, then phase portraits in Qi (i = 1, 2, 3, 4) do not have an unstable manifold. So (3.5)-(3.8) have no solutions in the corresponding Qi (i = 1, 2, 3, 4), which implies that (e) in Table 2, (e) in Table 3, (e) in Table 4 and (e) in Table 5 hold. When a2 > 0, on the one hand, from (e) in Table 2, (e) in Table 3 and (e) in Table 4, we know that a1 > a4 > b1 > c2 (i.e., a1 > c2). On the other hand, from (e) in Table 5, we get a1 < c2, which is a contradiction. Similarly, when a2 < 0, from (e) in Table 2, (e) in Table 3 and (e) in Table 4, we know a1 < a4 < b1 < c2 (i.e., a1 < c2). But from (e) in Table 5, we get a1 > c2. This yields another contradiction. Therefore, N + 1234 is an unstable node. When κ ∈ I0 N+ 1234 , we can obtain that (h) in Table 2, (h) in Table 3, (h) in Table 4 and (h) in Table 5 hold, which implies that the associated phase portrait has four unstable manifolds. Therefore, the phase portrait consists of four parabolic sectors, see Figure 2(a). Because there may be an elliptical sector between two manifolds in the principal directions of N+ and N−, respectively, we shall consider the number of elliptic sectors for types of N+ 123N − 4 , N+ 13N − 24 and N+ 12N − 34. (b) When κ ∈ I0 N+ 123N − 4 , we can obtain that (f) in Table 2, (g) in Table 3, (a) in Table 4 and (e) in Table 5 hold, which implies that the associated phase portrait has four unstable manifolds. Therefore, the phase portrait consists of four parabolic sectors, see Figure 2(b). By the same EJDE-2025/42 CLASSIFICATION OF BOUNDARY-EQUILIBRIA 13 Table 4. The number and the distribution of solutions of (3.7) in Q3 Number of solutions Distribution of solutions Conditions two solutions ∆3 > 0 two solutions inQ1 except the x-axis (a) −b2/c1 > 0, −(b1 − c2)/c1 > 0 one solution in Q1 except the x-axis, the other one on the x-axis (b) −b2/c1 = 0, −(b1 − c2)/c1 > 0 one solution in Q1 except the x-axis, the other one outside Q1 (c) −b2/c1 < 0 one solution on the x-axis, the other one outside Q1 (d) −b2/c1 = 0, −(b1 − c2)/c1 < 0 two solutions outside Q1 (e) −b2/c1 > 0, −(b1 − c2)/c1 < 0 one solution in Q1 except the y-axis, the other one on the y-axis (f) −c1/b2 = 0, (b1 − c2)/b2 > 0 one solution on the y-axis, the other one outside Q1 (g) −c1/b2 = 0, (b1 − c2)/b2 < 0 one solution on the x-axis, the other one on the y-axis (h) c1 = 0, b2 = 0 one solution ∆3 = 0 the solution in Q1 except the x-axis (i) −b2/c1 > 0, −(b1 − c2)/c1 > 0 the solution on the x-axis (j) −b2/c1 = 0, −(b1 − c2)/c1 = 0 the solution outside Q1 (k) −b2/c1 > 0, −(b1 − c2)/c1 < 0 the solution on the y-axis (l) −c1/b2 = 0, (b1 − c2)/b2 = 0 way, we can