PIECEWISE LINEAR DIFFERENTIAL SYSTEMS WITH AN ALGEBRAIC LINE OF SEPARATION

We study the number of limit cycles of planar piecewise linear differential systems separated by a branch of an algebraic curve. We show that for each n ∈ N there exist piecewise linear differential systems separated by an algebraic curve of degree n having [n/2] hyperbolic limit cycles. Moreover, when n = 2, 3 we study in more detail the problem, considering a perturbation of a center and constructing examples with 4 and 5 limit cycles, respectively. These results follow by proving that the set of functions generating the first order averaged function associated to the problem is an extended complete Chebyshev system in a suitable interval.


Introduction and statement of the main results
For planar polynomial differential systems there still stands the unsolved Hilbert's 16th problem, whose second part asks for the maximum number of limit cycles and their distribution in terms of their degree, see for instance [9,11]. As we know, planar linear differential systems have no limit cycle, whereas the situation is different for piecewise linear differential systems. For them, in the case when they are defined in two zones separated by a straight line, it is known that there are examples with 3 limit cycles, see [4,7,8,13]. It is yet an open problem to know the maximum number of limit cycles that this type of systems can have.
We address to a very related question. Let PL n be the set of planar piecewise linear differential systems with two zones separated by a branch of an algebraic curve of degree n, C n , where in each of the two zones the differential system is linear. As usual, limit cycles will be periodic solutions isolated in the set of all periodic solutions. Here, the definition of solution in the two zones scenario is the one given in [5,6,15]. We remark that in this setting there are crossing and sliding limit cycles. In this paper, when we refer to limit cycles we will mean the ones of crossing type, even when we do not explicitly mention it. Recall that this type of periodic solutions cut the discontinuity curve only at finitely many points, both vector fields are transversal to this curve at each of these points and, moreover, both vectors fields point towards the same component of R 2 \ C n at each of them.
Let L n ∈ N ∪ {∞} be the maximum number of limit cycles that systems inside class PL n can have. Our first main result, proved in Section 2, implies that L n tends to infinity with n. Theorem 1. For n ∈ N, there exists a polynomial f n (x) of degree n and a piecewise linear differential system with two zones separated by the curve y = f n (x), such that the corresponding piecewise linear differential system has at least [n/2] hyperbolic limit cycles, where [·] denotes the integer part function. As a consequence, L n ≥ [n/2].
The above theorem does not aim to give the optimal lower bound for L n , but simply to prove that L n goes to infinity with n. As far as we know, this result is new.
It is worthwhile to comment here that there are already results proving that systems with two zones can have an arbitrary large number of limit cycles (or even infinitely many), see [3,16,18], but in all these previous works the separation curve is neither polynomial, nor analytic.
Using similar arguments as in the proof of Theorem 1 we can get the next result.

Proposition 2.
There exists an analytic function f (x) and a piecewise linear differential system with two zones separated by the curve y = f (x), such that the corresponding piecewise linear differential system has infinitely many limit cycles.
Notice that in particular, Theorem 1 gives no information when n = 1, although it is known that L 1 ≥ 3. Next, we study in more detail the cases where the separation curve is a branch C n of a quadratic curve (n = 2) or a cubic curve (n = 3). This curve is contained in the set {(x, y) | F n (x, y) = 0}, for some irreducible polynomial F n , of degree n.
For tackling this problem, and to fix a generic situation, we treat the case in which C n is tangent to the x-axis at the origin, and it is locally convex, see Figure 3. The region above C n is denoted by Ω + n and the one below is denoted by Ω − n . To improve the lower bounds for L n , n = 2, 3, we consider the piecewise linear differential system which is a perturbation of a global linear center. As we will see in Proposition 5, given any C n as above, if we denote by M Cn the first order Melnikov function associated to this vector field, it is a linear combination, with free parameters, of 6 functions. The expression of M Cn is already obtained in [12]. Our main result is based on proving that the functions defining M Cn form an extended complete Chebyshev system (ECT-system). In Section 3 we recall this concept and its utility in our context. Theorem 3. Let the separation curve C n be tangent to the x-axis at the origin and locally convex. Let M Cn be the first order Melnikov function associated to (1).
