Electronic Journal of Differential Equations, Vol. 2023 (2023), No. 83, pp. 1–57. ISSN: 1072-6691. URL: https://ejde.math.txstate.edu, https://ejde.math.unt.edu DOI: 10.58997/ejde.2023.83 GLOBAL ANALYSIS ON A CONTINUOUS PLANAR PIECEWISE LINEAR DIFFERENTIAL SYSTEM WITH THREE ZONES MAN JIA, YOUFENG SU, HEBAI CHEN Abstract. This article concerns the global dynamics of a continuous planar piecewise linear differential system with three zones. We give global phase portraits in the Poincaré disc and classify bifurcation diagrams under certain parametric conditions, when the dynamics of central linear zone is anti-saddle. Rich dynamical behaviors are demonstrated, from which we observe homoclinic loops appearing in three linear zones and limit cycles occurring in three linear zones which surround a node or node-focus. 1. Introduction In several scientific fields, piecewise linear differential systems have attracted a lot of attention from a rather diverse group of scientists such as physicists and math- ematicians [1, 11, 12, 17, 25, 26, 30, 33]. This is because, in addition to academic- theoretical significance [2, 4, 13, 14, 16, 18, 23, 31, 32, 36, 37], the study of piecewise linear differential systems has practical applications [5, 6, 15, 19, 20, 28, 34, 35]. Piecewise linear differential systems can model a large number of nonlinear prob- lems arising in physics and engineering such as design of electric circuits [1, 3], some memristor oscillators [4, 5, 8, 9, 15, 23, 26, 27, 34] and FitzHugh-Nagumo system [30, 31, 35]. Although piecewise linear differential systems may be considered as some of the most tractable nonlinear ordinary differential equations, they display various rich and interesting dynamical behaviours, with all the dynamics of gen- eral smooth nonlinear systems (such as limit cycles, homoclinic loops, heteroclinic loops, strangle attractors and so on), and with special dynamical behaviors (such as jump bifurcation, grazing bifurcation, sliding bifurcation, singular continuous systems and so on) [15, 17]. In this article, we are interested in continuous planar piecewise linear (CPWL) differential systems. A CPWL differential system with three zones separated by two parallel lines is of the Liénard form: dx dt = F (x)− y, dy dt = g(x)− α, (1.1) 2020 Mathematics Subject Classification. 34C05, 34C23, 37G15, 37G20. Key words and phrases. Piecewise linear system; global phase portrait; bifurcation; invariant manifold; limit cycle; homoclinic loop. ©2023. This work is licensed under a CC BY 4.0 license. Submitted June 6, 2023. Published December 10, 2023. 1 2 M. JIA, Y. SU, H. CHEN EJDE-2023/83 where F (x) =  tr(x− 1) + tc, if x > 1, tcx, if − 1 ≤ x ≤ 1, tl(x+ 1)− tc, if x < −1, g(x) =  dr(x− 1) + dc, if x > 1, dcx, if − 1 ≤ x ≤ 1, dl(x+ 1)− dc, if x < −1, with three open linear zones in the plane R2: Sl := {(x, y) ∈ R2 : x < −1}, Sc := {(x, y) ∈ R2 : −1 < x < 1}, Sr := {(x, y) ∈ R2 : x > 1} by two straight lines Γl := {(x, y) ∈ R2 : x = −1} and Γr := {(x, y) ∈ R2 : x = 1}. Considerable attention has been devoted to characterizing global dynamics of system (1.1) [6, 7, 9, 10, 17, 20, 24, 25, 26, 27, 29]. Jia-Su-Chen [21] once investigated global dynamics of system (1.1) in the region: G := {(tr, tc, tl, dr, dc, dl, α) ∈ R7 : trtl > 0, drdl < 0}, where G is divided into four parametric regions: G1 := {(tr, tc, tl, dr, dc, dl, α) ∈ R7 : tr > 0, tl > 0, dr > 0, dl < 0}, G2 := {(tr, tc, tl, dr, dc, dl, α) ∈ R7 : tr > 0, tl > 0, dr < 0, dl > 0}, G3 := {(tr, tc, tl, dr, dc, dl, α) ∈ R7 : tr < 0, tl < 0, dr > 0, dl < 0}, G4 := {(tr, tc, tl, dr, dc, dl, α) ∈ R7 : tr < 0, tl < 0, dr < 0, dl > 0}. By a proper transformation, the regions G2, G3 and G4 can be changed into the region G1. So all discussions for system (1.1) were restricted in the region G1. When dc ≤ 0, global phase portraits in the Poincaré disc and bifurcation diagrams of system (1.1) in the region G1 were presented [21]. Therefore, in this study we continue to explore global dynamics of system (1.1) in the region G1 when dc > 0. The article is organized as follows. In Section 2, we state our main results of system (1.1) with dc > 0 in the region G1. To study the local dynamical behaviors of system (1.1), we introduce some preliminary results in Section 3. Local dynamics of system (1.1) are investigated in Section 4. Nonlocal dynamics of system (1.1) are explored in Section 5. The proofs of our main results are presented in Section 6, while numerical phase portraits are demonstrated in Section 7. A brief conclusion is given in Section 8. 2. Main results In this section, we summarize our main results of system (1.1) with dc > 0 in the region G1, i.e., the bifurcation diagram in the (α, tc)-plane and global phase portraits in the Poincaré disc. Notice that the condition t2r − 4dr < 0 (resp. = 0 or > 0) implies that the dynamics of right linear zone of system (1.1) is a focus (resp. improper node or bidirectional node) (see Lemma 4.1 for more details). Moreover, EJDE-2023/83 GLOBAL ANALYSIS ON A PLANAR PIECEWISE LINEAR SYSTEMS 3 since system (1.1) has different qualitative properties of equilibrium points at infin- ity for the three conditions t2r − 4dr < 0, t2r − 4dr = 0 and t2r − 4dr > 0 (see Lemma 4.2 for more details), our main results are achieved in the following three regions: G11 := { (tr, tc, tl, dr, dc, dl, α) ∈ R7 : tr > 0, tl > 0, dr > 0, dc > 0, dl < 0, t2r − 4dr < 0 } ⊂ G1, G12 := { (tr, tc, tl, dr, dc, dl, α) ∈ R7 : tr > 0, tl > 0, dr > 0, dc > 0, dl < 0, t2r − 4dr = 0 } ⊂ G1, G13 := { (tr, tc, tl, dr, dc, dl, α) ∈ R7 : tr > 0, tl > 0, dr > 0, dc > 0, dl < 0, t2r − 4dr > 0 } ⊂ G1. System (1.1) has two equilibrium points El : ((α + dc)/dl − 1, tl(α + dc)/dl − tc) and Ec : (α/dc, tcα/dc) when dc > 0, −dc < α < dc and tc < 0 in G1, where El is a saddle, Ec is a stable node for tc 2 − 4dc ≥ 0 or a stable focus for tc 2 − 4dc < 0. Theorems 2.1-2.3 and Theorems 2.4-2.6 are presented according to the nonexistence of the homoclinic loop appearing in three linear zones which surrounds the stable node Ec. For simplicity, we set t∗c := − tr(α− dc) + tr √ (α− dc)2 + 4αdr − dr dl (α+ dc)2 2dr + tl(α+ dc) 2dl t∗∗∗c := − tr(α− dc + √ 4αdr + (α− dc)2) 2dr , where t∗c and t∗∗∗c are continuous functions on α when tr, dr, dc and dl are fixed. Figure 1. Bifurcation diagram in the (α, tc)-plane of system (1.1) as (tr, tc, tl, dr, dc, dl, α) ∈ G11. Theorem 2.1. When (tr, tc, tl, dr, dc, dl, α) ∈ G11, the bifurcation diagram of sys- tem (1.1) in the (α, tc)-plane consists of the following bifurcation curves: (a) Boundary equilibrium bifurcation curves: BE11 = {(α, tc) ∈ R2 : α = −dc}, BE12 = {(α, tc) ∈ R2 : α = dc}. 4 M. JIA, Y. SU, H. CHEN EJDE-2023/83 (b) Homoclinic bifurcation curves: HL11 = {(α, tc) ∈ R2 : −dc < α ≤ dc, tc = φ(α)}, HL12 = {(α, tc) ∈ R2 : α > dc, tc = ϕ(α)}. (c) Double limit cycle bifurcation curve: DL1 = {(α, tc) ∈ R2 : α > dc, tc = h(α)}, where the function tc = φ(α) is continuous, monotonic and satisfies max{t∗c ,−2 √ dc} < φ(α) < 0 for − dc < α < dc, max{t∗c ,−2 √ dc} < φ(α) < −tr √ dc/dr for α = dc, and the function tc = h(α) is continuous satisfying ϕ(α) < h(α) < t∗∗∗c for α > dc. Moreover, the bifurcation diagram and global phase portraits in the Poincaré disc of system (1.1) in G11 are respectively shown in Figures 1 and 2, where −dc < α∗ < dc, and I = {(α, tc) ∈ R2 : α < −dc}, II = {(α, tc) ∈ R2 : −dc < α < dc, tc ≥ 2 √ dc}, III = {(α, tc) ∈ R2 : −dc < α < dc, 0 < tc < 2 √ dc}, IV = {(α, tc) ∈ R2 : −dc < α < dc, tc = 0}, V = {(α, tc) ∈ R2 : −dc < α < dc, φ(α) < tc < 0}, V I = {(α, tc) ∈ R2 : −dc < α < dc, −2 √ dc < tc < φ(α)}, V II = {(α, tc) ∈ R2 : −dc < α < dc, tc ≤ −2 √ dc}, V III = {(α, tc) ∈ R2 : α > dc, tc > h(α)}, IX = {(α, tc) ∈ R2 : α > dc, ϕ(α) < tc < h(α)}, X = {(α, tc) ∈ R2 : α > dc, tc < ϕ(α)}, BE121 = {(α, tc) ∈ R2 : α = dc, tc ≥ 2 √ dc}, BE122 = {(α, tc) ∈ R2 : α = dc, −tr √ dc/dr < tc < 2 √ dc}, BE123 = {(α, tc) ∈ R2 : α = dc, tc = −tr √ dc/dr}, BE124 = {(α, tc) ∈ R2 : α = dc, φ(α) < tc < −tr √ dc/dr}, BE125 = {(α, tc) ∈ R2 : α = dc, −2 √ dc < tc < φ(α)}, BE126 = {(α, tc) ∈ R2 : α = dc, tc ≤ −2 √ dc}, HL111 = {(α, tc) ∈ R2 : −dc < α < α∗ < dc, tc = φ(α)}, HL112 = {(α, tc) ∈ R2 : −dc < α = α∗ < dc, tc = φ(α)}, HL113 = {(α, tc) ∈ R2 : −dc < α∗ < α < dc, tc = φ(α)}, HL114 = {(α, tc) ∈ R2 : α = dc, tc = φ(α)}. In Figure 2, the stable limit cycle is marked by red color, the center is marked by baby blue color, the unstable limit cycle is marked by dark blue color, the semi- stable limit cycle is marked by yellow color and the unstable homoclinic loop is marked by green color. When there is a unique limit cycle (stable or unstable) in EJDE-2023/83 GLOBAL ANALYSIS ON A PLANAR PIECEWISE LINEAR SYSTEMS 5 I II III IV V VI VII VIII IX X BE11 BE121 BE122 BE123 BE124 BE125 BE126 HL111 HL112 HL113 HL114 HL12 DL1 Figure 2. Global phase portraits in the Poincaré disc of system (1.1) as (tr, tc, tl, dr, dc, dl, α) ∈ G11. 6 M. JIA, Y. SU, H. CHEN EJDE-2023/83 Theorem 2.1, it involves two or three linear zones, see global phase portraits in the Poincaré disc of V , X, BE124 and HL12. When there are two limit cycles, the inner limit cycle involves two or three linear zones and the outer limit cycle involves three linear zones, see global phase portrait in the Poincaré disc of IX. The semi-stable limit cycle involves three linear zones, see global phase portrait in the Poincaré disc of DL1. The homoclinic loop involves two (resp. three) linear zones, see global phase portraits in the Poincaré disc of HL111 and HL112 (resp. HL113, HL114 and HL12), and the homoclinic loop becomes tangent to Γr, see global phase portrait in the Poincaré disc of HL112. Figure 3. Bifurcation diagram in the (α, tc)-plane of system (1.1) as (tr, tc, tl, dr, dc, dl, α) ∈ G12. Theorem 2.2. When (tr, tc, tl, dr, dc, dl, α) ∈ G12, the bifurcation diagram of sys- tem (1.1) in the (α, tc)-plane consists of the following bifurcation curves: (a) Boundary equilibrium bifurcation curves: BE21 = {(α, tc) ∈ R2 : α = −dc}, BE22 = {(α, tc) ∈ R2 : α = dc}. (b) Homoclinic bifurcation curve: HL2 = {(α, tc) ∈ R2 : −dc < α < dc, tc = φ(α)}, where the function tc = φ(α) is continuous, monotonic, satisfying max{t∗c ,−2 √ dc} < φ(α) < 0 for −dc < α < dc. Moreover, the bifurcation diagram and global phase portraits in the Poincaré disc of system (1.1) in G12 are respectively shown in Figures 3 and 4, where −dc < α∗ < dc, and R1 = {(α, tc) ∈ R2 : α < −dc}, R2 = {(α, tc) ∈ R2 : −dc < α < dc, tc ≥ 2 √ dc}, R3 = {(α, tc) ∈ R2 : −dc < α < dc, 0 < tc < 2 √ dc}, R4 = {(α, tc) ∈ R2 : −dc < α < dc, tc = 0}, R5 = {(α, tc) ∈ R2 : −dc < α < dc, φ(α) < tc < 0}, R6 = {(α, tc) ∈ R2 : −dc < α < dc, −2 √ dc < tc < φ(α)}, EJDE-2023/83 GLOBAL ANALYSIS ON A PLANAR PIECEWISE LINEAR SYSTEMS 7 R7 = {(α, tc) ∈ R2 : −dc < α < dc, tc ≤ −2 √ dc}, R8 = {(α, tc) ∈ R2 : α > dc}, BE221 = {(α, tc) ∈ R2 : α = dc, tc ≥ 2 √ dc}, BE222 = {(α, tc) ∈ R2 : α = dc, tc < 2 √ dc}, HL21 = {(α, tc) ∈ R2 : −dc < α < α∗ < dc, tc = φ(α)}, HL22 = {(α, tc) ∈ R2 : −dc < α = α∗ < dc, tc = φ(α)}, HL23 = {(α, tc) ∈ R2 : −dc < α∗ < α < dc, tc = φ(α)}. R1 R2 R3 R4 R5 R6 R7 R8 BE21 BE221 BE222 HL21 HL22 HL23 Figure 4. Global phase portraits in the Poincaré disc of system (1.1) as (tr, tc, tl, dr, dc, dl, α) ∈ G12. 8 M. JIA, Y. SU, H. CHEN EJDE-2023/83 In Theorem 2.2, the unstable limit cycle involves two or three linear zones, see global phase portrait in the Poincaré disc of R5. The homoclinic loop involves two (resp. three) linear zones, see global phase portraits in the Poincaré disc of HL21 and HL22 (resp. HL23). The homoclinic loop becomes tangent to Γr, see global phase portrait in the Poincaré disc of HL22. Figure 5. Bifurcation diagram in the (α, tc)-plane of system (1.1) as (tr, tc, tl, dr, dc, dl, α) ∈ G13. Theorem 2.3. When (tr, tc, tl, dr, dc, dl, α) ∈ G13, the bifurcation diagram of sys- tem (1.1) in the (α, tc)-plane consists of the following bifurcation curves: (a) Boundary equilibrium bifurcation curves: BE31 = {(α, tc) ∈ R2 : α = −dc}, BE32 = {(α, tc) ∈ R2 : α = dc}. (b) Homoclinic bifurcation curve: HL3 = {(α, tc) ∈ R2 : −dc < α < dc, tc = φ(α)}, where the function tc = φ(α) is continuous and monotonic satisfying max{t∗c ,−2 √ dc} < φ(α) < 0 for −dc < α < dc. Moreover, the bifurcation diagram and global phase portraits in the Poincaré disc of system (1.1) in G13 are respectively shown in Fig- ures 5 and 6, where −dc < α∗ < dc, and I1 = {(α, tc) ∈ R2 : α < −dc}, I2 = {(α, tc) ∈ R2 : −dc < α < dc, tc ≥ 2 √ dc}, I3 = {(α, tc) ∈ R2 : −dc < α < dc, 0 < tc < 2 √ dc}, I4 = {(α, tc) ∈ R2 : −dc < α < dc, tc = 0}, I5 = {(α, tc) ∈ R2 : −dc < α < dc, φ(α) < tc < 0}, I6 = {(α, tc) ∈ R2 : −dc < α < dc, −2 √ dc < tc < φ(α)}, I7 = {(α, tc) ∈ R2 : −dc < α < dc, tc ≤ −2 √ dc}, I8 = {(α, tc) ∈ R2 : α > dc}, BE321 = {(α, tc) ∈ R2 : α = dc, tc ≥ 2 √ dc}, EJDE-2023/83 GLOBAL ANALYSIS ON A PLANAR PIECEWISE LINEAR SYSTEMS 9 BE322 = {(α, tc) ∈ R2 : α = dc, tc < 2 √ dc}, HL31 = {(α, tc) ∈ R2 : −dc < α < α∗ < dc, tc = φ(α)}, HL32 = {(α, tc) ∈ R2 : −dc < α = α∗ < dc, tc = φ(α)}, HL33 = {(α, tc) ∈ R2 : −dc < α∗ < α < dc, tc = φ(α)}. I1 I2 I3 I4 I5 I6 I7 I8 BE31 BE321 BE322 HL31 HL32 HL33 Figure 6. Global phase portraits in the Poincaré disc of system (1.1) as (tr, tc, tl, dr, dc, dl, α) ∈ G13. Theorem 2.4. When (tr, tc, tl, dr, dc, dl, α) ∈ G11, the bifurcation diagram of sys- tem (1.1) in the (α, tc)-plane consists of the following bifurcation curves: (a) Boundary equilibrium bifurcation curves: BE41 = {(α, tc) ∈ R2 : α = −dc}, BE42 = {(α, tc) ∈ R2 : α = dc}. 