This is a preprint of: “On the limit cycles of planar discontinuous piecewise linear differential systems with a unique equilibrium”, Shimin Li, Jaume Llibre, Discrete Contin. Dyn. Syst. Ser. B, vol. 24(11), 5885–5901, 2019. DOI: [10.3934/dcdsb.2019111] On the limit cycles of planar discontinuous piecewise linear differential systems with a unique equilibrium Shimin Li · Jaume Llibre Abstract This paper deals with planar discontinuous piecewise linear differ- ential systems with two zones separated by a vertical straight line x = k. We assume that the left linear differential systems (x < k) and the right linear differential systems (x > k) sharing the same equilibrium, which is located at the origin O(0, 0) without loss of generality. Our results show that if k = 0, that is when the unique equilibrium O(0, 0) is located in the separating line, then the planar discontinuous piecewise linear differential systems have no limit cycles. While for the case k ̸= 0, we bound the number of limit cycles of these planar discontinuous piecewise linear differential systems, see Table 2. Keywords Limit cycle · piecewise discontinuous linear systems 1 Introduction and statement of the main results Planar piecewise linear differential systems are the natural extension of linear differential systems in order to investigate nonlinear dynamic. It is obvious that this class of piecewise linear differential systems with two zones separated by a straight line is the simplest class of these piecewise differential systems. Without loss of generality we can assume that the separating straight line is Shimin Li School of Mathematics and Statistics, Guangdong University of Finance and Economics, Guangzhou, 510320, P.R. China Tel: +8602084096905 Fax: +8602084096159 E-mail: lism1983@126.com Jaume Llibre Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bellaterra, Barcelona, Catalonia, Spain E-mail: jllibre@mat.uab.cat 2 x = k, then we have ( − ) ( ) ( −) a1,1 a− x b ( )   a− 1,2 − + 1− if x < k, ẋ a2,2 y b = ( 2,1 ) ( ) ( 2+ ) (1) ẏ   a+ a+ x b  1,1 1,2 + 1+ if x > k, a+ 2,1 a+ 2,2 y b2 where the dot denote the derivative with respect to t. We call systems (1) with x < k (resp. x > k) the left (resp. right) linear differential systems. In 1990 Lum and Chua [20] conjectured that a continuous piecewise linear differential systems (1) has at most one limit cycle. This conjecture has been solved by Freire et al [4] in 1998, for a shorter proof see [15]. While for the discontinuous piecewise linear differential systems (1) the situation becomes more complicate because such systems have twelve parameters in general. We denote ΣC = {(k, y)|(a− − − + + + 1,1 k + a1,2 y + b1 )(a1,1 k + a1,2 y + b1 ) > 0}, (2) 1,1 k + a1,2 y + b1 )(a1,1 k + a1,2 y + b1 ) 6 0}. ΣS = {(k, y)|(a− − − + + + The sets ΣC and ΣS are called the crossing set and sliding set of systems (1), respectively. If an isolated periodic orbit of systems (1) have sliding points, then it will be called a sliding limit cycle. Otherwise it will be called a crossing limit cycle. In order to simplify the analysis of the crossing limit cycles of discontin- uous piecewise linear differential systems (1), Freire, Ponce and Torres in [6] reduced the study of systems (1) to study the following Liénard piecewise linear differential systems with seven parameters. ( − )( ) ( )  T −1 x 0 ( )   D− 0 − − if x < 0, ẋ y a = ( + )( ) ( ) (3) ẏ   T −1 x −b  − if x > 0, D+ 0 y a+ where T ± and D± denote the traces and determinants of the right and left linear differential systems of (1), respectively. Later on Freire, Ponce and Torres in [7] obtain a more simple canonical form with just five parameters as follows ( )( ) ( ) 2γL −1 x 0 ( )    γ 2 − α2 0 − if x < 0, ẋ y aL = ( L )( ) ( ) (4) ẏ   2γ −1 x −b  2 R 2 − if x > 0, γR − β 0 y aR √ √ where the parameters α = sign(∆− ) ∈ {i, 0, 1}, β = sign(∆+ ) ∈ {i, 0, 1} with ∆± = (T ± )2 − 4D± . Applying the canonical forms (3) or (4), the number of crossing limit cycles for planar discontinuous piecewise linear differential systems (1) have been 3 Table 1 Lower bounds for the maximum number of limit cycles of