(a) If n = 2 and C 2 is a symmetric conic with respect to the y-axis then the functions defining M C 2 form an ECT-system formed by 4 functions in the whole interval of definition of M C 2 . Then L 2 ≥ 3. (b) There exist C 2 non symmetric conics with respect to the y-axis, being an ellipse, a parabola or a hyperbola, such that the corresponding functions defining M C 2 form an ECT-system formed by 5 functions in some interval (0, δ], for δ small enough. Consequently L 2 ≥ 4. (c) There exists C 3 , non symmetric cubic with respect to the y-axis, such that the functions defining M C 3 form an ECT-system formed by 6 functions in some interval (0, δ], for δ small enough. Then L 3 ≥ 5. The case in which the separation curve is the symmetric parabola, F 2 (x, y) = y−x 2 , is also studied in [14]. In that paper the authors also prove that 3 limit cycles appear. They use that the 4 functions defining the first order Melnikov function are linearly independent. Recently, in [2], a second order study has been developed and applied to the symmetric cubic curve, F 3 (x, y) = y − x 3 . In that work the lower bound for L 3 has been increased to 7.
2.1. Proof of Theorem 1. Take f n (x, ε) = εT n (x), with ε > 0 small enough, where T n (x) is the n-th Chebyshev polynomial of the first kind, i.e. for |x| ≤ 1, T n (x) = cos(n arccos x), and for |x| > 1, its analytic extension. It is known that T n (x) has degree n and all its roots are in [−1, 1]. Now y = f n (x, ε) separates the (x, y) plane in two zones, denoted by Ω + when y ≥ f n (x, ε) and Ω − when y ≤ f n (x, ε).
Inside Ω + and Ω − we consider respectively the linear differential system which form a piecewise linear differential system in the full plane. Some easy calculations show that system (2) has the first integrals H + (x, y) = 8y + x 2 − 4xy + 8y 2 and H − (x, y) = −2y + x 2 + y 2 , in Ω + and Ω − , respectively. Notice that the level curves H + (x, y) = a 2 and H − (x, y) = b 2 , with a > 0, b > 0, are ellipses and circles, respectively. Their intersection points with the x-axis are both symmetric with respect to the origin. So the piecewise linear differential system (2) with the separation curve y − f n (x, 0) = 0 has a center at the origin. We can check that the ellipses H + (x, y) = a 2 have positive slopes at the intersection points with y = 0, whereas the circles H − (x, y) = b 2 are perpendicular to y = 0.
Recall that the Chebyshev polynomial T n (x) has exactly n zeros, all located in the interval [−1, 1]. Moreover, for n even, the polynomial T n (x) is even, and its zeros are symmetric with respect to the origin. For n odd, the polynomial T n (x) is odd, and its n − 1 zeros except x = 0 are also symmetric with respect to the origin.
We claim that for ε > 0 small enough, the piecewise linear differential system (2) with the separation curve y − f n (x, ε) = 0 has at least [n/2] hyperbolic limit cycles. Moreover, when n is even, the system has exactly [n/2] limit cycles, all them hyperbolic. Figure 1 illustrates the separation curve and the limit cycles for n even. We now prove this claim. Set m = [(n−2)/2] and let ±x 0 , . . . , ±x m be the 2(m+1) not null zeros of f n (x, ε). Then Clearly, for each k ∈ {0, 1, . . . , m}, We now prove that Γ k is a hyperbolic limit cycle of the piecewise linear differential system (2). For doing so, we compute the derivative of the Poincaré map associated to the periodic orbit Γ k . We take the transversal sections of the flows of the two linear differential systems defined in (2) near P k and P −k as two small enough segments Σ k and Σ −k in y = f n (x, ε) centered at P k and P −k , respectively. Notice that for ε > 0 suitably small, both segments Σ k and Σ −k are transversal to the flows of the two linear systems. Denote by Π − k the Poincaré map from Σ −k to Σ k along the flow of the linear system in Ω − , and by Π + k the Poincaré map from Σ k to Σ −k along the flow of the linear system in Ω + . Then the composition is the Poincaré map of the piecewise linear differential system (2) near the periodic orbit Γ k defined on Σ −k . For computing the derivative of the Poincaré map Π