10 M. JIA, Y. SU, H. CHEN EJDE-2023/83 (b) Homoclinic bifurcation curves: HL41 = {(α, tc) ∈ R2 : −dc < α ≤ dc, tc = φ(α)}, HL42 = {(α, tc) ∈ R2 : α > dc, tc = ϕ(α)}. (c) Double limit cycle bifurcation curve: DL4 = {(α, tc) ∈ R2 : α > dc, tc = h(α)}, where the function tc = φ(α) is continuous and monotonic satisfying max{t∗c ,−2 √ dc} < φ(α) < 0 for −dc < α < α, φ(α) = −2 √ dc for α = α = φ−1(−2 √ dc), t∗c < φ(α) < −2 √ dc for α < α ≤ dc, the function tc = ϕ(α) is continuous and monotonic, and the function tc = h(α) is continuous satisfying ϕ(α) < h(α) < t∗∗∗c for α > dc. Moreover, the bifurcation diagram and global phase portraits in the Poincaré disc of system (1.1) in G11 are shown in Figure 7, where −dc < α∗ < α < dc, and G1 = {(α, tc) ∈ R2 : α < −dc}, G2 = {(α, tc) ∈ R2 : −dc < α < dc, tc ≥ 2 √ dc}, G3 = {(α, tc) ∈ R2 : −dc < α < dc, 0 < tc < 2 √ dc}, G4 = {(α, tc) ∈ R2 : −dc < α < dc, tc = 0}, G5 = {(α, tc) ∈ R2 : −dc < α < α, φ(α) < tc < 0}∪ {(α, tc) ∈ R2 : α < α < dc, −2 √ dc < tc < 0}, G6 = {(α, tc) ∈ R2 : −dc < α < α, −2 √ dc < tc < φ(α)}, G7 = {(α, tc) ∈ R2 : α < α < dc, φ(α) < tc ≤ −2 √ dc}, G8 = {(α, tc) ∈ R2 : −dc < α < α, tc ≤ −2 √ dc}∪ {(α, tc) ∈ R2 : α < α < dc, tc < φ(α)}, G9 = {(α, tc) ∈ R2 : α > dc, tc > h(α)}, G10 = {(α, tc) ∈ R2 : α > dc, ϕ(α) < tc < h(α)}, G11 = {(α, tc) ∈ R2 : α > dc, tc < ϕ(α)}, BE421 = {(α, tc) ∈ R2 : α = dc, tc ≥ 2 √ dc}, BE422 = {(α, tc) ∈ R2 : α = dc, −tr √ dc/dr < tc < 2 √ dc}, BE423 = {(α, tc) ∈ R2 : α = dc, tc = −tr √ dc/dr}, BE424 = {(α, tc) ∈ R2 : α = dc, −2 √ dc < tc < tr √ dc/dr}, BE425 = {(α, tc) ∈ R2 : α = dc, φ(α) < tc ≤ −2 √ dc}, BE426 = {(α, tc) ∈ R2 : α = dc, tc < φ(α)}, HL411 = {(α, tc) ∈ R2 : −dc < α < α∗ < α, tc = φ(α)}, HL412 = {(α, tc) ∈ R2 : −dc < α = α∗ < α, tc = φ(α)}, HL413 = {(α, tc) ∈ R2 : −dc < α∗ < α < α, tc = φ(α)}, HL414 = {(α, tc) ∈ R2 : −dc < α ≤ α < dc, tc = φ(α)}, HL415 = {(α, tc) ∈ R2 : α = dc, tc = φ(α)}. EJDE-2023/83 GLOBAL ANALYSIS ON A PLANAR PIECEWISE LINEAR SYSTEMS 11 Bifurcation diagram in the (α, tc)-plane of system (1.1) as (tr, tc, tl, dr, dc, dl, α) ∈ G11 G7 BE425 HL414 HL415 Figure 7. Bifurcation diagram and global phase portraits in the Poincaré disc of system (1.1) as (tr, tc, tl, dr, dc, dl, α) ∈ G11. As we observe, the unstable limit cycle involves two or three linear zones in the global phase portrait in the Poincaré disc of G7 or BE425. The unstable homoclinic loop involves three linear zones in the global phase portrait in the Poincaré disc of HL414 or HL415. The equilibrium point lying in Sc is a stable node in the global phase portrait in the Poincaré disc of G7 or HL414. However, in the global phase portrait in the Poincaré disc of BE425 or HL415, the equilibrium point lying on the switching line Γr is a stable node (as seen from Sc), and is an unstable focus (as seen from Sr). Compared with Theorem 2.1, the homoclinic bifurcation curve of system (1.1) in Theorem 2.4 is different in the region {(α, tc) ∈ R2| − dc < α ≤ dc} of the (α, tc)-plane, while the other conditions are the same. Global phase portraits in the Poincaré disc of BE41, BE421, BE422, BE423, BE424, BE426, HL411, HL412, HL413, HL42, DL4, G1, G2, G3, G4, G5, G6, G8, G9, G10 and G11 of Theorem 2.4 are the same as those in the Poincaré disc of BE11, BE121, BE122, BE123, BE124, BE126, HL111,HL112, HL113, HL12, DL1, I, II, III, IV , V , V I, V II, V III, IX and X of Theorem 2.1, respectively. Therefore, we omit the details and only present global phase portraits in the Poincaré disc of G7, BE425, HL414 and HL415 in Theorem 2.4. Theorem 2.5. When (tr, tc, tl, dr, dc, dl, α) ∈ G12, the bifurcation diagram of sys- tem (1.1) in the (α, tc)-plane consists of the following bifurcation curves: 12 M. JIA, Y. SU, H. CHEN EJDE-2023/83 (a) Boundary equilibrium bifurcation curves: BE51 = {(α, tc) ∈ R2 : α = −dc}, BE52 = {(α, tc) ∈ R2 : α = dc}. (b) Homoclinic bifurcation curve: HL5 = {(α, tc) ∈ R2 : −dc < α < dc, tc = φ(α)}, where the function tc = φ(α) is continuous and monotonic satisfying max{t∗c ,−2 √ dc} < φ(α) < 0 for −dc < α < α, φ(α) = −2 √ dc for α = α = φ−1(−2 √ dc), and t∗c < φ(α) < −2 √ dc for α < α < dc. Moreover, the bifurcation diagram and global phase portraits in the Poincaré disc of system (1.1) in G12 are shown in Figure 8, where −dc < α∗ < α < dc, and N1 = {(α, tc) ∈ R2 : α < −dc}, N2 = {(α, tc) ∈ R2 : −dc < α < dc, tc ≥ 2 √ dc}, N3 = {(α, tc) ∈ R2 : −dc < α < dc, 0 < tc < 2 √ dc}, N4 = {(α, tc) ∈ R2 : −dc < α < dc, tc = 0}, N5 = {(α, tc) ∈ R2 : −dc < α < α, φ(α) < tc < 0}∪ {(α, tc) ∈ R2 : α < α < dc, −2 √ dc < tc < 0}, N6 = {(α, tc) ∈ R2 : −dc < α < α, −2 √ dc < tc < φ(α)}, N7 = {(α, tc) ∈ R2 : α < α < dc, φ(α) < tc ≤ −2 √ dc}, N8 = {(α, tc) ∈ R2 : −dc < α < α, tc ≤ −2 √ dc} ∪ {(α, tc) ∈ R2 : α < α < dc, tc < φ(α)}, N9 = {(α, tc) ∈ R2 : α > dc}, BE521 = {(α, tc) ∈ R2 : α = dc, tc ≥ 2 √ dc}, BE522 = {(α, tc) ∈ R2 : α = dc, tc < 2 √ dc}, HL51 = {(α, tc) ∈ R2 : −dc < α < α∗ < α, tc = φ(α)}, HL52 = {(α, tc) ∈ R2 : −dc < α = α∗ < α, tc = φ(α)}, HL53 = {(α, tc) ∈ R2 : −dc < α∗ < α < α, tc = φ(α)}, HL54 = {(α, tc) ∈ R2 : −dc < α ≤ α < dc, tc = φ(α)}. Although homoclinic bifurcation curves of system (1.1) in Theorem 2.5 are dif- ferent from the ones described in Theorem 2.2 in the region {(α, tc) ∈ R2| − dc < α < dc} of the (α, tc)-plane, all other bifurcation curves of system (1.1) in Theorem 2.5 are the same as those shown in Theorem 2.2. It indicates that 14 global phase portraits in the Poincaré disc of Theorem 2.5 can be found in Theorem 2.2. In other words, global phase portraits in the Poincaré disc of BE51, BE521, BE522, HL51, HL52, HL53, N1, N2, N3, N4, N5, N6, N8 and N9 of Theorem 2.5 are the same as those in the Poincaré disc of BE21, BE221, BE222, HL211, HL212, HL213, R1, R2, R3, R4, R5, R6, R7 and R8 of Theorem 2.2, respectively. Hence, we only show global phase portraits in the Poincaré disc of N7 and HL514 in Theorem 2.5. EJDE-2023/83 GLOBAL ANALYSIS ON A PLANAR PIECEWISE LINEAR SYSTEMS 13 Bifurcation diagram in the (α, tc)-plane of system (1.1) as (tr, tc, tl, dr, dc, dl, α) ∈ G12 N7 HL54 Figure 8. Bifurcation diagram and global phase portraits in the Poincaré disc of system (1.1) as (tr, tc, tl, dr, dc, dl, α) ∈ G12. Theorem 2.6. When (tr, tc, tl, dr, dc, dl, α) ∈ G13, the bifurcation diagram of sys- tem (1.1) in the (α, tc)-plane consists of the following bifurcation curves: (a) Boundary equilibrium bifurcation curves: BE61 = {(α, tc) ∈ R2 : α = −dc}, BE62 = {(α, tc) ∈ R2 : α = dc}; (b) Homoclinic bifurcation curve: HL6 = {(α, tc) ∈ R2 : −dc < α < dc, tc = φ(α)}, where the function tc = φ(α) is continuous, monotonous and satisfies max{t∗c ,−2 √ dc} < φ(α) < 0 for − dc < α < α, φ(α) = −2 √ dc for α = α = φ−1(−2 √ dc), and t∗c < φ(α) < −2 √ dc for α < α < dc. Moreover, the bifurcation diagram and global phase portraits in the Poincaré disc of system (1.1) in G13 are shown in Figure 9, where −dc < α∗ < α < dc, and U1 = {(α, tc) ∈ R2 : α < −dc}, U2 = {(α, tc) ∈ R2 : −dc < α < dc, tc ≥ 2 √ dc}, U3 = {(α, tc) ∈ R2 : −dc < α < dc, 0 < tc < 2 √ dc}, U4 = {(α, tc) ∈ R2 : −dc < α < dc, tc = 0}, 14 M. JIA, Y. SU, H. CHEN EJDE-2023/83 Bifurcation diagram in the (α, tc)-plane of system (1.1) as (tr, tc, tl, dr, dc, dl, α) ∈ G13 U7 HL64 Figure 9. Bifurcation diagram and global phase portraits in the Poincaré disc of system (1.1) as (tr, tc, tl, dr, dc, dl, α) ∈ G13. U5 = {(α, tc) ∈ R2 : −dc < α < α, φ(α) < tc < 0} ∪ {(α, tc) ∈ R2 : α < α < dc, −2 √ dc < tc < 0}, U6 = {(α, tc) ∈ R2 : −dc < α < α, −2 √ dc < tc < φ(α)}, U7 = {(α, tc) ∈ R2 : α < α < dc, φ(α) < tc ≤ −2 √ dc}, U8 = {(α, tc) ∈ R2 : −dc < α < α, tc ≤ −2 √ dc} ∪ {(α, tc) ∈ R2 : α < α < dc, tc < φ(α)}, U9 = {(α, tc) ∈ R2 : α > dc}, BE621 = {(α, tc) ∈ R2 : α = dc, tc ≥ 2 √ dc}, BE622 = {(α, tc) ∈ R2 : α = dc, tc < 2 √ dc}, HL61 = {(α, tc) ∈ R2 : −dc < α < α∗ < α, tc = φ(α)}, HL62 = {(α, tc) ∈ R2 : −dc < α = α∗ < α, tc = φ(α)}, HL63 = {(α, tc) ∈ R2 : −dc < α∗ < α < α, tc = φ(α)}, HL64 = {(α, tc) ∈ R2 : −dc < α ≤ α < dc, tc = φ(α)}. EJDE-2023/83 GLOBAL ANALYSIS ON A PLANAR PIECEWISE LINEAR SYSTEMS 15 In Theorem 2.6, the homoclinic bifurcation curves of system (1.1) are different from the ones described in Theorem 2.3 in the region {(α, tc) ∈ R2|−dc < α < dc} of the (α, tc)-plane. However, the other bifurcation curves of system (1.1) in Theorem 2.6 are the same as those described in Theorem 2.3 under the same conditions. Therefore, 14 global phase portraits in the Poincaré disc of Theorem 2.6 can be found in Theorem 2.3. In other words, global phase portraits in the Poincaré disc of BE61, BE621, BE622, HL61, HL62, HL63, U1, U2, U3, U4, U5, U6, U8 and U9 of Theorem 2.6 are the same as the corresponding ones in the Poincaré disc of BE31, BE321, BE322, HL311, HL312, HL313 and the regions I1, I2, I3, I4, I5, I6, I7, I8 of Theorem 2.3, respectively. Thus, we only illustrate global phase portraits in the Poincaré disc of U7 and HL614 as given in Theorem 2.6. In addition, note that system (1.1) is analytic in R2\{Γl ∪ Γr} and Lipschitz continuous in R2. Therefore, classical theorems on the existence, uniqueness and continuity of solutions hold for system (1.1) under the initial conditions [26]. 3. Preliminaries We consider a continuous planar piecewise linear differential system in the Liénard form ẋ = F (x)− y, ẏ = g(x), (3.1) where the functions F (x) and g(x) satisfy the following four conditions for x ∈ (a, b) with a < 0 < b: (C1) both F (x) and g(x) are Lipschitz continuous with respect to x; (C2) F (0) = g(0) = 0 and xg(x) > 0 for x 6= 0; (C3) there exists a unique switching line x = 0; (C4) the unique equilibrium point (0, 0), is a stable focus as seen from the left linear zone of the switching line x = 0, and is an unstable focus as seen from the right linear zone of the switching line x = 0. Let z(x) := ∫ x 0 g(s)ds with z1 := z(b) and z2 := z(a). Obviously, we have z(x) > 0 for x ∈ (a, 0) ∪ (0, b) and z(x) = 0 for x = 0. Denote two branches of the inverse of z(x) as x1(z) and x2(z) when x ≥ 0 and x ≤ 0, respectively. Set F1(z) := F (x1(z)), F2(z) := F (x2(z)). (3.2) Re-write system (3.1) as dz dy = z′(x)dx dy = g(x)dx dy = F1(z)− y, z ∈ (0, z1) (3.3) for x > 0, and dz dy = z′(x)dx dy = g(x)dx dy = F2(z)− y, z ∈ (0, z2) (3.4) for x < 0. From (3.1)-(3.2), we have the following result. Proposition 3.1. If conditions (C1)–(C4) hold, then we have (i) The unique equilibrium point O(0, 0) of system (3.1) is an unstable focus if and only if F1(z) > F2(z) for 0 < z � 1. (ii) The unique equilibrium point O(0, 0) of system (3.1) is a stable focus if and only if F1(z) < F2(z) for 0 < z � 1. (iii) The unique equilibrium point O(0, 0) of system (3.1) is a center if and only if F1(z) = F2(z) for 0 < z � 1. 16 M. JIA, Y. SU, H. CHEN EJDE-2023/83 Proof. Note that the existence, uniqueness and continuity of the solution of system (3.1) holds under the initial conditions, so the qualitative property of O(0, 0) de- pends on the qualitative property of the left-hand and right-hand linear zones of the switching line x = 0. It follows from condition (C4) that O(0, 0) is a focus or center. To prove that O(0, 0) of system (3.1) is an unstable focus if and only if F1(z) > F2(z) for 0 < z � 1, we suppose that O(0, 0) is a center, and take a closed orbit denoted by γ in a small neighborhood of O(0, 0). We denote by A and B the intersection points of γ with the positive and negative y-axis respectively, see Figure 10(a). Furthermore, we denote the corresponding integral curves of systems (3.3) and (3.4) by γ1 and γ2 respectively. Evidently, γ1 and γ2 are connected by A and B. We denote by P the intersection point of γ1 with the curve F1(z), as shown in Figure 10(b). If F1(z) > F2(z) holds in a small neighborhood of O(0, 0), by the comparison theorem [11, Corollary 6.3] to systems (3.3) and (3.4) we know that γ2 passing through A intersects the curve F1(z) at C and γ2 passing through B intersects the curve F1(z) at D, where C lies on the right side of P and D lies on the left side of P . This indicates that system (3.1) has no integral curves connecting A and B. This contradicts the existence of γ. In other words, O(0, 0) is a focus if F1(z) > F2(z) holds in a small neighborhood of O(0, 0). It is easy to see that O(0, 0) is an unstable focus if and only if F1(z) > F2(z) for 0 < z � 1. (a) (b) Figure 10. (a) Closed orbit γ of system (3.1); (b) Closed orbits γ1 and γ2 of systems (3.3) and (3.4) when F1(z) > F2(z). Similarly, we can prove that O(0, 0) is a stable focus if and only if F1(z) < F2(z) for 0 < z � 1. If F1(z) = F2(z) in a small neighborhood of O(0, 0), then γ1 and γ2 will coincide. This implies that O(0, 0) is a center if and only if F1(z) = F2(z) for 0 < z � 1. � (C5) The unique equilibrium point (0, 0) is an unstable focus as seen from the left linear zone of the switching line x = 0, and is a stable focus as seen from the right linear zone of the switching line x = 0. Proceeding in an analogous manner, we can derive the dual result of Proposition 3.1 as follows. Proposition 3.2. If conditions (C1)–(C3), (C5) hold, then we have EJDE-2023/83 GLOBAL ANALYSIS ON A PLANAR PIECEWISE LINEAR SYSTEMS 17 (i) The unique equilibrium point O(0, 0) of system (3.1) is a stable focus if and only if F1(z) > F2(z) for 0 < z � 1. (ii) The unique equilibrium point O(0, 0) of system (3.1) is an unstable focus if and only if F1(z) < F2(z) for 0 < z � 1. (iii) The unique equilibrium point O(0, 0) of system (3.1) is a center if and only if F1(z) = F2(z) for 0 < z � 1. 