discontinuous piecewise linear differential systems (3) known up to now. F , S and N denote a linear differential systems having a focus or a center, a saddle and a node, respectively. In the column there is the linear differential systems on x > k, and on the row the linear differential systems in x < k. F S N F 3 3 3 S 3 2 2 N 3 2 2 studied in several papers, see for instance [1, 7, 9, 10, 12, 18]. In summary, the maximum known number of limit cycles of planar discontinuous piecewise linear differential systems (3) up to now are given in Table 1. In this table F, S and N denotes linear differential systems with a focus or center, a saddle and a node, respectively. And for instance, when in the table we intersect the column S with the row N, and we obtain the number 2, this means that systems (3) having in x < 0 a linear saddle and in x > 0 a node the maximum number of limit cycles that we know for such systems is 2. From Table 1 a natural open question arise: Is 3 the upper bound for the maximum number of limit cycles that a discontinuous piecewise linear differ- ential systems (3) with a straight line of separation can have? There are several papers [2,3,6,13,14,19] which investigate the upper bound- s of limit cycles of systems (3) under some special conditions. Euzébio and Llibre [2] proved that if one of the linear differential systems of (3) has its equilibrium point on the line of discontinuity, that is a− a+ = 0, then systems (3) have at most four limit cycles. Later on, Llibre, Novaes and Teixeira [13, 14] reduced these upper bounds of [2] to two limit cycles and proved that this upper bound is reached. Giannakopoulos and Pliete [3] showed that systems (3) with Z2 symmetry have at most two crossing limit cycles. In [14], and later on in [19] it is proved that if one of the two linear differential systems (3) is a center, the maximum number of crossing limit cycles is two, and that this upper bound is reached. Freire, Ponce and Torres [6] investigated systems (3) with a maximal crossing set (that is, b = 0), and with a focus-focus dynamics, they proved that if either a+ 6 0 6 a− or a− a+ > 0, then systems (3) have at most one limit cycle. Recently Ponce, Ros and Vela [21] showed that systems (3) of the focus-saddle type with b = 0 have at most one limit cycle. In the present paper we consider the number of crossing limit cycles for discontinuous piecewise linear differential systems (1) sharing a unique non- degenerate equilibrium. Without loss of generality we assume that the unique equilibrium is located at the origin O(0, 0) and k > 0. In order to suppose that the orientation of limit cycles are counter clockwise, we impose that a−1,2 = −1 4 and a+ 1,2 < 0. Thus systems (1) become ( − )( ) a1,1 −1 x ( )   a− if x < k, ẋ a− y = ( 2,1 2,2 + )( ) (5) ẏ   a+ a1,2 x  1,1 if x > k, a+ 2,1 a+ 2,2 y with D± ̸= 0. According to the orientation of the flows of systems (5), it is easy to check that if the left linear differential systems of (5) is one type of nodes, including ′ diagonal node with distinct eigenvalues (N), non-diagonal node (N ) or diago- ∗ nal node with equal eigenvalues (N ), or if the right linear differential systems of (5) is a saddle (S) or a diagonal node with equal eigenvalues (N∗ ), then the crossing limit cycles can not exist. So we just need to study the cases that the left linear differential systems of (5) is a focus (F), a center (C) and a saddle (S), and the right linear differential systems of (5) is a focus (F), a center (C), ′ a diagonal node (N) or a non-diagonal node (N ), see Table 2. Our first result is concerned with the maximum number of limit cycles of systems (5) whose unique equilibrium O(0, 0) located in the separating line, that is k = 0. Theorem 1 Planar discontinuous piecewise linear differential systems (5) with k = 0 have no limit cycles. It is worth to remark that from [14] we can conclude that systems (3) have at most one limit cycle when the equilibrium of both the left and the right linear differential systems are located in the separating line x = 0. Thus we reduce the upper bounds of limit cycles of systems (3) when both the left and the right linear differential systems share a unique equilibrium which is located at the origin O(0, 0). The proof of Theorem 1 will be given in section 2. We denote by N (L, R) the maximum number of crossing limit cycles that systems (5) with L and R dynamics in x < k and x > k respectively. Note that a system (5) is not symmetry with respect to separating line x = k > 0, because the unique equilibrium is located in the left zone x < k, thus in general N (L, R) ̸= N (R, L), see Table 2. Our second main result is study the number of limit cycles of systems (5) with k > 0. Theorem 2 The following statements hold for a planar discontinuous piece- wise linear differential systems (5) with k > 0. ′ (i) N (F, F) > 3, N (F, C) > 2, N (F, N) > 1, N (F, N ) > 1; ′ (ii) N (C, F) 6 1, N (C, C) 6 0, N (C, N) 6 1, N (C, N ) 6 1; ′ (iii) N (S, F) > 1, N (S, C) > 1, N (S, N) > 1, N (S, N ) > 1. These results are summarized in Table 2. 5 Table 2 The lower bounds for the maximum number of limit cycles of discontinuous piece- wise linear differential systems (5) with k > 0. See Theorem 2. ′ F C N N F 3 2 1 1 C 1 0 1 1 S 1 1 1 1 The proof of Theorem 2 will be given in section 3. From statement (ii) of Theorem 2, we know that systems (5) with k > 0 have at most one crossing limit cycle when the left linear differential systems of (5) is a center. While for the other cases we only give examples to show the lower bounds of limit cycles of systems (5) with k > 0. Huan and Yang [8] investigated the number of limit cycles of systems (5) with k = 1 and of focus-focus type. They provided strong numerical evidence that those systems can have three limit cycles. Later on Llibre and Ponce in [16] gave an analytic proof that the following discontinuous piecewise linear differential systems ( )( )  4/3 −20/3 x ( )   377/750 −26/15 if x < 1, ẋ y = ( )( ) (6) ẏ   19/50 −1 x  if x > 1, 1 19/50 y of focus-focus type sharing the same equilibrium point has three limit cycles. The organization of the rest paper is as follows. In section 2 we prove Theorem 1. In section 3 we divide the proof of Theorem 2 into three parts according if the left linear differential systems of (5) is a focus, a center or a saddle. 2 Proof of Theorem 1 In this section we consider the number of crossing limit cycles for systems (5) with k = 0. Thus we consider the planar piecewise discontinuous linear differential systems (5). It is obvious that the unique equilibrium is located in the separating line x = 0. In order to have the possibility of existence of limit cycles we need only to study the cases for focus or center, that is, D± > 0 and (T ± )2 − 4D± < 0. We give a well known result on the necessary condition for the existence of crossing limit cycles of systems (5). Proposition 1 If a planar discontinuous piecewise linear differential system (5) has a crossing limit cycle Γ = Γ + ∪ Γ − that intersects the separating line x = 0 at the two points (0, y0 ) and (0, y1 ), then T − σ − + T + σ + = 0, (7) 6 y y y0 y0 y2 Γ− Γ+ Ω− Ω+ x x y1 y1 x=1 Fig 1.1 Fig 1.2 Fig. 1 Fig 1.1. A limit cycle of systems (5). Fig 1.2. Poincaré map of systems (5). where σ ± are denote the areas of Ω ± respectively, see Figure 1.1 The proof of Proposition 1 is deduced from the Green’s formula, see for instance [5]. According to Proposition 1 a necessary condition for the existence of a crossing limit cycle for a system (5) is either T + T − < 0 or T + = T − = 0. Proof of T heorem 1. Let X = x, Y = a− 2,2 x + y when x ≤ 0 and X = + −1 a2,2 + x, Y = − + x + y when x ≥ 0. Note that both changes of variables a1,2 a1,2 coincide on x = 0, and that the straight line x = 0 remains invariant by this change of variables. Then in the new variables systems (5) becomes ( − )( ) T −1 X ( )    D− 0 if X < 0, Ẋ Y = ( + )( ) (8) Ẏ   T −1 X  if X > 0. D+ 0 Y First we consider the focus-center type. It is obvious that T + = 0 and T − ̸= 0. By Proposition 1 we obtain that systems (5) have no limit cycles for the focus-center type. The proof of the center-focus is similar to the one of the focus-center. Second we study the focus-focus type. Applying the change of variables given in Proposition 4.1 of [5], we can write systems (8) into the canonical form (4) with aL = aR = b = 0. According with Theorem 4.3 of [5] we know that systems (8) and consequently systems (5) have no limit cycles. Finally we consider the center-center type, that is T ± = 0, thus we cannot use Proposition 1, but from Theorem 3 of [17] or Theorem 1 of [19] it follows that such systems have no limit cycles. This completes the proof of Theorem 1. 