deduce that the phase portrait of N+ 123N − 4 consists of two hyperbolic sectors when κ ∈ I ′ N+ 123N − 4 , see Figure 2(c). We now prove that when κ ∈ I1 N+ 123N − 4 , O is the type of N+ 123N − 4 and the phase portrait consists of one elliptical sector, one hyperbolic sector and two parabolic sectors as shown in Figure 2(d). When κ ∈ I1 N+ 123N − 4 , we can obtain that (f) in Table 2, (g) in Table 3, (e) in Table 4 and (a) in Table 5 hold, which implies that the associated phase portrait has two unstable manifolds and two stable manifolds. By the condition of κ1 := (a1, a2, a3, a4) ∈ I1 N+ 123N − 4 , we know a2 = 0 > a3 and a4 > a1 > 0, implying that O in Q1 is an unstable bidirectional node, and the characteristic direction D̄1 1 (resp. D̄2 1) is principal (resp. non-principal). Except for the orbits along directions D̄1 1 and D̄2 1, the rest of the orbits are tangent to the unstable manifold in the direction of D̄1 1 at the origin as t → −∞. By the condition of κ2 := (b1, a2, b2, a4) ∈ I1 N+ 123N − 4 , we know a2 = 0, a4 > b1 > 0 and b2 < 0, implying that O in Q2 is an unstable bidirectional node and the characteristic direction D̄1 2 (resp. D̄2 2) is principal (resp. non-principal), where D̄1 2 = ( b1 − a4 − √ (b1 − a4)2 + 4a2b2, 2b2 ) and D̄2 2 = ( b1−a4+ √ (b1 − a4)2 + 4a2b2, 2b2 ) . Except for the orbits along directions D̄1 2 and D̄2 2, the rest of the orbits are tangent to the unstable manifold in the direction of D̄1 2 at the origin as t → −∞. By the condition of κ3 := (b1, c1, b2, c2) ∈ I1 N+ 123N − 4 , we know b1 > −c2 > 0 and b2 < 0 < c1, implying that O in Q3 is an unstable bidirectional node. Since −c1(b1 − c2) < 0 and c1b2 < 0, the unstable manifolds with the principal direction and the non-principal direction do not pass through Q3. And for b2 < 0, we get ẏ > 0 when y = 0 and x < 0. It means that orbits in Q3 rotate clockwise. By the condition of κ4 := (a1, c1, a3, c2) ∈ I1 N+ 123N − 4 , we know 0 > −a1 > c2, c1 > 0 and a3 < 0. It follows that O in Q4 is a stable bidirectional node, the characteristic direction of D1 4 14 X. YANG, J. ZHOU EJDE-2025/42 Table 5. The number and the distribution of solutions of (3.8) in Q4 Number of solutions Distribution of solutions Conditions two solutions ∆4 > 0 two solutions in Q2 (a) −a3/c1 > 0, −(a1 − c2)/c1 < 0 one solution in Q2, the other one on the x-axis (b) −a3/c1 = 0, −(a1 − c2)/c1 < 0 one solution in Q2, the other one outside Q2 except the x-axis (c) −a3/c1 < 0 two solutions outside Q2 and one of which on the x-axis (d) −a3/c1 = 0, −(a1 − c2)/c1 > 0 two solutions outside Q2 except the x-axis (e) −a3/c1 > 0, −(a1 − c2)/c1 > 