4. Local dynamics of system (1.1) Lemma 4.1. When dc > 0 in G1, system (1.1) has no equilibrium points if α < −dc; one equilibrium point Ecl if α = −dc; two equilibrium points E1 and Ec if −dc < α < dc; two equilibrium points E1 and Ecr if α = dc; and two equilibrium points El and Er if α > dc. The qualitative properties of these equilibrium points are shown in Table 1, where El : ((α + dc)/dl − 1, tl(α + dc)/dl − tc) lies in Sl, Ec : (α/dc, tcα/dc) lies in Sc, Er : ((α − dc)/dr + 1, tr(α − dc)/dr + tc) lies in Sr, Ecl : (−1,−tc) lies on the left switching line Γl, and Ecr : (1, tc) lies on the right switching line Γr. Table 1. Qualitative properties of finite equilibrium points of sys- tem (1.1) with dc > 0 in G1 Possibilities tr, tc, dr, dc, α Number, Stability and Type α < −dc 0 tc < 0 tc 2 − 4dc < 0 1, Ecl cusp (Fig. 11(a)) tc 2 − 4dc ≥ 0 1, Ecl saddle-node (Fig. 11(b)) α = −dc tc = 0 1, Ecl cusp (Fig. 11(c)) tc > 0 tc 2 − 4dc < 0 1, Ecl cusp (Fig. 11(d)) tc 2 − 4dc ≥ 0 1, Ecl saddle-node (Fig. 11(e)) tc < 0 tc 2 − 4dc < 0 2, E1 saddle, Ec stable focus tc 2 − 4dc ≥ 0 2, E1 saddle, Ec stable node −dc < α < dc tc = 0 2, E1 saddle, Ec center tc > 0 tc 2 − 4dc < 0 2, E1 saddle, Ec unstable focus dc > 0 tc 2 − 4dc ≥ 0 2, E1 saddle, Ec unstable node tc < 0 tc 2 − 4dc < 0 tr 2 − 4dr < 0 2, E1 saddle, Ecr a focus or center tr 2 − 4dr ≥ 0 2, E1 saddle, Ecr focus-node (Fig. 11(f)) tc 2 − 4dc ≥ 0 tr 2 − 4dr < 0 2, E1 saddle, Ecr node-focus (Fig. 11(g)) tr 2 − 4dr ≥ 0 2, E1 saddle, Ecr node-node (Fig. 11(h)) α = dc tc = 0 tr 2 − 4dr < 0 2, E1 saddle, Ecr unstable focus (Fig. 11(i)) tr 2 − 4dr ≥ 0 2, E1 saddle, Ecr focus-node (Fig. 11(j)) tc > 0 tc 2 − 4dc < 0 tr 2 − 4dr < 0 2, E1 saddle, Ecr unstable focus tr 2 − 4dr ≥ 0 2, E1 saddle, Ecr focus-node (Fig. 11(k)) tc 2 − 4dc ≥ 0 tr 2 − 4dr < 0 2, E1 saddle, Ecr node-focus (Fig. 11(l)) tr 2 − 4dr ≥ 0 2, E1 saddle, Ecr unstable node α > dc tr 2 − 4dr < 0 2, E1 saddle, Er unstable focus tr 2 − 4dr ≥ 0 2, E1 saddle, Er unstable node Proof. As we know, the number of equilibrium points of system (1.1) is determined by the number of roots of g(x) = α. If dc > 0 in G1, by a direct calculation, we see that system (1.1) exhibits two equilibrium points El and Er for α > dc, two equilibrium points El and Ecr for α = dc, two equilibrium points El and Ec for −dc < α < dc, one equilibrium point Ecl for α = −dc, and no equilibrium points 18 M. JIA, Y. SU, H. CHEN EJDE-2023/83 for α < −dc, as shown in Table 1. The Jacobian matrices at El, Ec and Er have the forms respectively JE1 := [ t1 −1 d1 0 ] , JEc := [ tc −1 dc 0 ] , JEr := [ tr −1 dr 0 ] . From det JE1 = d1 < 0, det JEc = dc > 0 and det JEr = dr > 0, it follows that El is a saddle and Ec and Er are anti-saddles, as shown in Table 1. (a) tc < 0, tc 2 − 4dc < 0 (b) tc < 0, tc 2 − 4dc ≥ 0 (c) tc = 0 (d) tc > 0, tc 2 − 4dc < 0 (e) tc > 0, tc 2 − 4dc ≥ 0 (f) tc < 0, tc 2 − 4dc < 0, tr 2 − 4dr ≥ 0 (g) tc < 0, tc 2 − 4dc ≥ 0, tr 2 − 4dr < 0 (h) tc < 0, tc 2 − 4dc ≥ 0, tr 2 − 4dr ≥ 0 (i) tc = 0, tr 2 − 4dr < 0 (j) tc = 0, tr 2 − 4dr ≥ 0 (k) tc > 0, tc 2 − 4dc < 0, tr 2 − 4dr ≥ 0 (l) tc > 0, tc 2 − 4dc ≥ 0, tr 2 − 4dr < 0 Figure 11. Qualitative properties of Ecl and Ecr of system (1.1) with dc > 0 in G1 EJDE-2023/83 GLOBAL ANALYSIS ON A PLANAR PIECEWISE LINEAR SYSTEMS 19 It follows that the qualitative property of Ecl (resp. Ecr) depends on the qual- itative property of the left-hand and right-hand linear zones of the switching line Γl (resp. Γr). The equilibrium point Ecr is an unstable focus as seen from Sc, and is an unstable focus as seen from Sr for tc > 0, tc 2 − 4dc < 0 and tr 2 − 4dr < 0. Therefore, Ecr is an unstable focus in the above case. The equilibrium point Ecr is a stable focus as seen from Sc, but is an unstable focus as seen from Sr for tc < 0, tc 2 − 4dc < 0 and tr 2 − 4dr < 0. Then, Ecr can be a focus or center (see Lemma 5.9). The equilibrium point Ecr is a stable node as seen from Sc, and is an unstable node as seen from Sr. So Ecr is a node-node, as shown in Figure 11(h). More similar results are illustrated in Figure 11 as well. � Lemma 4.2. [21] Equilibrium points at infinity of system (1.1) in the Poincaré disc are shown in Figure 12 in the region G1. t2r − 4dr < 0 t2r − 4dr = 0 t2r − 4dr > 0 Figure 12. Equilibrium points at infinity of system (1.1) in the Poincaré disc in G1 5. Nonlocal dynamics of system (1.1) In this section, we study limit cycles and homoclinic loops of system (1.1) with dc > 0 in the region G1. By Lemma 4.1, Ecl lies on the left switching line Γl and is a half of a saddle in Sl. Namely, system (1.1) has two invariant lines in Sl. We know that system (1.1) has no limit cycles and homoclinic loops when α = −dc < 0. Therefore, we only need to investigate limit cycles and homoclinic loops of system (1.1) when α > −dc and dc > 0. When dc > 0, we separate our discussions into three cases −dc < a < dc, a = dc and α > dc, respectively. 5.1. Limit cycles and homoclinic loops for dc > 0 and −dc < α < dc in G1. According to Lemma 4.1, system (1.1) has two equilibrium points El and Ec when dc > 0 and −dc < α < dc in G1. Moreover, E1 is a saddle and Ec is a node for t2c − 4dc ≥ 0, tc 6= 0 or a focus for t2c − 4dc < 0, tc 6= 0 or a center for tc = 0. Set m := α/dc. Then, the coordinates of Ec can be rewritten as (m, tcm). By making a linear transformation x→ x+m, y → y + tcm, system (1.1) reduces to dx dt = F̃ (x)− y, dy dt = g̃(x), (5.1) 20 M. JIA, Y. SU, H. CHEN EJDE-2023/83 where F̃ (x) =  tr(x+m− 1) + tc(−m+ 1), if x > −m+ 1, tcx, if −m− 1 ≤ x ≤ −m+ 1, tl(x+m+ 1) + tc(−m− 1), if x < −m− 1, g̃(x) =  dr(x+m− 1) + dc(−m+ 1), if x > −m+ 1, dcx, if −m− 1 ≤ x ≤ −m+ 1, dl(x+m+ 1) + dc(−m− 1), if x < −m− 1. It is easy to see that Ec of system (1.1) is moved to O(0, 0) of system (5.1) and El of system (1.1) is moved to Nl(dc(m + 1)/dl −m − 1, tldc(m + 1)/dl − tc(m + 1)) of system (5.1). Clearly, we have −m + 1 > 0 and −m − 1 < 0 because of dc > 0 and −dc < α < dc. The plane R2 is divided into three open linear zones S̃l := {(x, y) ∈ R2 : x < −m− 1}, S̃c := {(x, y) ∈ R2 : −m− 1 < x < −m+ 1}, S̃r := {(x, y) ∈ R2 : x > −m+ 1} by two straight lines Γ̃l := {(x, y) ∈ R2 : x = −m − 1} and Γ̃r := {(x, y) ∈ R2 : x = −m + 1}. For simplicity, for system (5.1) we still use F and g to represent F̃ and g̃, respectively. Note that systems (1.1) and (5.1) are topologically equivalent. Therefore, we only need to study limit cycles and homoclinic loops of system (5.1) to obtain the corresponding results for system (1.1) by a translation. In the following lemma, we show the nonexistence of limit cycles and homoclinic loops of system (1.1) with dc > 0 and −dc < α < dc in G1. Lemma 5.1. Assume that dc > 0 and −dc < α < dc in G1. System (1.1) exhibits neither limit cycles nor homoclinic loops when tc ≥ 0 or tc ≤ t∗c , where t∗c := − tr(α− dc) + tr √ (α− dc)2 + 4αdr − dr dl (α+ dc)2 2dr + tl(α+ dc) 2dl . Proof. It follows from Lemma 4.1 that system (5.1) has two equilibrium points Nl and O when dc > 0 and −dc < α < dc in G1. Moreover, Nl is a saddle, and O is a node for t2c − 4dc ≥ 0, tc 6= 0 or a focus for t2c − 4dc < 0, tc 6= 0 or a center for tc = 0. To study the nonexistence of limit cycles and homoclinic loops of system (5.1), we set a generalized Filippov transformation z(x) := ∫ x 0 g(s)ds, where x1(z) and x2(z) are the branches of the inverse of z(x) for x ≥ 0 and x < 0 respectively. Denote the abscissa of saddle point Nl by xNl := dc(m+1)/dl−m−1. We know that limit cycles of system (5.1) must only surround O by the index theory [38, Chapter 4] if they exist. Therefore, it suffices to consider x ∈ (xNl ,+∞). From EJDE-2023/83 GLOBAL ANALYSIS ON A PLANAR PIECEWISE LINEAR SYSTEMS 21 system (5.1) it follows that z(x) =  drx 2 2 + (dr − dc)(m− 1)x+ 1 2 (dr − dc)(m− 1)2 if x > −m+ 1, dcx 2 2 if −m− 1 ≤ x ≤ −m+ 1, dlx 2 2 + (dl − dc)(m+ 1)x+ 1 2 (dl − dc)(m+ 1)2 if xNl < x < −m− 1. (5.2) Moreover, z(x) ∈  (dc(−m+1)2 2 ,+∞), if x > −m+ 1, [0, dc(−m+1)2 2 ], if 0 ≤ x ≤ −m+ 1, (0, dc(m+1)2 2 ], if −m− 1 ≤ x < 0, (dc(m+1)2 2 , zNl ), if xNl < x < −m− 1, where zNl := dc(dl − dc)(m+ 1)2/(2dl). It follows from (5.2) that x1(z) = {√ 2z dc , if 0 ≤ x ≤ −m+ 1, h1(z)−m+ 1, if x > −m+ 1, (5.3) x2(z) = { − √ 2z dc , if −m− 1 ≤ x < 0, h2(z)−m− 1, if xNl < x < −m− 1, (5.4) with h1(z) := −dc(−m+ 1) + √ (−drdc + d2 c)(−m+ 1)2 + 2drz dr , h2(z) := −dc(−m− 1)− √ (−dldc + d2 c)(−m− 1)2 + 2dlz dl . Let F1(z) := F (x1(z)) and F2(z) := F (x2(z)). Proving that F1(z) > F2(z) or F1(z) < F2(z) for z ∈ (0, zNl ), is equivalent to that there are no limit cycles and homoclinic loops [27, Section 6]. Based on (5.3) and (5.4), we derive F1(z) = tc √ 2z dc , if 0 ≤ z ≤ dc(−m+1)2 2 , trh1(z) + tc(−m+ 1), if z > dc(−m+1)2 2 , (5.5) and F2(z) = −tc √ 2z dc , if 0 < z ≤ dc(m+1)2 2 , tlh2(z) + tc(−m− 1), if dc(m+1)2 2 < z < zNl . (5.6) From (5.5) and (5.6) it follows that F1(z)− F2(z) =  2tc √ 2z dc , if 0 < z ≤ dc(−m+1)2 2 , trh1(z) + tc(−m+ 1) + tc √ 2z dc , if dc(−m+1)2 2 < z ≤ dc(m+1)2 2 , trh1(z)− tlh2(z) + 2tc, if dc(m+1)2 2 < z < zNl (5.7) 22 M. JIA, Y. SU, H. CHEN EJDE-2023/83 for 0 < α < dc, and F1(z)− F2(z) = 2tc √ 2z dc , if 0 < z ≤ dc 2 , tr −dc+ √ −drdc+d2c+2drz dr − tl dc− √ −dldc+d2c+2dlz dl + 2tc, if dc 2 < z < zNl (5.8) for α = 0, and F1(z)− F2(z) =  2tc √ 2z dc , if 0 < z ≤ dc(m+1)2 2 , tc √ 2z dc − tlh2(z) + tc(m+ 1), if dc(m+1)2 2 < z ≤ dc(−m+1)2 2 , trh1(z)− tlh2(z) + 2tc, if dc(−m+1)2 2 < z < zNl (5.9) for −dc < α < 0. We consider the value of F1(z) − F2(z) for 0 < α < dc by three cases tc > 0, tc = 0 and tc < 0, see (5.7). Case (a1) When tc > 0, we have F1(z) − F2(z) > 0 for z ∈ (0, dc(−m+ 1)2/2], and F ′1(z)− F ′2(z) = tr√ (−drdc + d2 c)(−m+ 1)2 + 2drz + tc√ 2dcz > 0 for z ∈ (dc(−m+ 1)2/2, dc(m+ 1)2/2], and F ′1(z)− F ′2(z) = tr√ (−drdc + d2 c)(−m+ 1)2 + 2drz + tl√ (−dldc + d2 c)(−m− 1)2 + 2dlz > 0 for z ∈ (dc(m+ 1)2/2, zNl ). Therefore, we have F1(z)− F2(z) > 0 for z ∈ (0, zNl ). According to [27, Section 6], we know that system (5.1) has no limit cycles and homoclinic loops for tc > 0, 0 < α < dc and dc > 0. Case (a2) When tc = 0, we have F1(z)−F2(z) = 0 for z ∈ (0, dc(−m+ 1)2/2] and F ′1(z)−F ′2(z) > 0 for z ∈ (dc(−m+ 1)2/2, zNl ). Hence, we obtain F1(z)−F2(z) ≥ 0 for z ∈ (0, zNl ), which implies that system (5.1) has no limit cycles and homoclinic loops according to [27, Section 6] when tc = 0, 0 < α < dc and dc > 0. Case (a3) When tc < 0, we have F1(z) − F2(z) < 0 for z ∈ (0, dc(−m+ 1)2/2]. For z ∈ (dc(−m+ 1)2/2, dc(m+ 1)2/2], a routine computation gives rise to F ′1(z)− F ′2(z) = −tc √ 2drz + tc √ (−drdc + d2 c)(−m+ 1)2 + 2drz√ (−drdc + d2 c)(−m+ 1)2 + 2drz · √ 2dcz < 0, if tc < −tr, = 0, if tc = −tr, > 0, if tc > −tr for t2rdc − t2cdr = 0; and F ′1(z)− F ′2(z)  < 0, if z < z0, = 0, if z = z0, > 0, if z > z0 EJDE-2023/83 GLOBAL ANALYSIS ON A PLANAR PIECEWISE LINEAR SYSTEMS 23 for t2rdc − t2cdr > 0; and F ′1(z)− F ′2(z)  > 0, if z < z0, = 0, if z = z0, < 0, if z > z0 for t2rdc − t2cdr < 0, where z0 := t2c(−drdc + d2 c)(−m+ 1)2 2(t2rdc − t2cdr) . (5.10) If t2rdc − t2cdr < 0 and z0 ∈ (dc(−m+ 1)2/2, dc(m+ 1)2/2], then (F1(z)− F2(z))|z=z0 = tr −dc(−m+ 1) + √ (−drdc + d2 c)(−m+ 1)2 + 2drz0 dr + tc(−m+ 1) + tc √ 2z0 dc = (−trdc + tcdr − √ (−dr + dc)(t2rdc − t2cdr) dr ) (−m+ 1) < 0. If tc < tc, then (F1(z)− F2(z))| z= dc(m+1)2 2 = tr −dc(−m+ 1) + √ d2 c(−m+ 1)2 + 4drdcm dr + tc(−m+ 1) + tc(m+ 1) = tr −dc(−m+ 1) + √ d2 c(−m+ 1)2 + 4drdcm dr + 2tc < 0, where tc := − tr(α− dc) + tr √ (α− dc)2 + 4αdr 2dr . As z ∈ (dc(m+ 1)2/2, zNl ), if tc < t∗c , then F ′1(z)− F ′2(z) > 0 and (F1(z)− F2(z))|z=zNl = tr dc(m− 1) + √ d2 c(−m+ 1)2 + 4drdcm− drd2c dl (m+ 1)2 dr − tldc(m+ 1) dl + 2tc < 0. It is easy to check that t∗c < tc. So we have F1(z) − F2(z) < 0 for z ∈ (0, zNl ) if tc < t∗c . According to [27, Section 6], system (5.1) exhibits no limit cycles and homoclinic loops when tc ≤ t∗c , 0 < α < dc and dc > 0. We now consider the value of F1(z) − F2(z) for α = 0 by three cases tc > 0, tc = 0 and tc < 0, see (5.8). Case (b1) When tc > 0, we have F1(z)− F2(z) > 0 for z ∈ (0, dc/2], and F ′1(z)− F ′2(z) = tr√ −drdc + d2 c + 2drz + tl√ −dldc + d2 c + 2dlz > 0, for z ∈ (dc/2, dc(dl − dc)/(2dl)). Therefore, F1(z)− F2(z) > 0 for z in the interval (0, dc(dl − dc)/(2dl)). It follows from [27, Section 6] that system (5.1) exhibits no limit cycles and homoclinic loops when tc > 0, α = 0 and dc > 0. Case (b2) When tc = 0, we have F1(z)− F2(z) = 0 for z ∈ (0, dc/2] and F ′1(z)− F ′2(z) > 0 for z ∈ (dc/2, zNl ). It implies that F1(z) − F2(z) ≥ 0 for z ∈ (0, zNl ). 