7 3 Proof of Theorem 2 For systems (5) with k > 0 we take k = 1 without loss of generality. In order to investigate the crossing limit cycles of systems (5), we construct the left Poincaré map and the right Poincaré map of systems (5) as follows. Assume that the orbits starting at the point (1, y0 ) with y0 > 0 go into the left zone x < 1 under the flow of the left linear differential systems. If these orbits can reach x = 1 again at some point (1, y1 ) with y1 < 0 after some time t− > 0, then we can define a left Poincaré map y1 = PL (y0 ), y0 > 0. (9) Similarly the orbits of systems (5) starting at the point (1, y1 ) with y1 < 0 will go into the right zone x > 1 under the flow of the right linear differential systems. If the orbits can go back to x = 1 again after some time t+ > 0 and intersect the line x = 1 at (1, y2 ) with y2 > 0, then we can define a right Poincaré map y2 = PR (y1 ), y1 < 0. (10) Composite the left Poincaré map PL and the right Poincaré map PR , we obtain the full Poincaré map y2 = P (y0 ) = PR ◦ PL (y0 ), y0 > 0. (11) It is obvious that the zeros of F (y0 ) , P (y0 ) − y0 = PR ◦ PL (y0 ) − y0 , y0 > 0, (12) correspondence to the limit cycles of the discontinuous piecewise linear differ- ential systems (5), see Figure 1.2. Since systems (5) have a vertical separating line x = 1 we need to use linear change of variables which preserve the vertical line x = 1, for obtaining the canonical forms of a piecewise linear differential systems (5). Proposition 2 Consider the following planar linear differential systems ( ) ( )( ) ẋ a1,1 a1,2 x = . (13) ẏ a2,1 a2,2 y After a vertical line-preserving linear change of variables and a time-rescaling, then systems (13) becomes one of the following three linear differential systems: (i) either a focus (resp. a center) of the form ( ) ( )( ) ẋ A −1 x = , (14) ẏ 1 A y with A ̸= 0 (resp. A=0); 8 (ii) or a saddle (resp. a diagonal node) of the form ( ) ( )( ) ẋ A −1 x = , (15) ẏ −1 A y with |A| < 1 (resp. |A| > 1); (iii) or a non-diagonal node of the form ( ) ( )( ) ẋ −1 −1 x = . (16) ẏ 0 −1 y For a proof of Proposition 2 see Proposition 4.3.1 of [14]. We divide the proof of Theorem 2 into three cases according if the left linear differential systems of (5) is a center, a saddle or a focus. 3.1 The left linear differential systems of (5) is of center type In this subsection we assume that the left linear differential systems of (5) is of center type. Using the canonical form of (14), we have ( )( ) 0 −1 x ( )    1 if x < 1. ẋ 0 y = ( + + )( ) (17) ẏ   a a1,2 x  1,1 if x > 1, a+ + 2,1 a2,2 y with a+ 1,2 < 0. When we analyze the crossing limit cycles of discontinuous piecewise linear systems (5), we cannot choose the canonical forms given in Proposition 2 in both the left zone and the right zone. Thus we need to assume that the right zone with a general canonical form as follows. Proposition 3 The origin of a planar linear differential systems (13) in the right zone is (i) a general focus (resp. a general center) when   ( ) a b ( ) ẋ  x =  −(a − c)2 − d2 if x > 1, (18) ẏ 2c − a y b with b < 0 and c ̸= 0 (resp. b < 0 and c = 0); (ii) a general diagonal node (resp. a general non-diagonal node) when   ( ) a b ( ) ẋ  x =  −(a − c)2 + d2 if x > 1, (19) ẏ 2c − a y b with c2 > d2 > 0 and b < 0 (resp. d = 0 and b < 0). 