0 one solution in Q2, the other one on the y-axis (f) −c1/a3 = 0, (a1 − c2)/a3 < 0 two solutions outside Q2 and one of which on the y-axis (g) −c1/a3 = 0, (a1 − c2)/a3 > 0 one solution on the x-axis and the other one on the y-axis (h) c1 = 0, a3 = 0 one solution ∆4 = 0 the solution in Q2 (i) −a3/c1 > 0, −(a1 − c2)/c1 < 0 the solution on the x-axis (j) −a3/c1 = 0, −(a1 − c2)/c1 = 0 the solution outside Q2 except the x-axis (k) −a3/c1 > 0, −(a1 − c2)/c1 > 0 the solution on the y-axis (l) −c1/a3 = 0, (a1 − c2)/a3 = 0 (resp. D2 4) is principal (resp. non-principal), where D1 4 = ( − 2c1, a1 − c2 − √ (a1 − c2)2 + 4c1a3 ) and D2 4 = ( − 2c1, a1 − c2 + √ (a1 − c2)2 + 4c1a3 ) . Except for the orbits along the directions of D1 4 and D2 4, the rest of the orbits are tangent to the stable manifold in the direction of D1 4 at the origin as t → +∞. To sum up, there is an elliptic sector in which orbits start from Q1 and are tangent to the direction D̄1 1 at the origin as t → −∞, then enter Q4, and are tangent to the origin with the direction of D1 4 as t → +∞. There is a hyperbolic sector in which orbits start from Q4, then cross Q3 and enter Q2, and get away from the origin as |t| → +∞. There are two parabolic sectors in which orbits are tangent to the directions of D̄1 1 and D1 4 at the origin in Q1 and Q4 respectively as t → −∞. The phase portrait is shown in Figure 2(d). By an analogous way, we can deduce that the phase portrait of N+ 123N − 4 consists of two elliptical sectors and two parabolic sectors, see Figure 2(e). Proceeding in a similarly manner, we arrive at the desired results (c) and (d). □ Proof of Theorem 2.9. To prove that the origin O(0, 0) of system (1.2) is monodromic if and only if G(θ) ̸≡ 0 with G(θ) := g(cos θ, sin θ) cos θ − f(cos θ, sin θ) sin θ, we use polar coordinates (x, y) = (r cos θ, r sin θ). Then system (1.3) reduces to ṙ = r (f(cos θ, sin θ) cos θ + g(cos θ, sin θ) sin θ) := rA(θ), θ̇ = g(cos θ, sin θ) cos θ − f(cos θ, sin θ) sin θ = G(θ), which leads to 1 r dr dθ = A(θ) G(θ) . EJDE-2025/42 CLASSIFICATION OF BOUNDARY-EQUILIBRIA 15 Then we have ∫ r r1 dr r = ∫ θ θ1 A(θ) G(θ) dθ, implying that | ln r − ln r1| ≤ M |θ − θ1| < +∞, where M = max0≤θ≤2π |A(θ)|/min0≤θ≤2π |G(θ)|. Hence, r and r1 cannot independently tend to 0. Thus O is monodromic. To show that G(θ) ̸≡ 0 if and only if condition (2.2) holds, we note that G(θ) =  −f(0, 1), if cos θ = 0, sin θ = 1, f(0,−1), if cos θ = 0, sin θ = −1, cos2 θ (g(1, tan θ)− tan θf(1, tan θ)) , if cos θ > 0, cos2 θ (−g(−1,− tan θ) + tan θf(−1,− tan θ)) , if cos θ < 0, and G(θ) ̸≡ 0 is equivalent to (i) f(0, 1) = ã2 + b̃2 ̸= 