24 M. JIA, Y. SU, H. CHEN EJDE-2023/83 Therefore, system (5.1) exhibits no limit cycles and homoclinic loops [27, Section 6], when tc = 0, α = 0 and dc > 0. Case (b3) When tc < 0, it follows that F1(z) − F2(z) < 0 for z ∈ (0, dc/2] and F ′1(z)− F ′2(z) > 0 for z ∈ (dc/2, dc(dl − dc)/(2dl)). Furthermore, we have (F1(z)− F2(z))| z= dc(dl−dc) 2dl = tr −dc + √ d2 c − drd2c dl dr − tldc dl + 2tc < 0, when tc < − −trdc + tr √ d2 c − drd2c dl 2dr + tldc 2dl . Note that t∗c with α = 0 is t∗c − ( −trdc + tr √ d2 c − drd2 c/dl ) /(2dr) + tldc/(2dl). Therefore, we obtain F1(z) − F2(z) < 0 for z ∈ (0, dc(dl − dc)/(2dl)) if tc < t∗c . Based on [27, Section 6], system (5.1) has no limit cycles and homoclinic loops when tc ≤ t∗c , α = 0 and dc > 0. Next, we discuss the sign of F1(z)−F2(z) for −dc < α < 0 by three cases tc > 0, tc = 0 and tc < 0, see (5.9). Case (c1) When tc > 0, we have F1(z)− F2(z) > 0 for z ∈ (0, dc(m+ 1)2/2], and F ′1(z)− F ′2(z) = tl√ (−dldc + d2 c)(−m− 1)2 + 2dlz + tc√ 2dcz > 0, for z ∈ (dc(m+ 1)2/2, dc(−m+ 1)2/2], and F ′1(z)− F ′2(z) = tr√ (−drdc + d2 c)(−m+ 1)2 + 2drz + tl√ (−dldc + d2 c)(−m− 1)2 + 2dlz > 0 for z ∈ (dc(−m+ 1)2/2, zNl ). Thus, we obtain F1(z) − F2(z) > 0 for z ∈ (0, zNl ). From [27, Section 6], it follows that system (5.1) exhibits no limit cycles and ho- moclinic loops for tc > 0, −dc < α < 0 and dc > 0. Case (c2). When tc = 0, we find F1(z)− F2(z) = 0 for z ∈ (0, dc(m+ 1)2/2] and F ′1(z)−F ′2(z) > 0 for z ∈ (dc(m+ 1)2/2, zNl ), which means that F1(z)−F2(z) ≥ 0 for z ∈ (0, zNl ). Then, system (5.1) has no limit cycles and homoclinic loops according to [27, Section 6], when tc = 0, −dc < α < 0 and dc > 0. Case (c3). When tc < 0, we find F1(z)− F2(z) < 0 for z ∈ (0, dc(m+ 1)2/2]. For z ∈ (dc(m+ 1)2/2, dc(−m+ 1)2/2], a routine computation gives rise to F ′1(z)− F ′2(z)  < 0, if z < z0, = 0, if z = z0, > 0, if z > z0, because t2l dc − t2cdl > 0, where z0 is given in (5.10). When tc < t̂c, it follows that (F1(z)− F2(z))| z= dc(−m+1)2 2 = tc(−m+ 1)− tc(−m− 1)− tl −dc(−m− 1)− √ d2 c(m+ 1)2 − 4dldcm dl EJDE-2023/83 GLOBAL ANALYSIS ON A PLANAR PIECEWISE LINEAR SYSTEMS 25 = tl dc(−m− 1) + √ d2 c(m+ 1)2 − 4dldcm dl + 2tc < 0, where t̂c := tl(α+ dc)− tl √ (α+ dc)2 − 4αdl 2dl . For z ∈ (dc(m+ 1)2/2, zNl ), it holds that F ′1(z)− F ′2(z) > 0. If tc < t∗c , then F1(z)− F2(z)| z= dc(dl−dc)(m+1)2 2dl < 0. We see that t∗c < t̂c implies F1(z)−F2(z) < 0 for z ∈ (0, zNl ) if tc < t∗c . According to [27, Section 6], system (5.1) exhibits no limit cycles and homoclinic loops when tc ≤ t∗c , −dc < α < 0 and dc > 0. From Cases (a1), (a2), (b1), (b2), (c1), and (c2), we obtain that system (1.1) exhibits neither limit cycles nor homoclinic loops when tc ≥ 0, −dc < α < dc and dc > 0 in G1. System (1.1) has neither limit cycles nor homoclinic loops by Cases (a3), (b3) and (c3), when tc ≤ t∗c , −dc < α < dc, and dc > 0 in G1. � From Lemma 5.1, it suffices to study the existence of limit cycles of system (1.1) when dc > 0, −dc < α < dc and t∗c < tc < 0 in G1. From Lemma 4.1, we know that system (1.1) has two equilibrium points El and Ec when dc > 0, −dc < α < dc and t∗c < tc < 0 in G1. Moreover, E1 is a saddle and Ec is a stable node for t2c − 4dc ≥ 0 or a stable focus for t2c−4dc < 0. Since Ec lies in Sc and limit cycles of system (1.1) must only surround Ec by the index theory [38, Chapter 4], we obtain the existence of small limit cycles of system (1.1) when dc > 0, −dc < α < dc and t∗c < tc < 0 in G1 as follows. Lemma 5.2. Assume that dc > 0 and −dc < α < dc in G1. If t∗c < tc < 0, then the following two assertions hold. (a) System (1.1) exhibits no small limit cycles when tc 2 − 4dc ≥ 0. (b) System (1.1) exhibits at most one small limit cycle that is unstable when tc 2− 4dc < 0. The small limit cycle lies in Sl ∪Γl ∪Sc if tc > tldc/dl or in Sc ∪ Γr ∪ Sr if tc > −tr √ dc/dr. Proof. When dc > 0, −dc < α < dc and tc < 0 in G1, it follows from Lemma 4.1 that system (5.1) has two equilibrium points Nl and O, Nl is a saddle, and O is a stable node for tc 2−4dc ≥ 0 or a stable focus for tc 2−4dc < 0. Applying the index theory [38, Chapter 4], we know that limit cycle of system (5.1) must surround O if it exists. Therefore, we study the existence of small limit cycles of system (5.1) by analyzing the dynamical behavior of O. Given tc 2 − 4dc ≥ 0, since O is a stable node, system (5.1) has at least one invariant line in S̃c, namely, there is no small limit cycle. Thus we arrive at the desire result (a). Given tc 2 − 4dc < 0, we note that small limit cycle of system (5.1) lies in S̃l∪ Γ̃l∪S̃c or in S̃c∪ Γ̃r∪S̃r if it exists. We discuss it by classification in what follows. Firstly, we denote the small limit cycle of system (5.1) by Γ1. The small limit cycle Γ1 lies in S̃l ∪ Γ̃l ∪ S̃c, as shown in Figure 13(a). Moreover, A1, B1, C1, D1, E1, F1, G1, and H1 are points on Γ1, the points A1, G1 lie on the line x = 0, the points B1, F1 lie on the left switching line Γ̃l, the points C1, E1 lie on the line x = tc(m + 1)/tl − m − 1, and D1, H1 are the intersection points of Γ1 and the curve y = F (x). We claim that the intersection point of Γ1 and the negative x-axis lies in the region {x ∈ R|xNl < x < x∗}, where xNl := dc(m + 1)/dl − m − 1 is 26 M. JIA, Y. SU, H. CHEN EJDE-2023/83 Small limit cycle of Γ1 of (5.1) Γ1 lies in the (p− y)-plane Figure 13. Small limit cycles of system (5.1) in S̃l ∪ Γ̃l ∪ S̃c the abscissa of saddle point Nl and x∗ := tc(m+ 1)/tl −m− 1 satisfies F (x∗) = 0. Otherwise, we have ∮ Γ1 dE = g(x)F (x) < 0, when the intersection point of Γ1 and the negative x-axis lies in the region {x ∈ R|x∗ ≤ x < 0}, where E(x, y) := ∫ x 0 g(s)ds+ y2 2 is an energy function along the vector field of system (5.1). This contradicts the fact that ∮ Γ1 dE = 0. To prove that ∮ Γ1 F ′(x)dt > 0, (5.11) which indicates that the small limit cycle Γ1 is unstable and hyperbolic [7, Propo- sisition 3.1], we let p := F (x) and denote by x1(p) and x2(p) the branches of the inverse of p(x) for −m − 1 < x < −m + 1 and xNl < x < −m − 1, respectively. From system (5.1) it follows that x1(p) = p tc , for tc(−m+ 1) < p < −tc(m+ 1), x2(p) = p tl + (tc − tl)(m+ 1) tl , for − tc(m+ 1) + tldc(m+ 1) dl < p < −tc(m+ 1). (5.12) We re-write (5.1) as dy dp = λ1(p) p− y , p ∈ (tc(−m+ 1), −tc(m+ 1)), dy dp = λ2(p) p− y , p ∈ (−tc(m+ 1) + tldc(m+ 1) dl , −tc(m+ 1)), (5.13) EJDE-2023/83 GLOBAL ANALYSIS ON A PLANAR PIECEWISE LINEAR SYSTEMS 27 where λ1(p) := g(x1) F ′(x1) = dcp t2c , λ2(p) := g(x2) F ′(x2) = dlp t2l + (tcdl − tldc)(m+ 1) t2l . If λ1(p) = λ2(p), then p0 := t2c(tcdl − tldc)(m+ 1) t2l dc − t2cdl . (5.14) From the sign of ẏ of system (5.1), we know that yA1 > yB1 > yC1 > 0 and 0 > yE1 > yF1 > yG1 , where yA1 , yB1 , yC1 , yE1 , yF1 and yG1 are the ordinates of A1, B1, C1, E1, F1 and G1, respectively. It means that p0 < 0 if it exists. Then, tc > tldc/dl. Denote x∗1 := x1(p0) and x∗2 := x2(p0). Evidently, we have xNl < x∗2 < tc(m+ 1) tl −m− 1 < 0 < x∗1 < −m+ 1. (5.15) We claim that p0 exists and p0 < 0, i.e., Γ1 intersects with the lines x = x∗1 and x = x∗2. Otherwise, it follows from the comparison theorem [11, Corollary 6.3] that it contradicts the existence of the small limit cycle Γ1. We denote Γ1 := ∪4 i=1Γ1i, where Γ11, Γ12, Γ13 and Γ14 are the parts of Γ1 located in the regions 0 ≤ x < −m+ 1, −m− 1 ≤ x < 0, x∗ ≤ x < −m− 1, and xNl < x < x∗, respectively. The small limit cycle Γ1 in the (p, y)-plane is indicated in Figure 13(b). We have∫ Γ12 F ′(x)dt+ ∫ Γ13 F ′(x)dt = ∫ −tc(m+1) 0 dp p− y1 − ∫ −tc(m+1) 0 dp p− z1 − ∫ −tc(m+1) 0 dp p− y2 + ∫ −tc(m+1) 0 dp p− z2 = ∫ −tc(m+1) 0 (y1 − y2) (p− y1)(p− y2) + ∫ −tc(m+1) 0 (z2 − z1) (p− z1)(p− z2) > 0, (5.16) where y1 and y2 represent the orbit segments Γ12 and Γ13 respectively, which lie above y = p, and z1 and z2 represent the orbit segments Γ12 and Γ13 respectively, which lie below y = p. We now compare two integral curves Γ11 and Γ14. By a change of coordinate transformation p → µp, y → µy with µ = yD1 /yH1 , Γ11 (ÂD1G) is an orbit segment crossing through D1, where p ≤ 0, and A and G are points on Γ11 and both lie on the y-axis. The functions y1 and z1 represent the orbit segments Γ11 respectively, which lie above and below the line y = p. Therefore,∫ Γ11 F ′(x)dt = ∫ ̂G1H1A1 F ′(x)dt = ∫ ĜD1A F ′(x)dt = ∫ Γ11 F ′(x)dt. (5.17) Note that y1(pD1) = y2(pD1) = z1(pD1) = z2(pD1), y1(0) > y2(0) and z1(0) < z2(0). By the uniqueness of zero of the equation λ1(p) = λ2(p), we have y1(p) > 28 M. JIA, Y. SU, H. CHEN EJDE-2023/83 y2(p) and z1(p) < z2(p) for p ∈ (pD1 , 0). Then∫ Γ11 F ′(x)dt+ ∫ Γ14 F ′(x)dt = − ∫ pD1 0 dp p− y1 + ∫ pD1 0 dp p− z1 + ∫ pD1 0 dp p− y2 − ∫ pD1 0 dp p− z2 = ∫ pD1 0 (y2 − y1) (p− y1)(p− y2) + ∫ pD1 0 (z1 − z2) (p− z1)(p− z2) > 0. (5.18) So, we see that (5.11) follows from (5.16)-(5.18) immediately when tc > tldc/dl. We claim that system (5.1) has at most one small limit cycle in S̃l ∪ Γ̃l ∪ S̃c. Otherwise, there are at least two small limit cycles denoted by Γs1 and Γs2 in S̃l ∪ Γ̃l ∪ S̃c, where Γs1 and Γs2 are adjacent to each other. From (5.11) it follows that ∮ Γs1 F ′(x)dt > 0, ∮ Γs2 F ′(x)dt > 0. In other words, Γs1 and Γs2 are all unstable [7, Proposisition 3.1]. According to the Poincaré-Bendixson Theorem, there exists a stable limit cycle between Γs1 and Γs2, which contradicts (5.11). Small limit cycle Γ2 of (5.1) Γ2 lies in the (p, y)-plane Figure 14. Small limit cycles of system (5.1) in S̃c ∪ Γ̃r ∪ S̃r Secondly, we take the small limit cycle of system (5.1) that lies in S̃c ∪ Γ̃r ∪ S̃r, denoted by Γ2, as shown in Figure 14(a). Note that A2, B2, C2, D2, E2, F2, G2, H2 are points on Γ2, the points A2, G2 lie on the line x = 0, the points B2, F2 lie on the right switching line Γ̃r, the points C2, E2 lie on the line x = tc(m− 1)/tr −m+ 1, and D2, H2 are the intersection points of Γ2 and the curve y = F (x). Set p := F (x). Then, the limit cycle Γ2 in the (p, y)-plane is displayed in Figure 14(b). Similarly, we can prove that ∮ Γ2 F ′(x)dt > 0, (5.19) if tc > −tr √ dc/dr, and system (5.1) has at most one small limit cycle in S̃c∪Γ̃r∪S̃r. In addition, it is impossible for a limit cycle to lie in S̃c ∪ Γ̃r ∪ S̃r, when a limit cycle lies in S̃l ∪ Γ̃l ∪ S̃c. The proof is complete. � EJDE-2023/83 GLOBAL ANALYSIS ON A PLANAR PIECEWISE LINEAR SYSTEMS 29 Lemma 5.3. Assume that dc > 0 and −dc < α < dc in G1. If t∗c < tc < 0, then the following two assertions hold: (a) System (1.1) exhibits at most one limit cycle when tc 2−4dc ≥ 0. Moreover, the limit cycle is large if it exists. (b) System (1.1) exhibits at most two limit cycles when tc 2−4dc < 0. Moreover, it consists of a small limit cycle and a large limit cycle if they exist. (a) (b) Figure 15. (a) Two large limit cycles Γ3 and Γ4 of system (5.1); (b) Orbit segments B̂3D3F3 and B̂4D4F4. Proof. It follows from Lemma 4.1 that system (5.1) has two equilibrium points Nl and O as dc > 0, −dc < α < dc and tc < 0 in G1. A straightforward calculation shows that Nl is a saddle, and O is a stable node for tc 2− 4dc ≥ 0 or a stable focus for tc 2−4dc < 0. By the index theory [38, Chapter 4], we know that the limit cycle of system (5.1) only surrounds O if it exists. Based on Lemma 5.2, we continue to investigate the existence of large limit cycles of system (5.1). We take any two large limit cycles of system (5.1) denoted by Γ3 and Γ4 that are adjacent to each other, as indicated in Figure 15(a). The large limit cycle Γ3 is the innermost one, Ai, Bi, Ci, Di, Ei, Fi, Gi, Hi, Ii, Ji,Ki, Li are points on Γi, the abscissa of Bi, Fi is −m − 1, the abscissa of Ci, Ei is tc(m + 1)/tl − m − 1, the abscissa of Hi, Li is m− 1, and the abscissa of Ii,Ki is tc(m− 1)/tr −m+ 1 for i = 3, 4. Following [9], we now prove that ∮ Γ3 F ′(x)dt < ∮ Γ4 F ′(x)dt. (5.20) Let y3(x) and y4(x) represent the orbit segments ̂L3A3B3 and ̂L4A4B4, respec- tively. It follows that∫ ̂L3A3B3 F ′(x)dt− ∫ ̂L4A4B4 F ′(x)dt = ∫ −m−1 m−1 F ′(x) F (x)− y3(x) dx− ∫ −m−1 m−1 F ′(x) F (x)− y4(x) dx = ∫ −m−1 m−1 F ′(x)(y3(x)− y4(x)) (F (x)− y3(x))(F (x)− y4(x)) dx < 0, (5.21) 30 M. JIA, Y. SU, H. CHEN EJDE-2023/83 because of F ′(x) = tc < 0, y3(x)− y4(x) < 0 and (F (x)− y3(x))(F (x)− y4(x)) > 0 for x ∈ (−m− 1,m− 1). Similarly, we can obtain∫ ̂F3G3H3 F ′(x)dt− ∫ ̂F4G4H4 F ′(x)dt < 0. (5.22) Consider a system without the switching lines dx dt = tl(x+m+ 1) + tc(−m− 1)− y := F (x)− y, dy dt = dl(x+m+ 1) + dc(−m− 1) := g(x), (5.23) in the region {(x, y) ∈ R2 : x ≤ 0}. Since systems (5.1) and (5.23) are the same in the region {(x, y) ∈ R2 : x ≤ −m − 1}, we turn to study two orbit segments B̂3D3F3 and B̂4D4F4 in system (5.23) for simplicity, as displayed in Figure 15(b). We denote by Pi and Qi the first intersection point of the orbit segment B̂iDiFi and y-axis for i = 3, 4 respectively, as t decreases and increases. To show that∫ ̂P3D3Q3 F ′ (x)dt = ∫ ̂P4D4Q4 F ′ (x)dt, (5.24) using p = F (x) we re-write system (5.23) as dy dp = dlp+ tcdl(m+ 1)− tldc(m+ 1) t2l (p− y) . (5.25) By making a change of coordinate transformation p → µp, y → µy with µ = yD4/yD3 , ̂P3D3Q3 becomes an orbit segment crossing through D4 of system (5.25). Therefore, we obtain (5.24). A routine computation gives rise to∫ P̂3B3 F ′ (x)dt− ∫ P̂4B4 F ′ (x)dt = ∫ −m−1 0 F ′ (x) F (x)− y3(x) dx− ∫ −m−1 0 F ′ (x) F (x)− y4(x) dx = ∫ −m−1 0 F ′ (x) (y3(x)− y4(x))( F (x)− y3(x) ) ( F (x)− y4(x) )dx > 0, (5.26) where y3(x) and y4(x) represent the orbit segments P̂3B3 and P̂4B4 respectively. Combining (5.24) and (5.26) yields∫ ̂B3D3F3 F ′(x)dt− ∫ ̂B4D4F4 F ′(x)dt < 0. (5.27) Similarly, we have ∫ Ĥ3J3L3 F ′(x)dt− ∫ Ĥ4J4L4 F ′(x)dt < 0. (5.28) Hence, inequality (5.20) follows from (5.21)-(5.22) and (5.27)-(5.28) immediately. (a) When tc 2 − 4dc ≥ 0, we suppose that system (5.1) has at least two large limit cycles denoted by Γl1 and Γl2, where Γl1 and Γl2 are adjacent to each other and Γl1 lies in the interior of Γl2. By Lemma 5.2, system (5.1) has no small limit cycles EJDE-2023/83 GLOBAL ANALYSIS ON A PLANAR PIECEWISE LINEAR SYSTEMS 31 when tc 2− 4dc ≥ 0. It means that Γl1 is internally unstable. From (5.20) it follows that ∮ Γl1 F ′(x)dt = 0, ∮ Γl2 F ′(x)dt > 0. In other words, Γl1 is semi-stable (internally unstable and externally stable) and Γl2 is unstable [7, Proposisition 3.1]. When a1 < a2, we obtain∣∣∣∣F (x)|tc=a1 − y g(x) F (x)|tc=a2 − y g(x) ∣∣∣∣ (−m+ 1)g(x)(a1 − a2) ≤ 0, x > −m+ 1, xg(x)(a1 − a2) < 0, −m− 1 ≤ x ≤ −m+ 1, −(m+ 1)g(x)(a1 − a2) < 0, dc(m+ 1) dl −m− 1 < x < −m− 1, where the equality sign obviously cannot hold for the entire closed orbit of system (5.1). Hence, the vector field (F (x) − y, g(x)) of system (5.1) is rotated about tc by [22, Definition 1.6] or [38, Definition 3.3]. It follows from [38, Theorem 3.4] that the semi-stable limit cycle Γl1 will bifurcate into at least one unstable limit cycle and one stable limit cycle when tc varies in the suitable direction. This contradicts (5.20). Thus, we infer that system (5.1) has at most one limit cycle when tc 2 − 4dc ≥ 0. Moreover, the limit cycle is large if it exists. (b) When tc 2 − 4dc < 0, we know that system (5.1) has at most one small limit cycle by Lemma 5.2 and the small limit cycle is unstable if it exists. If system (5.1) has no small limit cycles for tc 2 − 4dc < 0, the proof of which there is at most one large limit cycle is the same as the proof of Part (a), Therefore, we omit it. We denote the small limit cycle of system (5.1) by γs. We only need to study the existence of large limit cycles when γs exists. Assume that system (5.1) exhibits at least three large limit cycles when γs exists, denoted by γl1, γl2 and γl3, where γl1 lies in the interior of γl2 and γl2 lies in the interior of γl3. Since the small limit cycle γs is unstable, by (5.20) we have∮ γl1 F ′(x)dt < 0, ∮ γl2 F ′(x)dt = 0, ∮ γl3 F ′(x)dt > 0. In other words, γl1 is stable, γl2 is semi-stable (internally unstable and externally stable) and γl3 is unstable [7, Proposisition 3.1]. When tc varies in the suitable direction, it follows from [38, Theorem 3.4] that the semi-stable limit cycle γl2 will bifurcate into at least one unstable limit cycle and one stable limit cycle. This contradicts (5.20). We then suppose that system (5.1) has two large limit cycles when γs exists, denoted by γ1 and γ2, where γ1 lies in the interior of γ2. Since the small limit cycle γs is unstable, we find that γ1 is internally stable. If γ1 or γ2 is semi-stable, then there are three large limit cycles by [38, Theorem 3.4] when tc varies in the suitable direction. This is a contradiction. Therefore, we infer that γ1 is stable and γ2 is unstable, i.e., ∮ γ1 F ′(x)dt < 0, ∮ γ2 F ′(x)dt > 0. According to [38, Theorem 3.3], it follows that the attracting limit cycle of system (5.1) contracts and the repelling limit cycle of system (5.1) expands when tc de- creases. Therefore, γs and γ1 will coincide and become a semi-stable limit cycle 32 M. JIA, Y. SU, H. CHEN EJDE-2023/83 γ (internally unstable and externally stable) when tc decreases to a certain value. If tc continues to decrease, then the semi-stable limit cycle γ will disappear by [38, Theorem 3.3] again. It implies that system (5.1) has one limit cycle γ2 when tc → −∞. This contradicts the nonexistence of limit cycles of system (1.1) when tc → −∞ (see Lemma 5.1). Therefore, we have arrived at the desired result. � From Lemma 5.1, we only need to study homoclinic loops of system (1.1) when dc > 0, −dc < α < dc and t∗c < tc < 0 in G1. It follows from Lemma 4.1 that El of system (1.1) is a saddle and Ec of system (1.1) is a stable focus for tc 2− 4dc < 0 or a stable node for tc 2−4dc ≥ 0 when dc > 0, −dc < α < dc and t∗c < tc < 0 in G1. It is notable that t∗c may be greater than or less than or equal to −2 √ dc. Therefore, we study homoclinic loops of system (1.1) for two cases max{t∗c ,−2 √ dc} < tc < 0 and t∗c < tc ≤ −2 √ dc. In the case of max{t∗c ,−2 √ dc} < tc < 0, homoclinic loops of system (1.1) may involve two or three linear zones because Ec is a stable focus. In the case of t∗c < tc ≤ −2 √ dc, homoclinic loops of system (1.1) only involve three linear zones because Ec is a stable node, so there is invariant line in Sc. Lemma 5.4. When dc > 0, −dc < α < dc and max{t∗c ,−2 √ dc} < tc < 0 in G1, there is a continuous function tc = φ(α) ∈ (max{t∗c ,−2 √ dc}, 0) and a value α∗ ∈ (−dc, dc) such that the following three assertions are true. (a) System (1.1) exhibits a unique homoclinic loop involving two linear zones when tc = φ(α) for α < α∗, and the homoclinic loop is unstable. (b) System (1.1) exhibits a unique homoclinic loop that is tangent to Γr when tc = φ(α) for α = α∗, and the homoclinic loop is unstable. (c) System (1.1) exhibits a unique homoclinic loop involving three linear zones when tc = φ(α) for α > α∗, and the homoclinic loop unstable. Proof. For simplicity, we consider a continuous planar piecewise linear differential system with two zones: dx dt = F (x)− y, dy dt = g(x), (5.29) where F (x) = { tcx, if x ≥ −m− 1, tl(x+m+ 1) + tc(−m− 1), if x < −m− 1, g(x) = { dcx, if x ≥ −m− 1, dl(x+m+ 1) + dc(−m− 1), if x < −m− 1. Note that systems (5.1) and (5.29) are the same in the region {(x, y) ∈ R2 : x ≤ −m + 1}. Therefore, we consider the existence, uniqueness and stability of homo- clinic loops of system (5.29) in the region {(x, y) ∈ R2 : x ≤ −m + 1}. System (5.29) has two equilibrium points Nl and O when dc > 0 and −dc < α < dc in G1, where Nl is a saddle and O is a stable focus if −2 √ dc < tc < 0. To prove the existence of homoclinic loops of system (5.29), we denote the eigen- values of the corresponding Jacobian matrix at the saddle point N1 by λ1 and λ2, where λ1 := tl − √ t2l − 4dl 2 , λ2 := tl + √ t2l − 4dl 2 , λ1 + λ2 = tl, λ1λ2 = dl. (5.30) EJDE-2023/83 GLOBAL ANALYSIS ON A PLANAR PIECEWISE LINEAR SYSTEMS 33 We then obtain that the λ1-eigenvector and λ2-eigenvector of JN1 := [ t1 −1 d1 0 ] = [ λ1 + λ2 −1 λ1λ2 0 ] are (1, λ2)T and (1, λ1)T respectively. Denote the intersection point of the linear λ1 (resp. λ2) invariant manifold that emanates from the saddle point Nl along the eigenvector (1, λ2)T (resp. (1, λ1)T ) with the line x = −m− 1 by E : (−m− 1, y1) (resp. A : (−m− 1, z1)). That is,[ α+dc λ1λ2 −m− 1 (λ1+λ2)(α+dc) λ1λ2 − tc(m+ 1) ] + ρ [ 1 λ2 ] = [ −m− 1 y1 ] (5.31) and [ α+dc λ1λ2 −m− 1 (λ1+λ2)(α+dc) λ1λ2 − tc(m+ 1) ] + ρ [ 1 λ2 ] = [ −m− 1 z1 ] , (5.32) where ρ = −α+dc λ1λ2 . It follows from (5.31) and (5.32) that z1 = α+ dc λ1 − tc(m+ 1), y1 = α+ dc λ2 − tc(m+ 1). We denote the eigenvalues of the stable focus point O of system (5.29) by σ±ωi, where σ = tc 2 , ω = √ 4dc − t2c 2 , σ2 + ω2 = dc. Set η := σ ω = tc√ 4dc−t2c . Let (xb, yb) be the initial point of an orbit of system (5.29) with x ≥ −m − 1 and (xf , yf ) be the terminal point after a time t. By the phase angle θ = −ωt, we have( xf yf ) = e−ηθ ( − 1 dc cos θ + η dc sin θ 1 dc sin θ + η dc cos θ sin θ ω cos θ ω )( xb yb ) . (5.33) When E : (−m− 1, y1) is the beginning point for an orbit, we want to know the point D : (0, y2) along the orbit as t decreases. From (5.33) it follows that( 0 y2 ) = e−ηθ ( − 1 dc cos θ + η dc sin θ 1 dc sin θ + η dc cos θ sin θ ω cos θ ω )( −m− 1 y1 ) (5.34) From the first coordinate of (5.34) we obtain tan θ = ηy1 +m+ 1 −y1 + ηm+ η , sin θ = √ (ηy1 +m+ 1)2 (η2 + 1)[y2 1 + (m+ 1)2] , cos θ = √ (−y1 + ηm+ η)2 (η2 + 1)[y2 1 + (m+ 1)2] . From the second coordinate of (5.34) we derive y2 = e−ηθ ( − m+ 1 ω √ (ηy1 +m+ 1)2 (η2 + 1)[y2 1 + (m+ 1)2] + y1 ω √ (−y1 + ηm+ η)2 (η2 + 1)[y2 1 + (m+ 1)2] ) . 34 M. JIA, Y. SU, H. CHEN EJDE-2023/83 When A : (−m− 1, z1) is the beginning point for an orbit, we want to know the point B (0, z2) along the orbit as t increases. From (5.33) it follows that( 0 z2 ) = e−ηθ ( − 1 dc cos θ + η dc sin θ 1 dc sin θ + η dc cos θ sin θ ω cos θ ω )( −m− 1 z1 ) . Similarly, we have z2 = e−ηθ (m+ 1 ω √ (ηz1 +m+ 1)2 (η2 + 1)[z2 1 + (m+ 1)2] + z1 ω √ (−z1 + ηm+ η)2 (η2 + 1)[z2 1 + (m+ 1)2] ) . We define the Poincaé map P : (0, z2)→ (0, z∗) for system (5.29) with x ≥ 0, where z2 and z∗ lie on the same orbit. We have P (z∗) = −e−ηπz2 with z∗ = −e−ηπe−ηθ (m+ 1 ω √ (ηz1 +m+ 1)2 (η2 + 1)[z2 1 + (m+ 1)2] + z1 ω √ (−z1 + ηm+ η)2 (η2 + 1)[z2 1 + (m+ 1)2] ) . If z∗ = y2, then e−ηπ (m+ 1 ω √ (ηz1 +m+ 1)2 (η2 + 1)[z2 1 + (m+ 1)2] + z1 ω √ (−z1 + ηm+ η)2 (η2 + 1)[z2 1 + (m+ 1)2] ) = −m+ 1 ω √ (ηy1 +m+ 1)2 (η2 + 1)[y2 1 + (m+ 1)2] + y1 ω √ (−y1 + ηm+ η)2 (η2 + 1)[y2 1 + (m+ 1)2] . (5.35) Thus, system (5.29) has a homoclinic loop. Note that system (5.29) is piecewise linear, Lipschitz continuous in R2 and ana- lytic in R2\{(x, y) ∈ R2 : x = −m−1}. By [12, Theorem 3.3] and λ+ +λ− = tl > 0, the homoclinic loop of system (5.29) is unstable. As a1 < a2, it follows that∣∣∣∣F (x)|tc=a1 − y g(x) F (x)|tc=a2 − y g(x) ∣∣∣∣ = { xg(x)(a1 − a2) ≤ 0 for x ≥ −m− 1, −(m+ 1)g(x)(a1 − a2) < 0, for dc(m+1) dl −m− 1 < x < −m− 1, where the equality sign cannot hold for the entire closed orbit of system (5.29). Hence, the vector field (F (x) − y, g(x)) of system (5.29) is rotated about tc [22, Definition 1.6] or [38, Definition 3.3]. From [21, Lemma 5.3], it follows that the manifolds W+ N and W−N of the saddle point Nl of system (5.29) rotate clockwise when tc increases and tl, dc, dl, α are fixed, where W+ N is the stable manifold of the right-hand side of Nl and W−N is the unstable manifold of the right-hand side of Nl. This indicates that system (5.29) exhibits at most one homoclinic loop. To prove that the amplitude of the homoclinic loop of system (5.29) increases as α increases, we use the transformation x→ x−m− 1, y → y − tc(m+ 1), EJDE-2023/83 GLOBAL ANALYSIS ON A PLANAR PIECEWISE LINEAR SYSTEMS 35 to change (5.29) to dx dt = F̂ (x)− y, dy dt = ĝ(x), (5.36) where F̂ (x) = { tcx, if x ≥ 0, tlx, if x < 0, ĝ(x) = { dc(x−m− 1), if x ≥ 0, dlx+ dc(−m− 1), if x < 0. By the scaling transformation x→ dc(m+ 1)x, y → dc(m+ 1)y, (5.37) system (5.36) reduces to dx dt = F (x)− y, dy dt = g(x), (5.38) where F (x) = { tcx, if x ≥ 0, tlx, if x < 0, g(x) = { dcx− 1, if x ≥ 0, dlx− 1, if x < 0. Evidently, system (5.38) is independent on α, so the homoclinic loop of system (5.38) is independent on α as well. Since systems (5.29) and (5.38) are topologically equivalent, it follows from the scaling transformation (5.37) that the amplitude of the homoclinic loop of system (5.29) is increasing as α increases. tc = φ(α) for α < α∗ tc = φ(α) for α = α∗ tc = φ(α) for α > α∗ Figure 16. Homoclinic loops of system (5.1) We denote by C:(xc, tcxc) the right intersection point of the homoclinic loop of system (5.29) with the curve y = F (x) If the point C is on the line x = −m + 1, then t2c 2 − dc = dctc(−m+ 1)ω e−ηπz2 . (5.39) By solving equations (5.35) and (5.39) and using the implicit function theorem, there exists a continuous function tc = φ(α) and a value α∗ such that system (5.29) has one homoclinic loop that is tangent to the line x = −m+ 1 when tc = φ(α) for α = α∗, where φ(α) ∈ (max{t∗c ,−2 √ dc}, 0) and α∗ ∈ (−dc, dc). Since systems (5.1) 36 M. JIA, Y. SU, H. CHEN EJDE-2023/83 and (5.29) are the same in the region {(x, y) ∈ R2 : x ≤ −m+ 1}, we can plot the homoclinic loop for system (5.1) which is unstable, see Figure 16(b). In view of the monotonicity of the amplitude of the homoclinic loop of system (5.1) in the region {(x, y) ∈ R2 : x ≤ −m+ 1}, we are led to the conclusion that the homoclinic loop of system (5.1) involves two linear zones when tc = φ(α) for α < α∗, as shown in Figure 16(a). In addition, given the existence, uniqueness and continuity of system (5.1) under the initial conditions, we can obtain that the homoclinic loop of system (5.1) involves three linear zones when tc = φ(α) for α > α∗, as shown in Figure 16(c). The proof of Lemma 5.4 is completed. � The following lemma is concerning the existence, uniqueness and stability of homoclinic loops of system (1.1) when dc > 0, −dc < α < dc and t∗c < tc ≤ −2 √ dc in G1. The proof is closely similar to the one of Lemma 5.4, so we omit it. Lemma 5.5. If dc > 0, −dc < α < dc and t∗c < tc ≤ −2 √ dc in G1, then system (1.1) exhibits a unique homoclinic loop involving three linear zones that is unstable when tc = φ(α) for α > α∗, where tc = φ(α) ∈ (t∗c ,−2 √ dc] is a continuous function and α∗ ∈ (−dc, dc) is a constant. For the uniqueness of limit cycle of system (1.1) with dc > 0 and −dc < α < dc, we have the following lemma. Lemma 5.6. If dc > 0, −dc < α < dc and max{t∗c ,−2 √ dc} < tc < 0 in G1, then there is a continuous function tc = φ(α) ∈ (max{t∗c ,−2 √ dc}, 0) such that (a) system (1.1) exhibits no limit cycles when tc ∈ (max{t∗c ,−2 √ dc}, φ(α)]; and (b) system (1.1) exhibits a unique limit cycle that is unstable when tc ∈ (φ(α), 0). Proof. When dc > 0, −dc < α < dc and max{t∗c ,−2 √ dc} < tc < 0 in G1, system (1.1) has two equilibrium points El and Ec by Lemma 4.1, where El is a saddle and Ec is a stable focus. Denote by W+ E1 and W−El the stable and unstable manifolds of the right-hand side of El of system (1.1), respectively. According to Lemma 5.4, system (1.1) exhibits a unique homoclinic loop that is unstable when tc = φ(α) ∈ (max{t∗c ,−2 √ dc}, 0), see Figure 17(b). However, the unstable homoclinic loop breaks when tc = φ(α)± ε, even ε > 0 is sufficiently small. According to [21, Lemma 5.3], the manifolds W+ El and W−El rotate clockwise when tc increases and tr, tl, dr, dc, dl, α are fixed. When tc = φ(α) − ε and tc = φ(α) + ε, the relative locations of the manifolds W+ El and W−El are depicted in Figure 17(a) and (c), respectively. By Lemma 5.4 again, we know that the unstable homoclinic loop involves two or three linear zones depending on α and α∗, where α∗ ∈ (−dc, dc). It implies that we need to consider the nonexistence of limit cycles of system (1.1) in the interior of the unstable homoclinic loop by two cases α ≤ α∗ and α > α∗. tc = φ(α)− ε tc = φ(α) tc = φ(α) + ε Figure 17. Homoclinic loop bifurcation of system (1.1) EJDE-2023/83 GLOBAL ANALYSIS ON A PLANAR PIECEWISE LINEAR SYSTEMS 37 For α ≤ α∗, from Lemma 5.4 we know that the unstable homoclinic loop of system (1.1) involves two linear zones when tc = φ(α) for α ≤ α∗, and the unstable homoclinic loop is tangent to Γr when tc = φ(α) for α = α∗. Now we claim that there is no limit cycle in the interior of the unstable homoclinic loop. Suppose that there exists a small limit cycle because system (1.1) has at most one small limit cycle from Lemma 5.2. Since Ec is a stable focus and the homoclinic loop is unstable, the small limit cycle is semi-stable by the Poincaré-Bendixson theorem. This contradicts the fact that the small limit cycle of system (1.1) is unstable by Lemma 5.2 if it exists. Therefore, system (1.1) exhibits no limit cycles when tc = φ(α) for α ≤ α∗. For α > α∗, it follows from Lemma 5.4 that the unstable homoclinic loop involves three linear zones when tc = φ(α) for α > α∗. We claim that there is no limit cycle in the interior of the unstable homoclinic loop. Otherwise, there exist two limit cycles because system (1.1) has at most two limit cycles from Lemma 5.3. Since Ec is a stable focus and the homoclinic loop is unstable, we infer that the two limit cycles consist of an unstable limit cycle and a stable limit cycle, where the unstable limit cycle lies in the interior of the stable limit cycle. When tc = φ(α) + ε, the relative locations of the manifolds W+ El and W−El are shown in Figure 17(c). However, by the Poincaré-Bendixson theorem, there exist three limit cycles when tc ∈ (φ(α), φ(α) + ε), which contradicts the fact that system (1.1) has at most two limit cycles according to Lemma 5.3. Therefore, system (1.1) exhibits no limit cycles when tc = φ(α) for α > α∗. � By an analogous argument, we can obtain the following result. Lemma 5.7. If dc > 0, −dc < α < dc and t∗c < tc ≤ −2 √ dc in G1, then there is a continuous function tc = φ(α) ∈ (t∗c ,−2 √ dc] such that (a) system (1.1) exhibits no limit cycles when φ(α) ∈ (t∗c , φ(α)]; and (b) system (1.1) exhibits a unique limit cycle that is unstable when tc ∈ (φ(α), −2 √ dc]. 5.2. Limit cycles and homoclinic loops for α = dc > 0 in G1. It follows from Lemma 4.1 that system (1.1) exhibits two equilibrium points El and Ecr when α = dc > 0 in G1, where El lies in Sl and Ecr lies on Γr. Since El is a saddle, it follows that limit cycles of system (1.1) must only surround Ecr by the index theory of [38, Chapter 4] and homoclinic loops of system (1.1) must involve three linear zones by the location of Ecr. Note that the condition t2r − 4dr < 0 (resp. = 0 or > 0) means that the dynamic of right linear zone of system (1.1) is a focus (resp. improper node or bidirectional node) by Lemma 4.1. Hence, there is at least one invariant line in Sr when t2r − 4dr ≥ 0. This implies that system (1.1) has no limit cycles and homoclinic loops when α = dc > 0 and t2r − 4dr ≥ 0. In other words, we only need to investigate limit cycles and homoclinic loops of system (1.1) when α = dc > 0 and t2r − 4dr < 0. With the translation transformation x → x + 1, y → y + tc, system (1.1) becomes dx dt = F̂ (x)− y, dy dt = ĝ(x), (5.40) where F̂ (x) =  trx, if x > 0, tcx, if − 2 ≤ x ≤ 0, tl(x+ 2)− 2tc, if x < −2, 38 M. JIA, Y. SU, H. CHEN EJDE-2023/83 ĝ(x) =  drx, if x > 0, dcx, if − 2 ≤ x ≤ 0, dl(x+ 2)− 2dc, if x < −2. That is, Ecr of system (1.1) is moved to O1(0, 0) of system (5.40) and El of system (1.1) is moved to Ml(2dc/dl − 2,−2tc + 2tldc/dl) of system (5.40). The plane R2 is divided into three open linear zones Ŝl = {(x, y) ∈ R2 : x < −2}, Ŝc = {(x, y) ∈ R2 : −2 < x < 0}, Ŝr = {(x, y) ∈ R2 : x > 0} by two straight lines Γ̂l = {(x, y) ∈ R2 : x = −2} and Γ̂r = {(x, y) ∈ R2 : x = 0}. For the sake of convenience, for system (5.40) we still use F and g to represent F̂ and ĝ, respectively. Clearly, system (5.40) is topologically equivalent to system (1.1), so for the existence of limit cycles and homoclinic loops of system (1.1) we may start by analyzing system (5.40). Lemma 5.8. If α = dc > 0 and t2r − 4dr < 0 in G1, then system (1.1) exhibits neither limit cycles nor homoclinic loops when tc ≥ −tr √ dc/dr or tc ≤ t∗∗c , where t∗∗c := −tr √ dcdl − d2 c drdl + tldc dl . Proof. When α = dc > 0 and t2r − 4dr < 0 in G1, it follows from Lemma 4.1 that system (5.40) exhibits two equilibrium points O1 and Ml, where Ml is a saddle. We define a generalized Filippov transformation z(x) := ∫ x 0 g(s)ds. Let x1(z) and x2(z) be the branches of the inverse of z(x) for x ≥ 0 and x < 0, respectively. Let F1(z) := F (x1(z)) and F2(z) := F (x2(z)). Denote the abscissa of Ml by xMl := 2dc/dl − 2. Then we obtain z(x) =  drx 2 2 ∈ [0,+∞), if x ≥ 0, dcx 2 2 ∈ (0, 2dc], if − 2 ≤ x < 0, dlx 2 2 + 2(dl − dc)x+ 2(dl − dc) ∈ (2dc, zMl ), if xMl < x < −2, (5.41) where zMl := 2dc − 2d2 c/dl. It follows from (5.41) that x1(z) = √ 2z dr , if z ≥ 0, (5.42) x2(z) = − √ 2z dc , if 0 < z ≤ 2dc, −2(dl−dc)− √ −4dc(dl−dc)+2dlz dl , if 2dc < z < zMl . (5.43) From (5.42) and (5.43) we deduce F1(z) = tr √ 2z dr , if z ≥ 0, EJDE-2023/83 GLOBAL ANALYSIS ON A PLANAR PIECEWISE LINEAR SYSTEMS 39 F2(z) = −tc √ 2z dc , if 0 < z ≤ 2dc, tl 2dc− √ −4dc(dl−dc)+2dlz dl − 2tc, if 2dc < z < zMl . Then we have F1(z)− F2(z) = √ 2z( tr√ dr + tc√ dc )  > 0, if tc > −tr √ dc dr , = 0, if tc = −tr √ dc dr , < 0, if tc < −tr √ dc dr (5.44) for z ∈ (0, 2dc], and F ′1(z)− F ′2(z) = tr√ 2drz + tl√ −4dc(dl − dc) + 2dlz > 0 (5.45) for z ∈ (2dc, zMl ). From (5.44) and (5.45) we find F1(z)−F2(z) ≥ 0 for z ∈ (0, zMl ) if tc ≥ −tr √ dc/dr. However, when tc < −tr √ dc/dr, F1(z) − F2(z) < 0 holds for z ∈ (0, zMl ) if and only if (F1(z)− F2(z))|z=zMl = 2tr √ dcdl − d2 c drdl − 2tldc dl + 2tc < 0. (5.46) Solving (5.46) gives tc < t∗∗c . It is easy to check t∗∗c < −tr √ dc/dr, so F1(z)−F2(z) < 0 holds for z ∈ (0, zMl ) if tc < t∗∗c . From [27, Section 6], it follows that system (5.40) with α = dc > 0 and t2r − 4dr < 0 has neither limit cycles nor homoclinic loops when tc ≥ −tr √ dc/dr or tc ≤ t∗∗c . � Note that in Lemma 4.1 it is unclear whether Ecr of system (1.1) is a focus or a center when α = dc > 0, tc < 0, tc 2 − 4dc < 0 and tr 2 − 4dr < 0 in G1. Now we are ready to answer this question. Lemma 5.9. If α = dc > 0, tc < 0, tc 2 − 4dc < 0 and tr 2 − 4dr < 0 in G1, then the equilibrium point Ecr of system (1.1) is an unstable focus for tc > −tr √ dc/dr or a center for tc = −tr √ dc/dr or a stable focus for tc < −tr √ dc/dr. Proof. From Lemma 4.1 we know that Ecr of system (1.1) lies on Γr. When α = dc > 0, tc < 0, tc 2 − 4dc < 0 and tr 2 − 4dr < 0 in G1, Ecr is a stable focus as seen from Sc, but is an unstable focus as seen from Sr, so is O1 of system (5.40). This implies condition (C4) of Proposition 3.1. In order to study the local dynamics of O1 of system (5.40), it suffice to consider system (5.40) in the region {(x, y) ∈ R2 : −2 < x < 2}. This implicitly implies that conditions (C1)–(C3) of Proposition 3.1 hold. According to Proposition 3.1, we know that O1 of system (5.40) is an unstable focus for tc > −tr √ dc/dr or a stable focus for tc < −tr √ dc/dr or a center for tc = −tr √ dc/dr by (5.44), so is Ecr of system (1.1). � By Lemma 5.8, it suffices to study limit cycles of system (1.1) when α = dc > 0, tr 2 − 4dr < 0 and t∗∗c < tc < −tr √ dc/dr in G1. It follows from Lemmas 4.1 and 5.9 that El of system (1.1) is a saddle and Ecr of system (1.1) is a stable focus for tc 2 − 4dc < 0 or node-focus (see Figure 11(g)) for tc 2 − 4dc ≥ 0 when α = dc > 0, tr 2 − 4dr < 0 and t∗∗c < tc < −tr √ dc/dr in G1. Obviously, we have −2 √ dc < −tr √ dc/dr because of tr 2 − 4dr < 0. However, t∗∗c may be greater than or less than or equal to −2 √ dc. Therefore, we study limit cycles of system 40 M. JIA, Y. SU, H. CHEN EJDE-2023/83 (1.1) by two cases max{t∗∗c ,−2 √ dc} < tc < −tr √ dc/dr and t∗∗c < tc ≤ −2 √ dc, while α = dc > 0 and tr 2 − 4dr < 0 in G1. In what follows, we discuss the existence of limit cycles of system (1.1) with α = dc > 0 and tr 2 − 4dr < 0 when max{t∗∗c ,−2 √ dc} < tc < −tr √ dc/dr (see Lemma 5.10) or t∗∗c < tc ≤ −2 √ dc (see Lemma 5.11) in G1. Lemma 5.10. If α = dc > 0 and tr 2 − 4dr < 0 in G1, and max{t∗∗c ,−2 √ dc} < tc < −tr √ dc/dr, then system (1.1) exhibits at most one limit cycle, which is large if it exists. Proof. When α = dc > 0, tr 2 − 4dr < 0 and max{t∗∗c , −2 √ dc} < tc < −tr √ dc/dr in G1, it follows from Lemmas 4.1 and 5.9 that El of system (1.1) is a saddle and Ecr of system (1.1) is a stable focus. By the index theory of [38, Chapter 4], we know that limit cycles of system (1.1) must only surround Ecr. Since Ecr lies on Γr and El lies in Sl, it follows that limit cycle of system (1.1) may be a small limit cycle or a large limit cycle if it exists. If it is the small limit cycle, it only lies in Sc ∪ Γr ∪ Sr. We claim that system (1.1) has no small limit cycles when α = dc > 0, tr 2−4dr < 0 and max{t∗∗c ,−2 √ dc} < tc < −tr √ dc/dr in G1. Otherwise, there is at least one small limit cycle in Sc ∪ Γr ∪ Sr. Recall that system (1.1) exhibits at most one small limit cycle that lies in Sc ∪ Γr ∪ Sr only if tc > −tr √ dc/dr by Lemma 5.2 when dc > 0, −dc < α < dc and max{t∗c ,−2 √ dc} < tc < 0 in G1. When dc > 0 and α→ dc, we know that t∗c changes to t∗∗c and the equilibrium point Ec of system (1.1) changes to Ecr by Lemma 4.1. In view of the existence, uniqueness and continuity of solutions of (1.1), there is at most one small limit cycle in Sc ∪ Γr ∪ Sr only if tc > −tr √ dc/dr when α = dc > 0, max{t∗∗c ,−2 √ dc} < tc < 0 in G1. This contradicts the condition tc < −tr √ dc/dr. To prove that (1.1) has at most one large limit cycle when α = dc > 0, tr 2−4dr < 0 and max{t∗∗c ,−2 √ dc} < tc < −tr √ dc/dr in G1, by way of contradiction we assume that there are two large limit cycles of system (1.1) denoted by Γ1 and Γ2, where Γ1 and Γ2 are adjacent to each other and Γ1 is the innermost one. As discussing in the proof of Lemma 5.3, we can prove that∮ Γ1 F ′(x)dt < ∮ Γ2 F ′(x)dt. Consequently, we obtain that system (1.1) has at most one large limit cycle if it exists. � When α = dc > 0, tr 2 − 4dr < 0 and t∗∗c < tc ≤ −2 √ dc in G1, from Lemma 4.1 we know that El of system (1.1) is a saddle and Ecr of system (1.1) is node-focus as shown in Figure 11(g). Proceeding in analogous manner, we obtain the following lemma. Lemma 5.11. When α = dc > 0 and tr 2 − 4dr < 0 in G1, if t∗∗c < tc ≤ −2 √ dc, then system (1.1) exhibits at most one limit cycle, which is large if it exists. From Lemma 5.8, for homoclinic loops of system (1.1) we need to consider the case of α = dc > 0, tr 2 − 4dr < 0 and t∗∗c < tc < −tr √ dc/dr in G1. Since E1 of system (1.1) lies in Sl and Ecr of system (1.1) lies on Γr, it follows that homoclinic loops of system (1.1) must involve three linear zones. Thus, we obtain the following result. EJDE-2023/83 GLOBAL ANALYSIS ON A PLANAR PIECEWISE LINEAR SYSTEMS 41 Lemma 5.12. If α = dc > 0, tr 2 − 4dr < 0 and max{t∗∗c ,−2 √ dc} < tc < −tr √ dc/dr in G1, then system (1.1) exhibits a unique homoclinic loop involving three linear zones that is unstable when tc = φ(α), where tc = φ(α) is a continuous function satisfying φ(α) ∈ (max{t∗∗c ,−2 √ dc},−tr √ dc/dr). Proof. When α = dc > 0, tr 2 − 4dr < 0 and max{t∗∗c ,−2 √ dc} < tc < −tr √ dc/dr in G1, system (5.40) exhibits two equilibrium points Ml and O1 by Lemmas 4.1 and 5.9, where Ml is a saddle and O1 is a stable focus. To prove the existence of homoclinic loops of system (5.40), we denote by W+ Ml and W−Ml the stable and unstable manifolds of the right-hand side of Ml of system (5.40) respectively. If tr 2 − 4dr < 0, there is no equilibrium point at infinity in the right half plane of system (5.40) by Lemma 4.2 (see Figure 12(a)). So the manifolds W+ Ml and W−Ml must intersect the curve y = F (x). Denote the first intersection point of W+ Ml (resp. W−Ml ) with the curve y = F (x) by P (xP , F (xP )) (resp. Q(xQ, F (xQ))). Evidently, system (5.40) has a homoclinic loop if and only if xP − xQ = 0. To show the existence of homoclinic loops of system (5.40), we need to prove xP − xQ > 0 for tc = t∗∗c and xP − xQ < 