9 Proof The eigenvalues of systems (13) with respect to the origin O(0, 0) are √ a1,1 + a2,2 ± (a1,1 − a2,2 )2 + 4a1,2 a2,1 λ± = . (20) 2 (i) If we impose a1,1 + a2,2 = 2c and (a1,1 − a2,2 )2 + 4a1,2 a2,1 = −4d2 , then the origin is a general focus (resp. center) with eigenvalues c±id when c ̸= 0 (resp. −(a1,1 − c)2 − d2 c = 0). From the above two equations we have a2,1 = and a1,2 a2,2 = 2c − a1,1 . Replace a1,1 = a, a1,2 = b into (13) then we obtain systems (18). (ii) We assume that a1,1 +a2,2 = 2c and (a1,1 −a2,2 )2 +4a1,2 a2,1 = 4d2 , then the origin is a general diagonal node (resp. non-diagonal node) with eigenvalues −(a1,1 − c)2 + d2 c±d when d ̸= 0 (resp. d = 0). It is easy to know that a2,1 = a1,2 and a2,2 = 2c−a1,1 . Replace a1,1 = a, a1,2 = b into (13) then we obtain systems (19). With the help of Proposition 2 and Proposition 3, we can prove the state- ment (ii) of Theorem 2. Proof of statement (ii) of T heorem 2. From the left linear differential systems of (17) we obtain the left Poincaré map PL (y0 ) = −y0 , y0 > 0. (21) In the following we need to study the right Poincaré map, we distinguish four cases. Case (i): First we consider the center-focus case of systems (17) satisfying (18)|c̸=0 . From elementary calculations we obtain the solution of systems (18)|c̸=0 starting at the point (1, y1 ) when t+ = 0:   ( ) d cos(dt+ ) + (a − c + by1 ) sin(dt+ ) x(t+ ) ect+   . (22) = d2 + (a − c)(a − c + by1 ) y(t+ ) d dy1 cos(dt+ ) − sin(dt+ ) b If x(t+ ) = 1 then from (22) we get the parametric representation of the right Poincaré map −a + c + d cot(dt+ ) − dect+ csc(dt+ ) y2 (t+ ) = . (23) b Note that y1 (t+ ) = −a + c − d cot(dt+ ) + de−ct+ csc(dt+ ). Recall that y1 = −y0 from (21), the zeros of F (y0 ) is equivalent to the zeros of G(t+ ) , y2 (t+ ) − y0 = y2 (t+ ) + y1 (t+ ) −2 (24) = ((a − c)g0 (t+ ) + dg1 (t+ )), b sin(dt+ ) 10 where t+ ∈ (0, π/d) in order that the function G(t+ ) be well defined, and g0 (t+ ) = sin(dt+ ), g1 (t+ ) = sinh(ct+ ). (25) In the following we will show that (a − c)g0 (t+ ) + dg1 (t+ ) = 0, (26) with t+ ∈ (0, π/d) has at most one zero. It is obvious that the coefficients a − c and d can be chosen arbitrarily. The Wronskian of the functions g0 (t+ ) and g1 (t+ ) is g0 (t+ ) g1 (t+ ) W (t+ ) , ′ ′ = c cosh(ct+ ) sin(dt+ ) − d cos(dt+ ) sinh(ct+ ). (27) g0 (t+ ) g1 (t+ ) ′ Since W (0) = 0 and W (t+ ) = (c2 + d2 ) sin(dt+ ) sinh(ct+ ) ̸= 0 if t+ ∈ (0, π/d), the Wronskian W (t+ ) ̸= 0. Thus G(t+ ) has at most one zero when t+ ∈ (0, π/d), and then, using Theorem 3 of the appendix, we can conclude that a system (17) satisfying (18)|c̸=0 has at most one crossing limit cycle. Case (ii): Second we consider the center-center case of systems (17) satisfying (18)|c=0 . Taking into account that c = 0 in (18) we have PR (y1 ) = −2a/b − y1 . Then we can obtain P (y0 ) = −2a/b + y0 because y1 = −y0 . Thus F (y0 ) = −2a/b and then a system (17)+(18)|c=0 has no crossing limit cycles. In fact, this result also follows from Theorem 3 of [17] or Theorem 1 of [19]. Case (iii): Now we consider the center-diagonal node case of systems (17)+(19)|d̸=0 . Similar to the case (i) we get the parametric representation of the right Poincaré map of systems (19)|d̸=0 as −a + c − d coth(dt+ ) + de−ct+ csch(dt+ ) y1 (t+ ) = , b (28) −a + c + d coth(dt+ ) − dect+ csch(dt+ ) y2 (t+ ) = , b and then we obtain 2 G(t+ ) , y1 (t+ ) + y2 (t+ ) = − (a − c + dcsch(dt+ ) sinh(ct+ )), t+ > 0. (29) b In the following we will prove that G(t+ ) has at most one zero in t+ > 0. It is obvious that ′ 2d G (t+ ) = (d coth(dt+ ) − c coth(ct+ )) . (30) b sinh(ct+ ) sinh(dt+ ) Since c2 > d2 we need to consider two cases c > d > 0 and c < d < 0. If c > d > 0, then c coth(ct+ ) > d coth(dt+ ) because coth(t+ ) is an increasing function with respect to t+ . If c < d < 0, then we have −c > −d > 0. From −c coth(−ct+ ) > −d coth(−dt+ ), we can deduce that c coth(ct+ ) > d coth(dt+ ) because coth(t) is an odd function with respect to t. Thus we ′ obtain that G (t+ ) ̸= 0, and then we can conclude that a system (17) satisfy- ing (19)|d̸=0 has at most one crossing limit cycle. 