0, f(0,−1) = −ã2 + b̃2 ̸= 0, and (ii) g(1, tan θ)− tan θf(1, tan θ) ̸= 0 and g(−1,− tan θ)− tan θf(−1,− tan θ) ̸= 0 when cos θ ̸= 0. By substituting u = tan θ, (ii) can be formulated as ã3 + b̃3 + (−ã1 + ã4 − b̃1 + b̃4)u− (ã2 − b̃2)u 2 ̸= 0, ã3 + b̃3 + (−ã1 + ã4 − b̃1 − b̃4)u− (ã2 + b̃2)u 2 ̸= 0, ã3 − b̃3 + (−ã1 + ã4 + b̃1 − b̃4)u− (ã2 + b̃2)u 2 ̸= 0, ã3 − b̃3 + (−ã1 + ã4 + b̃1 + b̃4)u− (ã2 − b̃2)u 2 ̸= 0. (3.11) Combining (i) and (3.11) leads to condition (2.2). From G(θ) ̸≡ 0, it follows that O of system (1.2) is monodromic. Then w = ln r(2π)− ln r(0) = ∫ 2π 0 h(cos θ, sin θ)dθ = ∫ 2π 0 f(cos θ, sin θ) cos θ + g(cos θ, sin θ) sin θ g(cos θ, sin θ) cos θ − f(cos θ, sin θ) sin θ dθ, where r(0) = r0 > 0. Additionally, for a2 < 0, the orbits near O of system (1.2) rotate anticlock- wise. When a < 0 and O of system (1.2) is a center (resp. a stable focus, an unstable focus), we observe that r(2π)− r0 = 0 (resp. < 0, > 0) holds for sufficiently small r0 > 0, indicating that O is a center (resp., a stable focus, an unstable focus). Similarly, when a < 0 and O of system (1.2) is a center (resp. a stable focus, an unstable focus), we have r(2π) − r0 = 0 (resp. > 0, < 0) for sufficiently small r0 > 0, indicating that O is a center (resp. a stable focus, an unstable focus). □ Proof of Theorem 2.10 (a), (b) and (e). For S123N + 4 , S12N + 34, S13N + 24 and S1N + 234, there are no elliptical sectors in their phase portraits. The number of hyperbolic sectors is analyzed in specific cases. Using the same method as in (b) of Theorem 2.8, we can prove that (a), (b) andf (e) hold. (c) For S12N + 3 N− 4 , the manifolds in the stable principal direction and unstable principal direction may exist simultaneously. Therefore, we consider the number of elliptical sectors in this case. We claim that there is at most one elliptical sector. Using the same method as in the proof of Theorem 2.8, we can obtain (c) immediately. To prove that it is impossible for the phase portrait of S12N + 3 N− 4 to have two elliptical sectors, we suppose that there are two elliptical sectors in the phase portrait of S12N + 3 N− 4 . Then the manifolds with the stable principal direction and unstable principal direction are in Q4 and Q3 respectively, i.e., equations (3.8) and (3.7) have solutions in Q4 and Q3 respectively, implying that (c) in Table 4, H2 3 > 0, (c) in Table 5 and H1 4 < 0, where H2 3 := −2c1(b1 − c2 + √ (b1 − c2)2 + 4c1b2) and H1 4 := −2c1(a1 − c2 − √ (a1 − c2)2 + 4c1a3). On the one hand, from (c) in Table 4 and H2 3 > 0, we can obtain c1 < 0 and b2 < 0. Then by T3 > 0 and D3 > 0, we can get b1 > 0 and c2 > 0. On the other hand, from (c) in Table 5 and H1 4 < 0, we can obtain c1 < 0 and a3 < 0. Then by T4 < 0 and D4 > 0, we can obtain a1 < 0 and c2 < 0. This is a contradiction. 