0 for tc = −tr √ dc/dr. tc = t∗∗c tc = −tr √ dc/dr tc = φ(α) Figure 18. Homoclinic loops of system (5.40) When tc = t∗∗c , it follows from Lemma 5.8 that system (5.40) exhibits neither limit cycles nor homoclinic loops. So xP − xQ 6= 0 and it must be xP − xQ > 0, as shown in Figure 18(a). Otherwise, there exists at least one limit cycle by the Poincaré-Bendixson theorem. This yields a contradiction. When tc = −tr √ dc/dr, from Lemma 5.8 system (5.40) exhibits neither limit cycles nor homoclinic loops, i.e., xP − xQ 6= 0. Form (5.44) and (5.45), we see F1(z)−F2(z) = 0 for z ∈ (0, 2dc] and F1(z)−F2(z) > 0 for z ∈ (2dc, 2dc− 2d2 c/dl]. We can directly get xP − xQ < 0 as we discussed for Proposition 3.1, see Figure 18(b). There are some values tc ∈ (max{t∗∗c ,−2 √ dc}, −tr √ dc/dr) so that xP −xQ = 0 for system (5.40) by the mean value theorem. Therefore, we obtain the existence of homoclinic loops of system (5.40). For the uniqueness of homoclinic loops of system (5.40), by following [21, Lemma 5.3], we know that the manifolds W+ Ml and W−Ml of Ml of system (5.40) rotate clockwise when tc increases and tr, tl, dr, dc, dl, α are fixed. So there is a unique value tc ∈ (max{t∗∗c ,−2 √ dc},−tr √ dc/dr) satisfying xP − xQ = 0 for system (5.40). Namely, there is a continuous and monotonous function φ(α) on tc such 42 M. JIA, Y. SU, H. CHEN EJDE-2023/83 that (5.40) exhibits a unique homoclinic loop, as shown in Figure 18(c), where φ(α) ∈ (max{t∗∗c ,−2 √ dc},−tr √ dc/dr). For the stability of homoclinic loop of system (5.40), at the saddle point Ml of system (5.40), two eigenvalues of the associated Jacobian matrix are λ− and λ+ satisfying λ− + λ+ = tl. Due to tl > 0, by virtue of [12, Theorem 3.3], we know that the homoclinic loop of system (5.40) is unstable. � Similarly, we have the following lemma. Lemma 5.13. If α = dc > 0, tr 2 − 4dr < 0 and t∗∗c < tc ≤ −2 √ dc in G1, then system (1.1) exhibits a unique homoclinic loop involving three linear zones that is unstable when tc = φ(α), where tc = φ(α) is a continuous function satisfying φ(α) ∈ (t∗∗c ,−2 √ dc]. It is notable that system (1.1) also exhibits a unique homoclinic loop involving three linear zones that is unstable if α = dc > 0, tr 2 − 4dr < 0 and tc = φ(α), where tc = φ(α) is a continuous function satisfying φ(α) ∈ (t∗∗c , −tr √ dc/dr). The arguments are almost the same as we did for Lemma 5.12, so we skip them. Lemma 5.14. When α = dc > 0, tr 2 − 4dr < 0 and t∗∗c < tc < −tr √ dc/dr in G1, there is a continuous function tc = φ(α) ∈ (t∗∗c ,−tr √ dc/dr) such that the following two assertions hold. (a) System (1.1) exhibits no limit cycles when tc ∈ (t∗∗c , φ(α)]. (b) When tc ∈ (φ(α),−tr √ dc/dr), system (1.1) exhibits a unique limit cycle which is large and unstable. Proof. When α = dc > 0, tr 2 − 4dr < 0 and t∗∗c < tc < −tr √ dc/dr in G1, from Lemmas 4.1 and 5.9 it follows that El of system (1.1) is a saddle and Ecr of system (1.1) is a stable focus for tc 2 − 4dc < 0 or node-focus for tc 2 − 4dc ≥ 0, see Figure 11(g). We also know that system (1.1) has a unique homoclinic loop involving three linear zones that is unstable when α = dc > 0, tr 2 − 4dr < 0 and tc = φ(α) in G1. We claim that there is no limit cycle in the interior of the unstable homoclinic loop. Otherwise, there is a semi-stable limit cycle (internally unstable and externally stable) in the interior of the unstable homoclinic loop by the Poincaré-Bendixson theorem. Moreover, there are two limit cycles if the unstable homoclinic loop breaks. This contradicts Lemmas 5.10 and 5.11 Therefore, system (1.1) has no limit cycles when α = dc > 0, tr 2 − 4dr < 0 and tc = φ(α) in G1. Denote by W+ El and W−El the stable and unstable manifolds of the right-hand side of El of system (1.1) respectively. From the proof of Lemma 5.18, we know that the relative locations of the manifolds W+ El and W−El is shown in Figures 17(a) and 17(c) respectively when tc ∈ (t∗∗c , φ(α)) and tc ∈ (φ(α),−tr √ dc/dr). In view of the fact that the interior of the unstable homoclinic loop of system (1.1) exhibits no limit cycles, we arrive at the desired result. � 5.3. Limit cycles and homoclinic loops for α > dc > 0 in G1. By Lemma 4.1, we know that system (1.1) exhibits two equilibrium points El and Er when α > dc > 0 in G1, and El in Sl is a saddle and Er in Sr is an unstable focus for t2r − 4dr < 0 or an unstable node for t2r − 4dr ≥ 0. Therefore, limit cycle of system (1.1) must only surround Er by the index theory of [38, Chapter 4] which can be a small limit cycle or a large limit cycle. However, homoclinic loops of system (1.1) only involve three linear zones. Note that system (1.1) has no limit EJDE-2023/83 GLOBAL ANALYSIS ON A PLANAR PIECEWISE LINEAR SYSTEMS 43 cycles and homoclinic loops because of the invariant line in Sr when α > dc > 0 and t2r − 4dr ≥ 0 in G1. Therefore, we only need to investigate limit cycles and homoclinic loops of system (1.1) when α > dc > 0 and t2r − 4dr < 0 in G1. By setting e := (α− dc)/dr > 0, the coordinates of Er of system (1.1) can be represented by (e+ 1, tre+ tc). By the transformation x→ x+ e+ 1, y → y + tre+ tc, we change system (1.1) to dx dt = F (x)− y, dy dt = g(x), (5.47) where F (x) =  trx, if x > −e, tcx+ (tc − tr)e, if − e− 2 ≤ x ≤ −e, tl(x+ e+ 2)− tre− 2tc, if x < −e− 2, g(x) =  drx, if x > −e, dcx+ (dc − dr)e, if − e− 2 ≤ x ≤ −e, dl(x+ e+ 2)− dre− 2dc, if x < −e− 2. As we see, Er of system (1.1) changes to O2(0, 0) of system (5.47) and El of system (1.1) changes to Pl((dr/dl − 1)e + 2dc/dl − 2, (tldr/dl − tr)e − 2tc + 2tldc/dl) of system (5.47). The plane R2 can be divided into three open linear zones Sl = {(x, y) ∈ R2 : x < −e− 2}, Sc = {(x, y) ∈ R2 : −e− 2 < x < −e}, Sr = {(x, y) ∈ R2 : x > −e} by two straight lines Γl = {(x, y) ∈ R2 : x = −e − 2} and Γr = {(x, y) ∈ R2 : x = −e}. For convenience, for system (5.47) we still use F and g to represent F and g respectively. Since system (5.47) is topologically equivalent to system (1.1), we can study limit cycles and homoclinic loops of system (1.1) through analyzing system (5.47). Lemma 5.15. If α > dc > 0 and t2r − 4dr < 0 in G1, then system (1.1) exhibits neither limit cycles nor homoclinic loops when tc ≥ t∗∗∗c , where t∗∗∗c := − tr(α− dc + √ 4αdr + (α− dc)2) 2dr . Proof. When α > dc > 0 and t2r − 4dr < 0 in G1, system (5.47) has two equilibrium points Pl and O2, where Pl is a saddle and O2 is an unstable focus by Lemma 4.1. To prove the nonexistence of limit cycles and homoclinic loops of system (5.47), we define the Filippov transformation z(x) := ∫ x 0 g(s)ds. 44 M. JIA, Y. SU, H. CHEN EJDE-2023/83 It follows from system (5.47) that z(x) =  drx 2 2 , if x > −e, dc 2 (x+ e)2 − drex− dr 2 e 2, if − e− 2 ≤ x ≤ −e, dl 2 (x+ e+ 2)2 − (dre+ 2dc)x− dr 2 e 2 − 2dc − 2dce, if xPl < x < −e− 2, (5.48) and z(x) ∈  [0,+∞), if x ≥ 0, (0, dre 2 2 ), if − e < x < 0, [dre 2 2 , 2dc + 2dre+ dre 2 2 ], if − e− 2 ≤ x ≤ −e, (2dc + 2dre+ dre 2 2 , zPl ), if xPl < x < −e− 2, (5.49) where xPl := (dr/dl−1)e+2dc/dl−2, zPl := 2dc+2dre+dre 2/2−(dre+ 2dc) 2/(2dl). Let x1(z) and x2(z) be the branches of the inverse of z(x) for x ≥ 0 and xPl < x < 0 respectively. From (5.48) we derive x1(z) = √ 2z dr , if x ≥ 0, (5.50) x2(z) = − √ 2z dr , if − e < x < 0, (dr−dc)e− √ d2re 2−drdce2+2dcz dc , if − e− 2 ≤ x ≤ −e. (5.51) We define F1(z) := F (x1(z)) and F2(z) := F (x2(z)). It follows from (5.50) and (5.51) that F1(z) = tr √ 2z dr , if z > 0, F2(z) = −tr √ 2z dr , if 0 < z < dre 2 2 , tc dre− √ d2re 2−drdce2+2dcz dc − tre, if dre 2 2 ≤ z ≤ 2dc + 2dre+ dre 2 2 . By the two equalities above, we find that F1(z)− F2(z) = 2tr √ 2z dr , if 0 < z < dre 2 2 , tr √ 2z dr − tc dre− √ d2re 2−drdce2+2dcz dc + tre, if dre 2 2 ≤ z ≤ 2dc + 2dre+ dre 2 2 for z ∈ (0, 2dc + 2dre+ dre 2/2]. Clearly, from the above equality we have F1(z)− F2(z) > 0 when z ∈ (0, dre 2/2). When z ∈ [dre 2/2, 2dc + 2dre + dre 2/2], a direct calculation gives rise to F ′1(z)− F ′2(z) = tr √ d2 re 2 − drdce2 + 2dcz + tc √ 2drz√ d2 re 2 − drdce2 + 2dcz · √ 2drz > 0 (5.52) for tc ≥ 0. For tc < 0 we have F ′1(z)− F ′2(z) = tr √ d2 re 2 − drdce2 + 2dcz + tc √ 2drz√ d2 re 2 − drdce2 + 2dcz · √ 2drz  < 0, if z < z0, = 0, if z = z0, > 0, if z > z0, (5.53) EJDE-2023/83 GLOBAL ANALYSIS ON A PLANAR PIECEWISE LINEAR SYSTEMS 45 as t2rdc − t2cdr > 0, or F ′1(z)− F ′2(z) =  < 0, if dr < dc, = 0, if dr = dc, > 0, if dr > dc, (5.54) as t2rdc − t2cdr = 0, or F ′1(z)− F ′2(z)  > 0, if z < z0, = 0, if z = z0, < 0, if z > z0, (5.55) as t2rdc − t2cdr < 0, where z0 := t2r(drdce 2 − d2 re 2) 2(t2rdc − t2cdr) . (5.56) When z ∈ (2dc + 2dre+ dre 2/2, zPl ), from the monotonicity of F1(z) and F2(z), we have F1(z) > F1(2dc + 2dre+ dre 2/2), F2(z) < F2(2dc + 2dre+ dre 2/2). That is, F1(z)− F2(z) > 0. When tc ≥ 0, from (5.52) it follows that F1(z)−F2(z) > 0 for z ∈ (0, zPl ). When −tr √ dc/dr < tc < 0, we have (F1(z)− F2(z))|z=z0 = tr √ 2z0 dr − tc dre− √ d2 re 2 − drdce2 + 2dcz0 dc + tre > 0. That is, F1(z)−F2(z) > 0 for z ∈ (0, zPl ). When tc ≤ −tr √ dc/dr, from (5.54) and (5.55), we have F1(z)− F2(z) > 0 for z ∈ (0, zPl ) if and only if (F1(z)− F2(z))| z=2dc+2dre+ dre2 2 = tr √ 4dc + 4dre+ dre2 dr + 2tc + tre > 0. (5.57) Solving (5.57) leads to t∗∗∗c < tc ≤ −tr √ dc/dr. According to [27, Section 6], we know that system (5.47) exhibits neither limit cycles nor homoclinic loops when α > dc > 0, t2r − 4dr < 0 and tc ≥ t∗∗∗c in G1. � Similarly, we can obtain the following result. Lemma 5.16. If α > dc > 0, tr 2 − 4dr < 0 and tc < t∗∗∗c in G1, then system (1.1) exhibits at most two limit cycles. Following [21, Lemma 5.3], we have the following lemma. Lemma 5.17. if α > dc > 0, t2r − 4dr < 0 and tc < t∗∗∗c in G1, then system (1.1) exhibits a unique homoclinic loop that is unstable when tc = ϕ(α) ∈ (−∞, t∗∗∗c ), where the function tc = ϕ(α) is continuous and monotonous. Moreover, there exists a unique limit cycle that is stable in the interior of the homoclinic loop . Following [21, Lemma 5.4], we can further have the following lemma. Lemma 5.18. If α > dc > 0, t2r − 4dr < 0 and tc < t∗∗∗c in G1 then there is a continuous function tc = h(α) and a continuous and monotonous function tc = ϕ(α) such that the following assertions are true, where −∞ < ϕ(α) < h(α) < t∗∗∗c . (a) System (1.1) has a unique limit cycle that is stable when tc ∈ (−∞, ϕ(α)). 46 M. JIA, Y. SU, H. CHEN EJDE-2023/83 (b) System (1.1) has exactly two limit cycles when tc ∈ (ϕ(α), h(α)). The inner limit cycle is stable and the outer limit cycle is unstable. (c) System (1.1) has a unique limit cycle that is semi-stable when tc = h(α). (d) System (1.1) has no limit cycles when tc ∈ (h(α), t∗∗∗c ). 6. Proofs of main results Proof of Theorem 2.1. When the parameters lie in the region G11, it follows from Lemma 4.1 that system (1.1) has one equilibrium point Ecl for α = −dc and two equilibrium points El and Ecr for α = dc. If α = −dc ± ε or α = dc ± ε with small ε > 0, then Ecl or Ecr will vanish. Thus, BE11 and BE12 are the boundary equilibrium bifurcation curves. When the function tc = φ(α) is continuous and monotonous satisfying φ(α) ∈ (max{t∗c ,−2 √ dc}, 0) (resp. (max{t∗c ,−2 √ dc},−tr √ dc/dr) for −dc < α < dc (resp. α = dc), from Lemma 5.4 it follows that system (1.1) exhibits a unique homoclinic loop that is unstable (resp. Lemma 5.12) when tc = φ(α). When the function the function tc = ϕ(α) is continuous and monotonous satisfying ϕ(α) ∈ (−∞, t∗∗∗c ) for α > dc, from Lemma 5.17 it follows that system (1.1) has a unique homoclinic loop that is unstable when tc = ϕ(α). Therefore, HL11 and HL12 are the homoclinic bifurcation curves. From Lemma 5.18, it follows that system (1.1) exhibits a unique limit cycle that is semi-stable when tc = h(α), where the function tc = h(α) is continuous satisfying h(α) ∈ (ϕ(α), t∗∗∗c ) for α > dc. If tc = h(α) ± ε with small ε > 0, then the semi- stable limit cycle will vanish. Moreover, there are two limit cycles accompanied by the vanishing of the semi-stable limit cycle if ε < 0. Hence, we call DL the double limit cycle bifurcation curve. It follows from Lemma 5.6 that there is a unique limit cycle that is unstable when parameters lie in the region V . By Lemma 5.14, system (1.1) exhibits a unique limit cycle that is large and unstable when parameters lie in the region BE124. By Lemma 5.18, there are two limit cycles when parameters lie in the region IX. Moreover, the inner limit cycle is stable and the outer limit cycle is unstable. From Lemma 5.18 again, there is a unique limit cycle that is stable when parameters lie in the region X. By virtue of Lemmas 4.1 and 4.2, we depict global phase portraits of system (1.1) in the Poincaré disc. � Because the primary procedures in the proofs of Theorems 2.2–2.6 are closely similar to the proof of Theorem 2.1, we omit them to avoid unnecessary repetition and redundancy. 