11 Case (iv): Finally we consider the center-nondiagonal node case of systems (17)+(18)|d=0 . From direct computations we obtain the parametric represen- tation of the right Poincaré map of systems (19)|d=0 as (c − a)t+ + e−ct+ − 1 (c − a)t+ + 1 − ect+ y1 (t+ ) = , y2 (t+ ) = , t+ > 0. bt+ bt+ (31) Take G(t+ ) , y1 (t+ ) + y2 (t+ ) derivative with respect to t+ we have ′ sinh(ct+ ) − ct+ cosh(ct+ ) G (t+ ) = 2 , t+ > 0. (32) bt2+ ′ It is obvious that G1 (t+ ) = sinh(ct+ ) − ct+ cosh(ct+ ) ̸= 0 because G1 (t+ ) = ′ −c2 t+ sinh(ct+ ) ̸= 0 and G1 (0) = 0. From (32) we have G (t+ ) ̸= 0. From the above analysis, we can conclude that a system (17) satisfying (19)|d=0 has at most one crossing limit cycle. At the end of this subsection we give three discontinuous piecewise linear differential systems (5) having one crossing limit cycle of the type center-focus, center-node and center non-diagonal node. Example 1 Consider a system (17) with a+ + + + 1,1 = 1, a1,2 = −1, a2,1 = 5, a2,2 = −3, then this system is of center-focus type. It has exactly one limit cycle, which intersects the switching line x = 1 at the two points y0 ≈ 2.104005385670952, y1 = −y0 , with the times t− ≈ 4.028952638215224, t+ ≈ 1.4354222030070583, respectively, see Figure 2.1. Example 2 Consider a system (17) with a+ + + + 1,1 = a1,2 = −1, a2,1 = 15, a2,2 = 7, then this system is of center-node (diagonal) type. It has exactly one limit cycle, which intersects the switching line x = 1 at the two points y0 ≈ 2.2360679774997894, y1 = −y0 , with the times t− ≈ 3.9826613241577236, t+ ≈ 0.4812118250596032, respectively, see Figure 2.2. Example 3 Consider a system (17) with a+ + + + 1,1 = a1,2 = −1, a2,1 = 4, a2,2 = 3, then this system is of center-node (non-diagonal) type. It has exactly one limit cycle, which intersects the switching line x = 1 at the two points y0 ≈ 1.5927767117205982, y1 = −y0 , with the times t− ≈ 4.262862522751957, t+ ≈ 2.1773189849653076, respectively, see Figure 2.3. 12 3 3 2 2 1 1 3 2 1 1 2 3 3 2 1 1 2 3 1 1 2 2 3 3 Fig 2.1 Fig 2.2 3 2 1 3 2 1 1 2 3 1 2 3 Fig 2.3 Fig. 2 The unique limit cycle of some systems (17). Fig 2.1. Center-Focus type. Fig 2.2. Center-Node (diagonal) type. Fig 2.3. Center-Node (non-diagonal) type. 3.2 The left linear differential systems of (5) is of saddle type In this section we study the number of limit cycles of systems (5) with a saddle dynamics in the left zone. Using the canonical form of (15) we need to consider the following differ- ential systems ( )( ) A −1 x ( )    −1 A if x < 1, ẋ y = ( ) ( ) (33) ẏ   a+ a+ x  1,1 + 1,2 + if x > 1, a2,1 a2,2 y where |A| < 1 and a+ 1,2 < 0. 2 Example 4 Consider a system (33) with A = and a+ + + 1,1 = 0, a1,2 = −1, a2,1 = 5 5 + , a = 1, then this system is of saddle-focus type. It has exactly one limit 4 2,2 13 cycle, which intersects the switching line x = 1 at the two points y0 ≈ 0.9718001266500934, y1 ≈ −0.6557898066657364, with the times t− ≈ 2.9090827941080404, t+ ≈ 1.128550128416395, respectively, see Figure 3.1. Fig 3.1 Fig 3.2 Fig 3.3 Fig 3.4 Fig. 3 Limit cycles of some systems (33). Fig 3.1. Saddle-Focus type. Fig 3.2. Saddle-Center type. Fig 3.3. Saddle-Node (diagonal) type. Fig 3.4. Saddle-Node (non-diagonal) type. 1 1 + Example 5 Consider a system (33) with A = − and a+ 1,1 = − , a1,2 = 2 4 + 17 + 1 −1, a2,1 = ,a = , then this system is of saddle-center type. It has 16 2,2 4 14 exactly one limit cycle, which intersects the switching line x = 1 at the two points y0 ≈ 0.4716878364870325, y1 ≈ −0.9716878364870325, with the times t− ≈ 2.633915793849635, t+ ≈ 1.250267504826089, respectively, see Figure 3.2. 2 1 + Example 6 Consider a system (33) with A = and a+ + 1,1 = , a1,2 = −1, a2,1 = 5 2 5 2, a+ 2,2 = − , then this system is of saddle-node type. It has exactly one limit 2 cycle, which intersects the switching line x = 1 at the two points y0 ≈ 0.7998728207036497, y1 ≈ −0.14894096273908286, with the times t− ≈ 1.2483163101194772, t+ ≈ 1.1650952227781781, respectively, see Figure 3.3. 3 Example 7 Consider a system (33) with A = − and a+ + + 1,1 = a1,2 = −1, a2,1 = 5 9 + , a = −3, then this system is of saddle-node (non-diagonal) type. It has 4 2,2 exactly one limit cycle, which intersects the switching line x = 1 at the two points y0 ≈ 0.5292900080573865, y1 ≈ −0.9952630050743477, with the times t− ≈ 3.6107224418352737, t+ ≈ 0.6294144408512573, respectively, see Figure 3.4. 