ON THE LIMIT CYCLES OF PLANAR DISCONTINUOUS PIECEWISE LINEAR DIFFERENTIAL SYSTEMS WITH A UNIQUE EQUILIBRIUM

. This paper deals with planar discontinuous piecewise linear differential systems with two zones separated by a vertical straight line x = k . We assume that the left linear diﬀerential system ( x < k ) and the right linear diﬀerential system ( x > k ) share 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 on the line of discontinuity, then the discontinuous piecewise linear diﬀerential systems have no crossing limit cycles. While for the case k (cid:54) = 0 we provide lower and upper bounds for the number of limit cycles of these planar discontinuous piecewise linear diﬀerential systems depending on the type of their linear diﬀerential systems, i.e. if those systems have foci, centers, saddles or nodes, see Table 2.

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 dynamics. 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 separation straight line is x = k, then we have where the dot denote the derivative with respect to t. We call the linear differential system of (1) defined in x < k (resp. x > k) the left (resp. right) linear differential system. In 1990 Lum and Chua [22] conjectured that a continuous piecewise linear differential systems (1) has at most one limit cycle. This conjecture has been solved in the positive way by Freire et al [6] in 1998, for a shorter proof see [17]. While for the discontinuous piecewise linear differential systems (1) the situation becomes more complicate because such systems have in general twelve parameters.
In order to simplify the analysis of the crossing limit cycles of discontinuous piecewise linear differential systems (1), Freire, Ponce and Torres in [8] reduced the study of systems (1) to study the following Liénard piecewise linear differential systems with seven parameters.
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 [9] obtained a more simple canonical form with just five parameters as follows where the parameters α = sign(∆ − ) ∈ {i, 0, 1}, β = sign(∆ + ) ∈ {i, 0, 1} with ∆ ± = (T ± ) 2 − 4D ± . Using the canonical forms (3) or (4) the number of crossing limit cycles for planar discontinuous piecewise linear differential systems (1) have been studied in several papers, see for instance [1,9,11,12,14,20]. 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 having a focus or center, a saddle and a node, respectively, see the definitions of these equilibria in [3]. 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 we see that up to now we do not have discontinuous piecewise linear differential systems (3) with two pieces separated by one straight line having more than 3 limit cycles. Then a natural question is: Is 3 the upper bound for the maximum number of limit cycles that a discontinuous piecewise linear differential systems (3) with two pieces separated by one straight line can have? The answers to this question remains open.
There are several papers [4,5,8,15,16,21] which investigate the upper bounds of limit cycles of systems (3) under some special conditions. Euzébio and Llibre [4] proved that if one of the linear differential systems of (3) has its equilibrium point 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 > 0, and on the row the linear differential systems in x < 0.
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 [15,16] reduced these upper bounds of [4] to two limit cycles and proved that this upper bound is reached. Giannakopoulos and Pliete [5] showed that systems (3) with Z 2 symmetry have at most two crossing limit cycles. In [16], and later on in [21] 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 [8] investigated systems (3) with a maximal crossing set (that is, b = 0), and with a focus-focus dynamics, they proved that if either a + 0 a − or a − a + > 0, then systems (3) have at most one limit cycle. Recently Ponce, Ros and Vela [23] showed that systems (3) of the focus-saddle type with b = 0 have at most one limit cycle. The objective of this paper is to study the maximum number of crossing limit cycles for discontinuous piecewise linear differential systems (1) sharing a unique non-degenerate equilibrium, i.e. the eigenvalues of this equilibrium are different from 0. 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 counterclockwise, we impose that a − 1,2 = −1 and a + 1,2 < 0. Thus systems (1) become with D ± = 0. According to the orientation of the flows of systems (5), it is easy to check that if the left linear differential system of (5) is one type of nodes, including diagonal node with distinct eigenvalues (N), non-diagonal node (N ) or diagonal node with equal eigenvalues (N * ), or if the right linear differential system 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 system of (5) is a focus (F), a center (C) and a saddle (S), and the right linear differential system 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 on the limit cycles of systems (5) whose unique equilibrium O(0, 0) is located on the separation line, that is when k = 0. Table 2. The lower bounds for the maximum number of limit cycles of discontinuous piecewise linear differential systems (5) with k > 0. See Theorem 2.