16 X. YANG, J. ZHOU EJDE-2025/42 By an analogous argument, we can obtain (d), (f) and (g). □ The proofs of Theorems 2.11-2.13 are closely similar, to avoid unnecessary repetition, so we omit them. 4. Conclusions Chen et al. [7] presented the classification and local phase portraits of the boundary-equilibria of a two dimensional continuous piecewise linear system with a switching line. Based on their results, we considered a piecewise linear system with two intersecting switching lines. Since two intersecting switching lines divide Sδ(O) into four parts and system (1.2) has eight parameters, the classification becomes more complicated, which also greatly increases the amount of computations. This paper gives a complete classification of boundary-equilibria of system (1.2) such as S1234, N1234, M1234, S-N , S-M , N -M and S-N -M . In the further study, we will explore boundary- equilibria index of two dimensional continuous piecewise linear systems with more intersecting switching lines and boundary-equilibria index of n-dimensional continuous piecewise linear sys- tems. Appendix A For convenience, we compile a list of symbols. IS1234 := {κ ∈ R8 : Di < 0(i = 1, 2, 3, 4)}, I0S1234 := IS1234 ∩ { κ ∈ R8 : a1 > 0, a2 = a3 = b2 = c1 = 0, a4 < 0, b1 > 0, c2 < 0 } , I1S1234 := IS1234 ∩ { κ ∈ R8 : a1 > 0, a2 = b2 = c1 = 0, a3 > 0, a4 < 0, b1 > 0, c2 < 0 } , I2S1234 := IS1234 ∩ { κ ∈ R8 : a1 < 0, a2 = b2 = 0, 0 < a3 < a1c2/c1, a4 > 0, b1 < 0, c1 < 0, c2 > 0 } , I3S1234 := IS1234 ∩ { κ ∈ R8 : a1 < 0, a2 = 0, 0 < a3 < a1c2/c1, a4 > 0, b1 < 0, b2 < 0, c1 < 0, c2 > 0 } , I4S1234 := IS1234 ∩ {κ ∈ R8 : a1 = b1 = c2 = 0, a2 > 0, a3 > 0, a4 > 0, b2 > 0}, IN+ 1234 := {κ ∈ R8 : Ti > 0, Di > 0,∆i ≥ 0(i = 1, 2, 3, 4)}, I0 N+ 1234 := IN+ 1234 ∩ { κ ∈ R8 : a1 > a4 > 0, a1 > c2 > 0, a2 = a3 = b2 = c1 = 0, b1 > a4, b1 > c2 } , IN+ 123N − 4 := {κ ∈ R8 : Ti > 0(i = 1, 2, 3), Dj > 0,∆j ≥ 0(j = 1, 2, 3, 4), T4 < 0}, I0 N+ 123N − 4 := IN+ 123N − 4 ∩ { κ ∈ R8 : a3 > 0, b1 > −c2 > a1 > a4 > 0, b1c2/c1 < b2, a2 = 0, c1 < 0 } , I ′ N+ 123N − 4 := IN+ 123N − 4 ∩ { κ ∈ R8 : a1 < 0, a2 < 0, a3 > 0, a4 > 0, b1 > 0, b2 < 0, c1 > 0, c2 < 0 } , I1 N+ 123N − 4 := IN+ 123N − 4 ∩ { κ ∈ R8 : a2 = 0, a3 < 0, a4 > b1 > −c2 > a1 > 0, b2 < 0, c1 > 0 } , I2 N+ 123N − 4 := IN+ 123N − 4 ∩ { κ ∈ R8 : ∆1 = ∆3 = 0, a2 > 0, a3 < 0, b2 > 0, c1 < 0, a4 > −a1 > 0, b1 > −c2 > 0 } , IN+ 13N − 24 := { κ ∈ R8 : Ti > 0, Di > 0,∆i ≥ 0(i = 1, 3), Tj < 0, Dj > 0,∆j ≥ 0 (j = 2, 4) } , I0 N+ 13N − 