7. Numerical results In this section we perform numerical phase portraits for system (1.1) to demon- strate our theoretical results. In Theorem 2.1, system (1.1) has 23 global phase portraits in the Poincaré disc. Since there is no finite equilibrium point when parameters lie in the region I, we show the remaining numerical phase portraits of system (1.1) in Figure 19. Table 2 is offered to help facilitate understanding comparisons between global phase portraits in the Poincaré disc in Theorem 2.1 and numerical phase portraits in Figure 19. Similarly, we demonstrate numerical phase portraits of system (1.1) in Figures 20 and 21 with the associated Tables, as stated in Theorems 2.2 and 2.4. EJDE-2023/83 GLOBAL ANALYSIS ON A PLANAR PIECEWISE LINEAR SYSTEMS 47 Table 2. Numerical values chosen for plotting phase portraits of Theorem 2.1 Figure Case α tc tr tl dr dc dl 19 (a) II -0.5 2 1 1 1 1 -1 19 (b) III -0.5 1 1 1 1 1 -1 19 (c) IV -0.5 0 1 1 1 1 -1 19 (d) V -0.5 -0.1 1 1 1 1 -1 19 (e) VI -0.5 -0.2 1 1 1 1 -1 19 (f) VII -0.5 -2 1 1 1 1 -1 19 (g) VIII 1.2 -1 1 1 1 1 -1 19 (h) IV 1.2 -1.7 1 1 1 1 -1 19 (i) X 1.2 -4 1 1 1 1 -1 19 (j) BE11 -1 0 1 1 1 1 -1 19 (k) BE121 1 2 1 1 1 1 -1 19 (l) BE122 1 0 1 1 1 1 -1 19 (m) BE123 1 -1 1 1 1 1 -1 19 (n) BE124 1 -1.009 1 1 1 1 -1 19 (o) BE125 1 -1.8 1 1 1 1 -1 19 (p) BE216 1 -2 1 1 1 1 -1 19 (q) HL111 -0.5 -0.171 1 1 1 1 -1 19 (r) HL112 -0.21 -0.1712 1 1 1 1 -1 19 (s) HL113 0 -0.338 1 1 1 1 -1 19 (t) HL114 1 -1.61 1 1 1 1 -1 19 (u) HL12 1.2 -1.929 1 1 1 1 -1 19 (v) DL1 1.2 -1.3766 1 1 1 1 -1 Table 3. Numerical values chosen for plotting phase portraits of Theorem 2.2 Figure Case α tc tr tl dr dc dl 20 (a) R2 -0.5 2 2 1 1 1 -1 20 (b) R3 -0.5 1 2 1 1 1 -1 20 (c) R4 -0.5 0 2 1 1 1 -1 20 (d) R5 -0.5 -0.1 2 1 1 1 -1 20 (e) R6 -0.5 -0.2 2 1 1 1 -1 20 (f) R7 -0.5 -2 2 1 1 1 -1 20 (g) R8 1.2 -1 2 1 1 1 -1 20 (h) BE21 -1 0 2 1 1 1 -1 20 (i) BE221 1 2 2 1 1 1 -1 20 (j) BE222 1 -2 2 1 1 1 -1 20 (k) HL21 -0.5 -0.171 2 1 1 1 -1 20 (l) HL22 -0.21 -0.1712 2 1 1 1 -1 20 (m) HL23 0.1 -0.57 2 1 1 1 -1 8. Conclusion In this article, we studied global dynamics of a continuous planar piecewise linear differential system with three zones. Combining with results obtained in [21], we have achieved our goal to fully present complicated and rich dynamical behaviors through demonstrating global phase portraits in the Poincaré disc and bifurcation diagrams of system (1.1) for the case trtl > 0 and drdl < 0. 48 M. JIA, Y. SU, H. CHEN EJDE-2023/83 Table 4. Numerical values chosen for plotting phase portraits of The- orem 2.4 Figure Case α tc tr tl dr dc dl 21 (a) G5 0.5 -1 1 4 1 1 -1 21 (b) G6 0.2 -1.9 1 4 1 1 -1 21 (c) G7 0.8 -3 1 4 1 1 -1 21 (d) G8 0.8 -4 1 4 1 1 -1 21 (e) BE421 1 2 1 4 1 1 -1 21 (f) BE422 1 0 1 4 1 1 -1 21 (g) BE423 1 -1 1 4 1 1 -1 21 (h) BE424 1 -1.5 1 4 1 1 -1 21 (i) BE425 1 -3 1 4 1 1 -1 21 (j) BE425 1 -4 1 4 1 1 -1 21 (k) HL411 -0.6 -0.565 1 4 1 1 -1 21 (l) HL412 -0.446 -0.566 1 4 1 1 -1 21 (m) HL413 0 -1.38 1 4 1 1 -1 21 (n) HL414 0.8 -3.2 1 4 1 1 -1 21 (o) HL42 1 -3.7 1 4 1 1 -1 Specifically, we discussed limit cycles and homoclinic loops of system (1.1) with dc > 0 and −dc < α < dc in G1 in Section 5.1. The obtained results are totally distinguished from [21]. If φ(α) ∈ (max{t∗c ,−2 √ dc}, 0) (resp. φ(α) ∈ (t∗c ,−2 √ dc]), system (1.1) exhibits a unique limit cycle that is unstable surrounding Ec when tc ∈ (φ(α), 0) (resp. tc ∈ (φ(α),−2 √ dc]), see Lemma 5.6 (resp. Lemma 5.7). Ec lies in Sc and is a stable focus (resp. a stable node). However, homoclinic loops of system (1.1) may involve two or three linear zones. For example, system (1.1) exhibits a unique homoclinic loop involving two (resp. three) linear zones when tc = φ(α) for α < α∗ (resp. α > α∗) and exhibits a unique homoclinic loop that is tangent to Γr when tc = φ(α) for α = α∗, when φ(α) ∈ (max{t∗c ,−2 √ dc}, 0), see Lemmas 5.4-5.5. Limit cycles and homoclinic loops of system (1.1) with α = dc > 0 in G1 and with α > dc > 0 in G1 were explored in Sections 5.2 and 5.3 respectively. In the future, we will develop the methods described herein to the Liénard system with perturbations and present dynamical behaviors on limit cycles and homoclinic loops in a subsequent paper somewhere else. Acknowledgments. This work was supported by the National Natural Science Foundation of China (Nos. 12322109, 62173092, and 12171485), by the Science and Technology Innovation Program of Hunan Province (No. 2023RC3040), and by the China Postdoctoral Science Foundation (No. 2023M743969). References [1] J. H. Bonsel, R. H. B. Fey, H. Nijmeijer; Application of a dynamic vibration absorber to a piecewise linear beam system, Nonlinear Dyn., 37 (2004), 227–243. [2] J. Borghetti, G. S. Snider, P. J. Kuekes, J. J. Jang, D. R. Stewart, R. S. Williams; ‘Memristive’ switches enable ‘stateful’ logic operations via material implication, Nature, 468 (2010), 873– 876. [3] Q. Cao, M. Wiercigroch, E. E. Pavlovskaia, J. M. T. Thompson, C. Grebogi; Piecewise linear approach to an archetypal oscillator for smooth and discontinuous dynamics, Phil. Trans. R. Soc. A, 366 (2008), 635–652. [4] V. Carmona, E. Freire, E. Ponce, F. Torres; On simplifying and classifying piecewise linear systems, IEEE Trans. Circuits. Syst. I, 49 (2002), 609–620. EJDE-2023/83 GLOBAL ANALYSIS ON A PLANAR PIECEWISE LINEAR SYSTEMS 49 (α, tc) = (−0.5, 2) (α, tc) = (−0.5, 1) (α, tc) = (−0.5, 0) (α, tc) = (−0.5,−0.1) (α, tc) = (−0.5,−0.2) (α, tc) = (−0.5,−2) (α, tc) = (1.2,−1) (α, tc) = (1.2,−1.7) Figure 19. Numerical phase portraits of Theorem 2.1 with (tr, tl, dr, dc, dl) = (1, 1, 1, 1,−1) ∈ G11 50 M. JIA, Y. SU, H. CHEN EJDE-2023/83 (α, tc) = (1.2,−4) (α, tc) = (−1, 0) (α, tc) = (1, 2) (α, tc) = (1, 0) (α, tc) = (1,−1) (α, tc) = (1,−1.009) (α, tc) = (1,−1.8) (α, tc) = (1,−2) Figure 19. Numerical phase portraits of Theorem 2.1 with (tr, tl, dr, dc, dl) = (1, 1, 1, 1,−1) ∈ G11 EJDE-2023/83 GLOBAL ANALYSIS ON A PLANAR PIECEWISE LINEAR SYSTEMS 51 (α, tc) = (−0.5,−0.171) (α, tc) = (−0.21,−0.1712) (α, tc) = (0,−0.338) (α, tc) = (1,−1.61) (α, tc) = (1.2,−1.929) (α, tc) = (1.2,−1.3766) Figure 19. Numerical phase portraits of Theorem 2.1 with (tr, tl, dr, dc, dl) = (1, 1, 1, 1,−1) ∈ G11 [5] H. Chen; Global dynamics of memristor oscillators, Int. J. Bifurcation and Chaos, 26 (2016), 1650198, 1–29. [6] H. Chen, M. Jia, Y. Tang; A degenerate planar piecewise linear differential system with three zones, J. Differential Equations, 297 (2021) 433–468. [7] H. Chen, D. Li, J. Xie, Y. Yue; Limit cycles in planar continuous piecewise linear systems, Commun. Nonlinear Sci. Numer. Simulat., 47 (2017), 438–454. [8] H. Chen, X. Li; Global phase portraits of memristor oscillators, Int. J. Bifurcation and Chaos, 24 (2014), 1450152, 1–31. [9] H. Chen, Y. Tang; At most two limit cycles in a piecewise linear differential system with three zones and asymmetry, Physica D, 386-387 (2019), 23–30. [10] H. Chen, Y. Tang; A proof of Euzébio-Pazim-Ponce’s conjectures for a degenerate planar piecewise linear differential system with three zones, Physica D, 401 (2020), 132150, 1–22. [11] H. Chen, F. Wei, Y. Xia, D. Xiao; Global dynamics of an asymmetry piecewise linear differ- ential system: theory and applications, Bull. Sci. Math., 160 (2020), 102858, 1–43. 52 M. JIA, Y. SU, H. CHEN EJDE-2023/83 (α, tc) = (−0.5, 2) (α, tc) = (−0.5, 1) (α, tc) = (−0.5, 0) (α, tc) = (−0.5,−0.1) (α, tc) = (−0.5,−0.2) (α, tc) = (−0.5,−2) (α, tc) = (1.2,−1) (α, tc) = (−1, 0) Figure 20. Numerical phase portraits of Theorem 2.2 with (tr, tl, dr, dc, dl) = (2, 1, 1, 1,−1) ∈ G12 EJDE-2023/83 GLOBAL ANALYSIS ON A PLANAR PIECEWISE LINEAR SYSTEMS 53 (α, tc) = (1, 2) (α, tc) = (1,−2) (α, tc) = (−0.5,−0.171) (α, tc) = (−0.21,−0.1712) (α, tc) = (0.1,−0.57) Figure 20. Numerical phase portraits of Theorem 2.2 with (tr, tl, dr, dc, dl) = (2, 1, 1, 1,−1) ∈ G12 [12] S.-N. Chow, C. Li, D. Wang; Normal Forms and Bifurcation of Planar Vector Fields, Cam- bridge. Press, 1994. [13] L. O. Chua; Memristor: The missing circuit element, IEEE Trans. Circuit Theory, CT-18 (1971), 507–519. [14] F. Corinto, A. Ascoli, M. Gilli, Nonlinear dynamics of memristor oscillators, IEEE Trans. Ciruits Syst. I: Regul. Pap., 58 (2011), 1323–1336. [15] M. di Bernardo, C. J. Budd, A. R. Champneys, P. Kowalczyk; Piecewise-smooth Dynamical Systems: Theory and Applications, Springer-Verlag, London, 2008. [16] E. Diz-Pita, J. Llibre, M. V. Otero-Espinar; Phase portraits of a family of Kolmogorov systems depending on six parameters, Electron. J. Differential Equations, 2021 (2021), no. 35, 1-38. [17] R. Euzébio, R. Pazim, E. Ponce; Jump bifurcations in some degenerate planar piecewise linear differential systems with three zones, Physica D 325 (2016), 74–85. 54 M. JIA, Y. SU, H. CHEN EJDE-2023/83 (α, tc) = (0.5,−1) (α, tc) = (0.2,−1.9) (α, tc) = (0.8,−3) (α, tc) = (0.8,−4) (α, tc) = (1, 2) (α, tc) = (1, 0) (α, tc) = (1,−1) (α, tc) = (1,−1.5) Figure 21. Numerical phase portraits of Theorem 2.4 with (tr, tl, dr, dc, dl) = (1, 4, 1, 1,−1) ∈ G11 EJDE-2023/83 GLOBAL ANALYSIS ON A PLANAR PIECEWISE LINEAR SYSTEMS 55 (α, tc) = (1,−3) (α, tc) = (1,−4) (α, tc) = (−0.6,−0.565) (α, tc) = (−0.446,−0.566) (α, tc) = (0,−1.38) (α, tc) = (0.8,−3.2) (α, tc) = (1,−3.7) Figure 21. Numerical phase portraits of Theorem 2.4 with (tr, tl, dr, dc, dl) = (1, 4, 1, 1,−1) ∈ G11 56 M. JIA, Y. SU, H. CHEN EJDE-2023/83 [18] Z. Feng; Duffing-van der pol-type oscillator systems, Discrete Contin. Dyn. Syst. Ser. S, 7 (2014), 1231-1257. [19] E. Freire, E. Ponce, F. Rodrigo, F. Torres; Bifurcation sets of continuous piecewise linear systems with two zones, Int. J. Bifurcation and Chaos, 8 (1998), 2073–2097. [20] E. Freire, E. Ponce, F. Rodrigo, F. Torres; Bifurcation sets of symmetrical continuous piece- wise linear systems with three zones, Int. J. Bifurcation and Chaos, 12 (2002), 1675–1702. [21] M. Jia, Y. Su, H. Chen; Global studies on a continuous planar piecewise linear differential system with three zones, Nonlinear Dyn., (2022), https://doi.org/10.1007/s11071-022-08005- 1. [22] M. Han; Global behavior of limit cycles in rotated vector fields, J. Differential Equations, 151 (1999), 20–35. [23] M. Itoh, L. O. Chua; Memristor oscillator, Int. J. Bifurcation and Chaos, 18 (2008), 3183– 3206. [24] S. Li, J. Llibre; Phase portraits of piecewise linear continuous differential systems with two zones separated by a straight line, J. Differential Equations, 266 (2019), 8094–8109. [25] J. Llibre, M. Ordóñez, E. Ponce; On the exisentence and uniquness of limit cycles in planar continuous piecewise linear systems without symmetry, Nonlinear Anal. Real World Appl., 14 (2013), 2002–2012. [26] J. Llibre, E. Ponce, C. Valls; Uniqueness and non-uniqueness of limit cycles for piecewise linear differential systems with three zones and no symmetry, J. Nonlinear Sci., 25 (2015), 861–887. [27] J. Llibre, E. Ponce, C. Valls; Two limit cycles in Liénard piecewise linear differential systems, J. Nonlinear Sci., 29 (2019), 1499–1522. [28] J. Llibre, J. Sotomayor; Phase portraits of planar control systems, Nonlinear Anal., 27 (1996), 1177–1197. [29] J. Llibre, A. E. Teruel; Introduction to the Qualitative Theory of Differential Systems: Pla- nar, Symmetric and Continuous Piecewise Linear Systems, Birkhäuser Advanced Texts, Berlin, 2014. [30] H. P. McKean; Nagumo’s equation, Adv. Math., 4 (1970), 209–223. [31] H. P. McKean; Stabilization of solutions of a caricature of the Fitzhugh-Nagumo equation, Comm. Pure. Appl. Math., 36 (1983), 291–324. [32] L. P. Peng, Z. Feng; Limit cycles from a cubic reversible system via the third-order averaging method, Electron. J. Differential Equations, 2015 (2015), no. 111, 1-27. [33] E. Ponce, J. Ros, E. Vela; Limit cycle and boundary equilibrium bifurcations in continuous planar piecewise linear systems, Int. J. Bifurcation and Chaos, 25 (2015), 1530008. [34] E. Ponce, J. Ros, E. Vela; The boundary focus-saddle bifurcation in planar piecewise linear systems. Application to the analysis of memristor oscillators, Nonlinear Anal. Real World Appl., 43 (2018), 495–514. [35] J. Rinzel; Repetitive activity and Hopf bifurcation under point-Stimulation for a simple FitzHugh-Nagumo nerve conduction model, J. Math. Biology, 5 (1978), 363–382. [36] D. B. Strukov, G. S. Snider, D. R. Stewart, R. S. Williams; The missing memristor found, Nature, 453 (2008), 80–83. [37] P. Yao, H. Wu, B. Gao, J. Tang, Q. Zhang, W. Zhang, J.J. Yang, H. Qian; Fully hardware- implemented memristor convolutional neural network, Nature, 577 (2020), 641–646. [38] Z. Zhang, T. Ding, W. Huang, Z. Dong; Qualitative Theory of Differential Equations, Transl. Math. Monogr., Amer. Math. Soc., Providence, RI, 1992. Man Jia School of Mathematics and Statistics, HNP-LAMA, Central South University, Chang- sha, Hunan 410083, China. School of Mathematics and Statistics, Fuzhou University, Fuzhou, Fujian 350116, China Email address: jiaman9305@163.com Youfeng Su College of Computer and Data Science, Fuzhou University, Fuzhou, Fujian 350116, China Email address: yfsu@fzu.edu.cn EJDE-2023/83 GLOBAL ANALYSIS ON A PLANAR PIECEWISE LINEAR SYSTEMS 57 Hebai Chen (corresponding author) School of Mathematics and Statistics, HNP-LAMA, Central South University, Chang- sha, Hunan 410083, China Email address: chen hebai@csu.edu.cn 1. Introduction 2. Main results 3. Preliminaries 4. Local dynamics of system (??) 5. Nonlocal dynamics of system (??) 5.1. Limit cycles and homoclinic loops for dc>0 and -dc<< dc in G1 5.2. Limit cycles and homoclinic loops for =dc>0 in G1 5.3. Limit cycles and homoclinic loops for >dc >0 in G1 6. Proofs of main results 7. Numerical results 8. Conclusion Acknowledgments References