The proof of Theorem 1 will be given in section 2.
We remark that from [16] it follows 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 separation 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).
Using the notation of Table 1 we denote by L ∈ {F, C, S, N, N } the left linear differential system, when this system is either a focus, or a center, or a saddle, or a node with distinct eigenvalues, or a non-diagonalizale node, respectively. In a similar way we define R ∈ {F, C, S, N, N } for the right linear differential system. Then we denote by N (L, R) the maximum number of crossing limit cycles of 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 separation 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 on the number of limit cycles of systems (5) with k > 0.
Theorem 2. The following statements hold for a planar discontinuous piecewise linear differential systems (5) with k > 0.
These results are summarized in Table 2.
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 which provide lower bounds for the number of limit cycles of systems (5) with k > 0.
Planar piecewise linear differential systems (5) of focus-focus type have been studied by several authors, see for instance [2,10,18]. Huan and Yang [10] 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 [18] gave an analytic proof that the following discontinuous piecewise linear differential system 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 study the existence or non-existence of crossing limit cycles for systems (5) with k = 0. These systems have the unique equilibrium (0, 0) located on the separation line x = 0. Therefore, in order that they can have some limit cycle such equilibrium point must be either a focus or center for the left and for right linear differential system, that is, We give a well known result on the necessary condition for the existence of crossing limit cycles in systems (5) with k = 0.
Proposition 3. If a planar discontinuous piecewise linear differential system (5) has a crossing limit cycle Γ = Γ + ∪ Γ − that intersects the separation line x = 0 at the two points (0, y 0 ) and (0, y 1 ), then where σ ± denotes the area of Ω ± , respectively, see  The proof of Proposition 3 is uses the Green's formula, see for instance [7]. According to Proposition 3 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 Theorem 1. We do the changes of variables ( changes of variables coincide on x = 0, and that the straight line x = 0 remains invariant under these changes of variables. Then in the new variables systems (5) First we consider the focus-center systems (8). It is obvious that T + = 0 and T − = 0. By Proposition 3 we obtain that systems (8) have no limit cycles under these assumptions. The proof for the center-focus systems (8) is similar to the one of the focus-center.
Second we study the focus-focus systems (8). Applying the change of variables given in Proposition 4.1 of [7], we can write systems (8) into the canonical form (4) with a L = a R = b = 0. According with Theorem 4.3 of [7] it follows that systems (8) have no limit cycles.
Finally we consider the center-center systems (8), that is T ± = 0, thus we cannot use Proposition 3, but from Theorem 3 of [19] or Theorem 1 of [21] it follows that such systems have no limit cycles. This completes the proof of Theorem 1.
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) with k = 1, we shall use the Poincaré maps for the left and the right side of these systems. Assume that the orbits starting at the point (1, y 0 ) with y 0 > 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, y 1 ) with y 1 < 0 after some time t − > 0, then we can define a left Poincaré map y 1 = P L (y 0 ), y 0 > 0. (9) Similarly the orbits of systems (5) starting at the point (1, y 1 ) with y 1 < 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, y 2 ) with y 2 > 0, then we can define a right Poincaré map Composing the left Poincaré map P L with the right Poincaré map P R , we obtain the full Poincaré map It is obvious that the zeros of correspond to the limit cycles of the discontinuous piecewise linear differential systems (5) with k = 1, see Figure 1 in the left zone of the piecewise linear differential systems (5) with k = 1. After a vertical line-preserving linear change of variables and a time-rescaling, then system (13) becomes one of the following three linear differential systems: (i) either a focus (resp. a center) of the form with A = 0 (resp. A=0); (ii) or a saddle (resp. a diagonal node) of the form with |A| < 1 (resp. |A| > 1); (iii) or a non-diagonal node of the form For a proof of Proposition 4 see Proposition 4.3.1 of [16]. 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 system of (5) is a center. In this subsection we assume that the left linear differential system of (5) is a center. Using the canonical form (14), we have with a + 1,2 < 0. When we analyze the crossing limit cycles of the discontinuous piecewise linear systems (5), we cannot choose the canonical forms given in Proposition 4 in the left and the right zones simultaneously. Thus, for instance we assume that in the right zone we have a general linear differential system given in the following proposition.