24 := IN+ 13N − 24 ∩ { κ ∈ R8 : a1 < 0, a2 > 0, a3 < 0,−c2 < b1 < −a4 < 0, b2 < 0, c1 > 0 } , IN+ 12N − 34 := { κ ∈ R8 : Ti > 0, Di > 0,∆i ≥ 0(i = 1, 2), Tj < 0, Dj > 0,∆j ≥ 0 (j = 3, 4) } , I0 N+ 12N − 34 := IN+ 12N − 34 ∩ { κ ∈ R8 : a1 < 0, a2 < 0, a3 > 0, a4 > 0, b1 > 0, b2 < 0, c1 > 0, c2 < 0 } , I1 N+ 12N − 34 := IN+ 12N − 34 ∩ { κ ∈ R8 : a2 = 0, a1c2/c1 < a3,−c2 > a1 > a4, c1 < 0,∆3 = 0,−c2 > b1 > a4 > 0 } , I2 N+ 12N − 34 := IN+ 12N − 34 ∩ { κ ∈ R8 : a1 < 0, a2 > 0, a3 < 0, a4 > 0, b1 > 0, b2 > 0, c1 < 0, c2 < 0 } , IS123N + 4 := {κ ∈ R8 : Di < 0(i = 1, 2, 3), T4 > 0, D4 > 0,∆4 ≥ 0}, I0 S123N + 4 := IS123N + 4 ∩ { κ ∈ R8 : a1 > 0, a2 < 0, a3 < 0, a4 = 0, b1 = 0, b2 < 0, c1 < 0, c2 > 0 } , EJDE-2025/42 CLASSIFICATION OF BOUNDARY-EQUILIBRIA 17 IS12N + 34 := {κ ∈ R8 : Di < 0(i = 1, 2), Tj > 0, Dj > 0,∆j ≥ 0(j = 3, 4)}, I0 S12N + 34 := IS12N + 34 ∩ { κ ∈ R8 : a2 = 0,∆4 = 0, c1 < 0,∆3 = 0, c2 > a1 > 0, a4 < 0, b1 > c2 > 0 } , IS12N + 3 N− 4 := { κ ∈ R8 : Di < 0(i = 1, 2), T3 > 0, D3 > 0,∆3 ≥ 0, T4 < 0, D4 > 0,∆4 ≥ 0 } , I0 S12N + 3 N− 4 := IS12N + 3 N− 4 ∩ { κ ∈ R8 : a2 = 0, a4 < 0, c1 < 0, b1c2/c1 < b2,∆4 = 0, b1 > −c2 > a1 > 0 } , I ′ S12N + 3 N− 4 := { κ ∈ R8 : Di < 0(i = 2, 3), T1 < 0, D1 > 0,∆1 ≥ 0, T4 > 0, D4 > 0,∆4 ≥ 0 } , I1 S12N + 3 N− 4 := I ′ S12N + 3 N− 4 ∩ { κ ∈ R8 : a2 = 0,∆4 = 0, c1 < 0, b2 < b1c2/c1, b1 > 0,−c2 < a1 < a4 < 0 } , IS13N + 24 := {κ ∈ R8 : Di < 0(i = 1, 3), Tj > 0, Dj > 0,∆j ≥ 0(j = 2, 4)}, I0 S13N + 24 := IS13N + 24 ∩ { κ ∈ R8 : a2 = 0, b1 > a4 > 0, b2 < b1c2/c1, c1 < 0,−c2 < a1 < 0,∆4 = 0 } , IS13N + 2 N− 4 := { κ ∈ R8 : Di < 0(i = 1, 3), T2 > 0, D2 > 0,∆2 ≥ 0, T4 < 0, D4 > 0,∆4 ≥ 0 } , I0 S13N + 2 N− 4 := IS13N + 2 N− 4 ∩ { κ ∈ R8 : a1 < 0, a2 = 0, a3 > 0, b1 > a4 > 0, c1 > 0, b2 = 0, c2 < 0 } , I1 S13N + 2 N− 4 := IS13N + 2 N− 4 ∩ { κ ∈ R8 : a2 > 0, a3 < 0, b1 > −a4 > 0, b2 < 0, c1 > 0, c2 < −a1 < 0 } , IS1N + 234 := {κ ∈ R8 : D1 < 0, Tj > 0, Dj > 0,∆j ≥ 0(j = 2, 3, 4)}, I0 S1N + 234 := IS1N + 234 ∩ { κ ∈ R8 : a2 = 0, b1 > a4 > 0, c1 < 0, b1 > c2 > −a1 > 0,∆3 = 0,∆4 = 0 } , IS1N + 23N − 4 := { κ ∈ R8 : D1 < 0, Tj > 0, Dj > 0,∆j ≥ 0(j = 2, 3), T4 < 0, D4 > 0,∆4 ≥ 0 } , I0 S1N + 23N − 4 := IS1N + 23N − 4 ∩ { κ ∈ R8 : a2 = 0, a4 > b1 > −c2 > −a1 > 0, c1 < 0,∆3 = 0,∆4 = 0 } , I1 S1N + 23N − 4 := IS1N + 23N − 4 ∩ { κ ∈ R8 : a2 = 0, a4 > b1 > c2 > 0, c1 < 0,∆3 = 0, a1 < −c2 < 0,∆4 = 0 } , IS1N + 24N − 3 := { κ ∈ R8 : D1 < 0, Tj > 0, Dj > 0,∆j ≥ 0(j = 2, 4), T3 < 0, D3 > 0,∆3 ≥ 0 } , I0 S1N + 24N − 3 := IS1N + 24N − 3 ∩ { κ ∈ R8 : a1 > 0, a2 > 0, a3 > 0, a4 > 0, b1 < 0, b2 < 0, c1 < 0, c2 < 0 } , I1 S1N + 24N − 3 := IS1N + 24N − 3 ∩ { κ ∈ R8 : a1 > 0, a2 > 