3.3 The left linear differential systems of (5) of focus type In this subsection we consider the limit cycles of systems (5) with a focus dynamics in the left zone. We assume that the left linear differential systems of (5) is a general focus, then systems (5) become    a b ( )  ( )    −(a − c)2 − d2   x if x < 1, ẋ 2c − a y = ( b (34) ẏ   a+ + )( )   a x  1,1 1,2 if x > 1, a+ + 2,1 a2,2 y with b < 0 and a+ 1,2 < 0. 15 Example 8 The discontinuous piecewise linear differential system (6) is of focus-focus type, it is the systems studied in [16]. It has exactly three lim- it cycles Γi , i = 1, 2, 3. The smallest one Γ1 intersects the switching line x = 1 at the two points y0 ≈ 0.6188541651825252, y1 ≈ 0.1638262276270832, with the times t− ≈ 6.220096927394248, t+ ≈ 3.9178388598443292, respectively. The middle limit cycle Γ2 intersects the switching line x = 1 at the two points ỹ0 ≈ 0.9657998558466794, ỹ1 ≈ −0.08891408094287911, with the times t− ≈ 5.043121491547688, t+ ≈ 8.566832229211103, respectively. The biggest limit cycle Γ3 intersects the switching line x = 1 at the two points ȳ0 ≈ 1.6811945105189345, ȳ1 ≈ −0.48606164912659217, with the times t− ≈ 4.601444430614628, t+ ≈ 14.866328036550136, respectively, see Figure 4.1. 1 Example 9 Consider a system (34) with a = 1, b = −1, c = , d = 8, 10 + + + + a1,1 = a2,2 = 0, a1,2 = −1 and a2,1 = 1, then this system is of focus-center type. It has exactly two limit cycles Γ1 and Γ2 . The smallest one Γ1 intersects the switching line x = 1 at the two points y0 ≈ 2.8326371864989257, y1 = −y0 , with the times t− ≈ 0.7011989432509553, t+ ≈ 2.462853167680777, respectively. The largest limit cycle Γ2 intersects the switching line x = 1 at the two points ỹ0 ≈ 38.96308262220769, ỹ1 = −ỹ0 , with the times t− ≈ 0.4433513431734112, t+ ≈ 3.0902732789460954, respectively, see Figure 4.2. 16 2 40 1 20 8 6 4 2 2 10 20 30 40 1 20 2 40 Fig 4.1 Fig 4.2 15 20 10 10 5 2 2 4 6 3 2 1 1 2 3 10 5 20 10 Fig 4.3 Fig 4.4 Fig. 4 Limit cycles of some systems (33). Fig 4.1. Focus-Focus type. Fig 4.2. Focus-Center type. Fig 4.3. Focus-Node (diagonal) type. Fig 4.4. Focus-Node (non-diagonal) type. Example 10 Consider a system (34) with a = 2, b = c = −1, d = 5, a+ 1,1 = + + + a2,2 = 3 and a1,2 = a2,1 = −1, then this system is of focus-node type. It ha exactly one limit cycle, which intersects the switching line x = 1 at the two points y0 ≈ 15.74300241360837, y1 ≈ 1.2713151560165141, with the times t− ≈ 0.9506860894050889, t+ ≈ 0.9988106885099418, respectively, see Figure 4.3. Example 11 Consider a system (34) with a = c = 1, b = −1, d = 20, a+ 1,1 = a+ 2,2 = −1, a+ 1,2 = −1 and a + 2,1 = 0, then this system is of focus-node (no- diagonal) type. It has exactly one limit cycle, which intersects the switching 17 line x = 1 at the two points y0 ≈ −0.227958611935782, y1 ≈ −17.293759360246668, with the times t− ≈ 0.27908115601573263, t+ ≈ 4.328936898906302, respectively, see Figure 4.4. These examples complete the result presented in Table 2, and consequently prove Theorem 2. Appendix: Extended Complete Chebyshev systems The set ot functions (f0 , . . . , fn ) defined on the interval I form an Extended Chebyshev systems on I if and only if any nontrivial linear combination of these functions has at most n zeros counting their multiplicities and this number is reached. The functions (f0 , . . . , fn ) form an Extended Complete Chebyshev systems on I if and only if for any k ∈ {0, 1, . . . , n}, (f0 , . . . , fk ) form an Extended Chebyshev systems. Theorem 3 Let f0 , . . . , fn be analytic functions defined on an open interval I ⊂ R. Then the set of functions (f0 , . . . , fn ) form an Extended Complete Chebyshev systems on I if and only if for each k ∈ {0, 1, . . . , n} and all y ∈ I the Wronskian f0 (y) f1 (y) · · · fk (y) f0′ (y) f1′ (y) · · · fk′ (y) W (f0 , . . . , fk )(y) = .. .. .. .. ̸ 0. = . . . . (k) (k) (k) f0 (y) f1 (y) · · · fk (y) See the book [11] for a proof of the previous theorem. 