k , we use the next general formula (see [1]) where γ 0 (s) and γ 1 (s) are the local expressions of two transversal sections Σ 0 and Σ 1 to a class C 1 vector field X, T is the flying time from P = γ 0 (0) to Π(P ) = γ 1 (0), and Π : Σ 0 → Σ 1 is the map induced by the flow ϕ of X, see Figure 2. Here, as usual, ·, · is the inner product of two vectors and (u, v) ⊥ = (−v, u). Figure 2. The map Π Denote by X + and X − the two vector fields associated to systems defined in (2), respectively. We parameterize the two segments Σ k and Σ −k by ( , respectively, where δ > 0 is suitably small. Since the divergences of the two linear systems defined in (2) identically vanish, by using (3) we have It follows that By using the expression of x k , one gets that for k ∈ {0, 1, . . . , m}, A. GASULL, J. TORREGROSA, AND X. ZHANG As a consequence, The above computations imply: • For each k ∈ {0, 1, . . . , m} and ε > 0 suitably small, Γ k is a hyperbolic limit cycle of system (2). • For n even, Γ 2s are stable, and Γ 2 +1 are unstable for s, ∈ Z and 0 ≤ 2s, 2 + 1 ≤ m. • For n odd, Γ 2s are unstable, and Γ 2 +1 are stable for s, ∈ Z and 0 ≤ 2s, 2 + 1 ≤ m.
These facts prove the first part of the claim.
Let us prove the second part, i.e. that when n is even and ε > 0 is small enough, system (2), with the separation curve y − f n (x, ε) = 0, has exactly [n/2] limit cycles.
It suffices to prove that for n even system (2) has no more limit cycles than the [n/2] ones given above. Let Γ * be a limit cycle of the considered piecewise linear system. Then, Γ * intersects the separation curve y = εT n (x) at exactly two points, provided ε > 0 suitably small. Denote these two points by (v, εT n (v)) and (u, εT n (u)) with v < u. We know that Γ * is given by one piece of the level curve H + (x, y) = a 2 0 and one piece of n (x). With the above definitions, we have On the other hand, since H − (x, y) = b 2 0 and y = εT n (x) are both symmetric with respect to the y-axis, we must have v = −u. This implies that Hence u is a zero of T n and Γ * is one of the hyperbolic limit cycles Γ k 's, as we wanted to prove. This completes the proof of the theorem.

Proof of Proposition 2.
We choose the separation curve as y = ε cos x separating the (x, y) plane in two zones with Ω + := {(x, y)| y ≥ ε cos x} and Ω − := {(x, y)| y ≤ ε cos x}. The piecewise linear differential system with two zones separated by y = ε cos x is again (2).
Notice that at the intersection points on the x-axis of y = ε cos x with the level curves H + (x, y) = a 2 and H − (x, y) = b 2 , the slopes of the two curves H + (x, y) = a 2 and H − (x, y) = b 2 at the intersection points are whose absolute values are always larger than a positive constant. Using this fact together with similar arguments as in the proof of Theorem 1 we get that for ε > 0 small enough the piecewise linear differential system has infinitely many limit cycles, which pass through the points (−π/2 − kπ, π/2 + kπ) for k ∈ N. Thus, the proposition follows.

Preliminary results and proof of Theorem 3.
Recall that given n + 1 smooth functions g 0 , . . . , g n on an open interval I, it is said that (g 0 , . . . , g n ) is an ECT-system on I if for all k = 0, 1, 2, . . . , n, any nontrivial linear combination with γ j ∈ R has at most k isolated zeros on I counting multiplicities. We remark that it is easy to see that given a linear combination of n + 1 functions, as in (4) with k = n, there always exist γ j such that it has at least n zeros in I. When the functions form an ECT-system, we can moreover ensure that these n zeros are simple. When the function (4) corresponds to a first order Melnikov function, the condition that its zeros are simple is precisely the one needed to ensure that they give rise to limit cycles of the perturbed system. A very useful characterization of ECT-systems is the following (see [10]): 0≤i,j≤k is the Wronskian of (g 0 , . . . , g k ) at x ∈ I.
For the sake of completeness, and following [12,17], we include the main steps of the proof of the following proposition, that uses the averaging method to obtain the Melnikov function associated to (1).