0, a3 > 0, a4 > 0, b1 < 0, b2 < 0, c1 > 0, c2 > 0 } , I2 S1N + 24N − 3 := IS1N + 24N − 3 ∩ { κ ∈ R8 : a1 > 0, a2 < 0, a3 < 0, a4 > 0, b1 < 0, b2 > 0, c1 > 0, c2 < 0 } , IS123M4 := { κ ∈ R8 : a1 > 0, a2 = 0, a4 < 0, 0 < b2 < b1c2/c1, b1 > 0, c1 < 0, c2 < 0,∆4 < 0 } IS12M34 := { κ ∈ R8 : a1 > 0, a2 = 0, a4 < 0, b1 > 0, c1 < 0,∆3 < 0,∆4 < 0 } , IS13M24 := { κ ∈ R8 : a2 > 0, a3 > 0, D1 < 0, b2 < 0, c1 < 0, D3 < 0,∆3 < 0,∆4 < 0 } , IS1M234 := { κ ∈ R8 : a1 > 0, a2 < 0, a3 > 0, a4 < 0, D1 < 0,∆2 < 0,∆3 < 0,∆4 < 0 } , I ′S1M234 := { κ ∈ R8 : a1 > 0, a2 > 0, a1a4/a2 < a3 < 0, a4 < 0, b1 > 0, b2 < 0, c1 > 0, c2 < 0, D1 < 0,∆i < 0(i = 2, 3, 4) } , IM1N + 234 := {κ ∈ R8 : ∆1 < 0, Tj > 0, Dj > 0,∆j ≥ 0(j = 2, 3, 4)}, I0 M1N + 234 := IM1N + 234 ∩ { κ ∈ R8 : a1 > 0, a2 < 0, a3 > 0, a4 = 0, b1 > 0, b2 > 0, c1 < 0, c2 < 0 } , IM1N + 23N − 4 := { κ ∈ R8 : ∆1 < 0, Tj > 0, Dj > 0,∆j ≥ 0(j = 2, 3), T4 < 0, D4 > 0,∆4 ≥ 0 } , I0 M1N + 23N − 4 := IM1N + 23N − 4 ∩ { κ ∈ R8 : a1 < 0, a2 < 0, a3 > 0, a4 > 0, b1 > 0, b2 < 0, c1 > 0, c2 < 0 } , I1 M1N + 23N − 4 := IM1N + 23N − 4 ∩ { κ ∈ R8 : a1 < 0, a2 < 0, a3 > 0, a4 = 0, b1 > 0, b2 > 0, c1 < 0, c2 < 0,∆3 = 0,∆4 = 0 } , I2 M1N + 23N − 4 := IM1N + 23N − 4 ∩ { κ ∈ R8 : a1 < 0, a2 > 0, a3 < 0, a4 > 0, b1 > 0, b2 > 0, c1 < 0, c2 < 0 } , 18 X. YANG, J. ZHOU EJDE-2025/42 IM1N + 24N − 3 := { κ ∈ R8 : ∆1 < 0, Tj > 0, Dj > 0,∆j ≥ 0(j = 2, 4), T3 < 0, D3 > 0,∆3 ≥ 0 } , I0 M1N + 24N − 3 := IM1N + 24N − 3 ∩ { κ ∈ R8 : a1 < 0, a2 < 0, a3 > 0, a4 > 0, b1 < 0, b2 > 0, c1 < 0, c2 > 0 } , I1 M1N + 24N − 3 := IM1N + 24N − 3 ∩ { κ ∈ R8 : a1 > 0, a2 < 0, a3 > 0, a4 > 0, b1 = 0, b2 > 0, c1 < 0, c2 < 0 } , IM12N + 34 := {κ ∈ R8 : ∆i < 0(i = 1, 2), Tj > 0, Dj > 0,∆j ≥ 0(j = 3, 4)}, I0 M12N + 34 := IM12N + 34 ∩ { κ ∈ R8 : 0 < a1 < c2 < b1, a2 < 0, a3 > 0, a4 = 0, b2 > 0, c1 = 0 } , IM12N + 3 N− 4 := { κ ∈ R8 : ∆i < 0(i = 1, 2), T3 > 0, D3 > 0,∆3 ≥ 0, T4 < 0, D4 > 0,∆4 ≥ 0 } , I0 M12N + 3 N− 4 := IM12N + 3 N− 4 ∩ { κ ∈ R8 : a1 < 0, a2 < 0, a3 > 0, a4 = 0, b1 < c2, b2 > 0, c1 < 0, c2 > 0 } , I1 M12N + 3 N− 4 := IM12N + 3 N− 4 ∩ { κ ∈ R8 : a1 < 0, a2 < 0, a3 > 0, a4 = 0, b2 > 0, c1 < 0, 0 < c2 < b1 } , IM13N + 24 := {κ ∈ R8 : ∆i < 0(i = 1, 3), Tj > 0, Dj > 0,∆j ≥ 0(j = 2, 4)}, I0 M13N + 24 := IM13N + 24 ∩ { κ ∈ R8 : a1 < 0, a2 < 0, a3 > 0, 0 < b1 < a4, b2 > 0, c1 < 0, c2 > 0 } , IM13N + 2 N− 4 := { κ ∈ R8 : ∆i < 0(i = 1, 3), T2 > 0, D2 > 0,∆2 ≥ 0, T4 < 0, D4 > 0,∆4 ≥ 0 } , I0 M13N + 2 N− 4 := IM13N + 2 N− 4 ∩ { κ ∈ R8 : a1 < c2 < 0, a2 < 0, a3 > 0, b2 > 0, c1 < 0, 0 < b1 < a4 } , I1 M13N + 2 N− 4 := IM13N + 2 N− 4 ∩ { κ ∈ R8 : a1 < 0, a2 < 0, a3 > 0, 0 < b1 < a4, b2 > 0, c1 < 0, c2 = 0 } , I2 M13N + 2 N− 4 := IM13N + 2 N− 4 ∩ { κ ∈ R8 : a1 < 0, a2 > 0, a3 < 0, a4 > 0, b1 > 0, b2 > 0, c1 < 0, c2 < 0 } , IM123N + 4 := {κ ∈ R8 : ∆i < 0(i = 1, 2, 3), T4 > 0, D4 > 0,∆4 ≥ 0}, I0 M123N + 4 := IM123N + 4 ∩ { κ ∈ R8 : a1 < 0, a2 < 0, a3 > 0, a4 = 0, b1 = 0, b2 > 0, c1 < 0, c2 > 0 } , I ′ M123N + 4 := IM123N + 4 ∩ { κ ∈ R8 : a1 < 0, a2 > 0, a3 < 0, a4 = 0, b1 = 0, b2 < 0, c1 > 0, c2 > 0 } , IS12N + 3 M4 := {κ ∈ R8 : Di < 0(i = 1, 2), T3 > 0, D3 > 0,∆3 ≥ 0,∆4 < 0}, I0 S12N + 3 M4 := IS12N + 3 M4 ∩ { κ ∈ R8 : a1 > 0, a2 = 0, a3 > 0, a4 < 0, b2 = 0, c1 < 0, b1 = c2 > 0 } , IS13N + 2 M4 := {κ ∈ R8 : Di < 0(i = 1, 3), T2 > 0, D2 > 0,∆2 ≥ 0,∆4 < 0}, I0 S13N + 2 M4 := IS13N + 2 M4 ∩ { κ ∈ R8 : a1 < 0, a2 = 0, a3 > 0, b1 > a4 > 0, b2 = 0, c1 < 0, c2 < 0 } , IS3N + 12M4 := {κ ∈ R8 : Ti > 0, Di > 0,∆i ≥ 0(i = 1, 2), D3 < 0,∆4 < 0}, I0 S3N + 12M4 := IS3N + 12M4 ∩ { κ ∈ R8 : a1 > a4 > 0, a2 = 0, a3 > 0, c1 < 0, c2 < 0, b2 = 0, b1 > a4 > 0 } , IS3N + 1 N− 2 M4 := { κ ∈ R8 : T1 > 0, D1 > 0,∆1 ≥ 0, T2 < 0, D2 > 0,∆2 ≥ 0, D3 < 0,∆4 < 0 } , I0 S3N + 1 N− 2 M4 := IS3N + 1 N− 2 M4 ∩ { κ ∈ R8 : a1 < 0, a2 > 0, a3 < 0, a4 > 0 > b1, b2 < 0, c1 > 0, c2 > 0 } , I1 S3N + 1 N− 2 M4 := IS3N + 1 N− 2 M4 ∩ { κ ∈ R8 : a1 > 0, a2 < 0, a3 > 0, a4 = 0 > b1, b2 > 0, c1 < 0, c2 > 0 } , IS2N + 13M4 := {κ ∈ R8 : Ti > 0, Di > 0,∆i ≥ 0(i = 1, 3), D2 < 0,∆4 < 0}, I0 S2N + 13M4 := IS2N + 13M4 ∩ { κ ∈ R8 : a1 > a4 > 0, a2 = 0, a3 > 0,−c2 < b1 < 0, b2 > 0, c1 < 0 } , IS2N + 1 N− 3 M4 := { κ ∈ R8 : T1 > 0, D1 > 0,∆1 ≥ 0, D2 < 0, T3 < 0, D3 > 0,∆3 ≥ 0,∆4 < 0 } , I0 S2N + 1 N− 3 M4 := IS2N + 1 N− 3 M4 ∩ { κ ∈ R8 : a1 > a4 > 0, a2 = 0, a3 > 0, b1 < −c2 < 0, b2 > 0, c1 < 0 } , I1 S2N + 1 N− 3 M4 := IS2N + 1 N− 3 M4 ∩ { κ ∈ R8 : a3 > 0 > a2, a1 > −a4 > 0, b1 > 0, b2 > 0, c1 < 0, c2 < 0 } , IS1N + 2 M34 := {κ ∈ R8 : D1 < 0, T2 > 0, D2 > 0,∆2 ≥ 0,∆3 < 0,∆4 < 0}, I0 S1N + 2 M34 := IS1N + 2 M34 ∩ { κ ∈ R8 : a1 < 0, a2 = 0, a3 > 0, 0 < b1 < a4, b2 > 0, c1 < 0, c2 = 0 } , IS1N + 3 M24 := {κ ∈ R8 : D1 < 0,∆2 < 0, T3 > 0, D3 > 0,∆3 ≥ 0,∆4 < 0}, I0 S1N + 3 M24 := IS1N + 3 M24 ∩ { κ ∈ R8 : a1 = 0, a2 > 0, a3 > 0, a4 = 0, b1 > 0, b2 < 0, c1 < 0, c2 > 0 } . 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Xin Yang School of Mathematics and Statistics, MNP-LAMA, Central South University, Changsha, Hunan 410083, China Email address: xinyang@csu.edu.cn Jueliang Zhou (corresponding author) School of Mathematics and Statistics, MNP-LAMA, Central South University, Changsha, Hunan 410083, China Email address: zhou jueliang@csu.edu.cn 1. Introduction 2. Main results 3. Proofs of main results 4. Conclusions Appendix A Acknowledgements References