4 Acknowledgements The first author is partially supported by Science and Technology Program of Guangzhou (No.201707010426, 20180401350), Natural Science Foundation of Guangdong Province (2017A030313010) and China Scholarship Council (No. 201608440447). The second author is partially supported by the Ministerio de Economı́a, Industria y Competitividad, Agencia Estatal de Investigación grant MTM2016- 77278-P (FEDER), the Agència de Gestió d’Ajuts Universitaris i de Recerca grant 2017 SGR 1617, and the European project Dynamics-H2020-MSCA- RISE-2017-777911. 18 References 1. Braga, D.C., Mello, L.F.: Limit cycles in a family of discontinuous piecewise linear dif- ferential systems with two zones in the plane. Nonlinear Dyn. 73, 1283–1288 (2013) 2. Euzébio R.D., Llibre J.: On the number of limit cycles in discontinuous piecewise linear differential systems with two pieces separated by a straight line. J. Math. Anal. Appl. 424, 475–486 (2015) 3. Giannakopoulos F., Pliete K.: Planar systems of piecewise linear differential equations with a line of discontinuity. Nonlinearity 14, 611–1632 (2001) 4. Freire E., Ponce, E., Rodrigo F., Torres F.: bifurcation sets of continuous piecewise linear systems with two zones. Internat. J. Bifur. Chaos 8, 2073–2097 (1998) 5. Freire, E., Ponce E., Torres, F.: Canonical discontinuous planar piecewise linear systems. SIAM J. Appl. Dyn. Syst. 11, 181–211 (2012) 6. Freire, E., Ponce, E., Torres, F.: Planar Filippov systems with maximal crossing set and piecewise linear focus dynamics, Progrss and challenges in dynamical systems. Springer, NewYork (2013) 7. Freire, E., Ponce, E., Torres, F.: A general mechanism to generate three limit cycles in planar Filippov systems with two zones. Nonlinear Dyn. 78, 251–263 (2014) 8. Huan, S.M., Yang, X.S.: On the number of limit cycles in general planar piecewise linear systems. Discrete Contin. Dyn. Syst. 32, 2147–2164 (2012) 9. Huan, S.M., Yang, X.S.: Existence of limit cycles in general planar piecewise linear sys- tems of saddle-saddle dynamics. Nonlinear Anal. 92, 82–95 (2013) 10. Huan, S.M., Yang, X.S.: On the number of limit cycles in general planar piecewise linear systems of node-node types. J. Math. Anal. Appl. 411, 340–353 (2014) 11. Karlin, J., Studden, W.J.: T-systems: with applications in analysis and statistics, Pure Appl. Math. Interscience Publishers, NewYork, London, Sidney (1966) 12. Li, L.: Three crossing limit cycles in planar piecewise linear systems with saddle-focus type. Electron. J. Qual. Theory Differ. Equ. 70, 1–14 (2014) 13. Llibre, J., Novaes, D.D., Teixeira, M.A.: Limit cycles bifurcating from the periodic orbits of a discontinuous piecewise linear differential center with two zones. Internat. J. Bifur. Chaos (2015). https://doi.org/10.1142/S0218127415501448 14. Llibre, J., Novaes, D.D., Teixeira, M.A.: Maximum number of limit cycles for certain piecewise linear dynamical systems. Nonlinear Dyn. 82, 1159–1175 (2015) 15. Llibre, J., Ordóñez M., Ponce, E.: On the existence and uniqueness of limit cycles in a planar continuous piecewise linear systems without symmetry. Nonlinear Anal. Ser. B: Real World Appl. 14, 2002–2012 (2013) 16. Llibre J., Ponce, E.: Three limit cycles in discontinuous piecewise linear differential systems with two zones. Dynam. Contin. Discrete Impuls. Systems. Ser. B 19, 325–335 (2012) 17. Llibre J., Teixeira, M.A.: Piecewise linear differential systems with only centers can create limit cycles?. Nonlinear Dyn. 91, 249–255 (2018) 18. Llibre, J., Teixeira, M.A., Torregrosa, J.: Lower bounds for the maximum number of limit cycles of discontinuous piecewise linear differential systems with a straight line of separation. Internat. J. Bifur. Chaos (2013). https://doi.org/10.1142/S0218127413500661 19. Llibre J., Zhang, X.: Limit cycles for discontinuous planar piecewise linear differential systems separated by one straight line and having a center. J. Math. Anal. Appl. 467, 537–549 (2018) 20. Lum, R., Chua, L.O.: Global properties of continuous piecewise-linear vector fields. Part I: Simplest case in R2 . University of California at Berkeley, Memorandum UCB/ERL M90/22. (1990) 21. Ponce, E., Ros J., Vela, E.: The boundary focus-saddle bifurcation in planar piecewise linear systems. Application to the analysis of memristor oscillators. Nonlinear Anal. Ser. B: Real World Appl. 43, 495–514 (2018)