Proposition 5. The origin of a planar linear differential system (13) in the right zone is (i) a general focus (resp. a general center) when with b < 0 and c = 0 (resp. b < 0 and c = 0); (ii) a general diagonal node (resp. a general non-diagonal node) when
Using Propositions 4 and 5 we can prove statement (ii) of Theorem 2.
Proof. of statement (ii) of T heorem 2. From the left linear differential system of (17) we obtain the left Poincaré map In the study of the right Poincaré map we distinguish four cases.
Case (i). First we consider that system (17) is of center-focus type satisfying (18)| c =0 . The solution of system (18)| c =0 starting at the point (1, y 1 ) when t + = 0 is If x(t + ) = 1 then from (22) we get the parametric representation of the right Poincaré map Note that Recall that y 1 = −y 0 from (21), then the zeros of F (y 0 ) are the zeros of with t + ∈ (0, π/d) in order that the function G(t + ) be well defined, where g 0 (t + ) = sin(dt + ), g 1 (t + ) = sinh(ct + ).
Case (iii). Now we consider a systems (17) Then we obtain In the following we will prove that G(t + ) has at most one zero in t + > 0. It is obvious that Since c 2 > d 2 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) satisfying (19)| d =0 has at most one crossing limit cycle.

SHIMIN LI AND JAUME LLIBRE
For ending 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 the piecewise linear differential system (17) with a + 1,1 = 1, a + 1,2 = −1, a + 2,1 = 5, a + 2,2 = −3, then this system is of center-focus type. We claim that it has exactly one limit cycle, which starts at the point (1, y 0 ) with y 0 ≈ 2.104005385670952, enters in the half-plane x < 1 and after a time t − ≈ 4.028952638215224 reaches the switching line x = 1 at the point (1, −y 0 ), enters in the half-plane x > 1 and after a time t + ≈ 1.4354222030070583 reaches the point (1, y 0 ). See Figure 3.1. Now we prove the claim. The solution of the linear differential system in x > 1 starting at the point (1, y 0 ) at time zero is x + (t) = e −t (2 sin t + cos t − y 0 sin t), y + (t) = e −t (y 0 cos t − 2y 0 sin t + 5 sin t).
The solution of the linear differential system in x < 1 starting at the point (1, y 0 ) at time zero is x − (t) = cos t − y 0 sin t, y − (t) = y 0 cos t + sin t. Now the existence of a limit cycle as the one described in the claim must satisfy the following three equations where the unknowns are y 0 , t + and t − . From equation e 1 = 0 we obtain cos t − = (1 − y 2 0 )/(1 + y 2 0 ) and sin t − = −2y 2 0 /(1 + y 2 0 ). From equation e 2 = 0 we get y 0 = 2 − cot t + + e −t+ csc t + . Substituting cos t − , sin t − and y 0 in the equation e 3 = 0 we get a function of f (t + ) equal to zero. The graphic of this function is given in Figure 2. This function has the unique zero t + ≈ 1.4354222030070583 in the interval (0, π). From this value of t + we get the values of y 0 and t − of the claim. Hence the claim is proved.   Center-Node (non-diagonal) type.
The proofs of the remaining examples use the same kind of arguments than the proof of Example 1, and we omit them. In these remaining examples we shall use the notation introduced in Example 1.
Example 2. Consider a system (17) with a + 1,1 = a + 1,2 = −1, a + 2,1 = 15, a + 2,2 = 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 Example 3. Consider a system (17) with a + 1,1 = a + 1,2 = −1, a + 2,1 = 4, a + 2,2 = 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 3.2. The left linear differential system of (5) is of saddle type. In this subsection 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 differential systems where |A| < 1 and a + 1,2 < 0.
Appendix: Extended complete Chebyshev systems. The set ot functions (f 0 , . . . , f n ) 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 (f 0 , . . . , f n ) form an Extended Complete Chebyshev systems on I if and only if for any k ∈ {0, 1, . . . , n}, (f 0 , . . . , f k ) form an Extended Chebyshev systems. See the book [13] for a proof of the previous theorem.