Proposition 5. Let C n be the separation curve tangent to the x-axis at the origin and locally convex given in system (1). Then its first order Melnikov function M Cn is where γ k , k = 0, . . . , 5, are arbitrary real parameters and x 2 + y 2 = r > 0 parameterizes the closed orbits of the unperturbed system (ε = 0). Moreover, and f j (r) = cos(θ j (r)), where the functions θ j (r), j = 1, 2, are given by the cuts of C n and x 2 + y 2 = r 2 shown in Figure 3. The function M Cn is defined when both functions θ j (r) are well-defined.
Proof. In order to apply the averaging method we consider polar coordinates, (x, y) = (r cos θ, r sin θ), and system (1) writes as where G n (r, θ) is the expression of the separation curve F n (x, y) in polar coordinates. Notice that G n ≥ 0 and G n ≤ 0 corresponds to Ω + n and Ω − n , respectively, and 2 )/2 and α ± 4 = (a ± 2 + b ± 1 )/2. Observe also that the parameters α ± i can be chosen arbitrarily, because of the arbitrariness of a ± i and b ± j . The above non-autonomous differential equation is well defined on any interval r ∈ [r 0 , β] where r 0 > 0 is a fixed and small enough number and β depends on each specific F n . The first order averaged function (or Melnikov function) is (see [12]) cos(2θ 1 (r))−cos(2θ 2 (r)) r.
Writing the above expression in terms of f j (r) = cos(θ j (r)) we arrive to (5).
Lemma 6. (a) When C n is a symmetric curve with respect to the y-axis, that is when F n (−x, y) = F n (x, y), the function M Cn reduces to , where β k , k = 0, . . . , 3, are arbitrary real constants.
(b) When n = 2 one of the functions g 2 , g 3 , g 4 , and g 5 can be removed in the expression (5) of M Cn . In particular, when the coefficient of xy in F 2 is not zero then the function g 5 can be always removed.
Proof of Theorem 3 (a). For n = 2, by Lemma 6 we have to prove that the 4 functions that define (6) form an ECT-system in a suitable interval. Let us prove first that, under our hypotheses it is not restrictive to write the quadratic polynomial F 2 that defines C 2 , as F 2 (x, y) = x 2 + Ky 2 − y = 0. (7) In general, F 2 (x, y) = 2 i+j=0 d i,j x i y j . Since it passes through the origin, d 0,0 = 0. Moreover, the symmetry condition F 2 (−x, y) = F 2 (x, y) implies that d 1,0 = d 1,1 = 0. Additionally d 0,1 = 0, because otherwise F 2 would be a homogeneous conic. Multiplying F 2 by a constant, if necessary, we can assume that d 0,1 = −1, and the convexity condition forces that d 2,0 /d 0,1 < 0. In short, F 2 (x, y) = L 2 x 2 + d 0,2 y 2 − y. Finally, changing the scale, that is taking x 1 = λx and y 1 = λy for a suitable λ, we arrive again to a piecewise linear system as (1), but with F 2 as in (7).
To study M C 2 , when F 2 (x, y) = x 2 +Ky 2 −y, it is convenient to consider separately three cases: K < 1, K = 1, and K > 1. For the first case, we write K = 1 − d 2 , for d > 0. Then we choose C 2 to be a branch of the quadratic curve which is either a parabola if d = 1, or an ellipse if 0 < d < 1, or a hyperbola if d > 1. Notice that C 2 at the origin satisfies y (0) = 0, y (0) > 0, and it is locally convex.
For this curve C 2 , we have that θ 1 (r) = arccos(f 1 (r)) and we can compute f 1 (r) as follows. Using that (x, y) = rf 1 (r), r 1 − f 2 1 (r) , we get Thus, r (1−d 2 )r +d 2 rf 2 1 (r)− 1 − f 2 1 (r) = 0. Solving this last expression we obtain In order to simplify the computations, we can use a new variable u instead of r, to avoid the inner square root in f 1 , because 4d 2 r 2 + 1 = (1 + u 2 ) 2 /(4u 2 ). Now, 0 < u < 1 when C 2 is a parabola (d = 1) or a branch of hyperbola (d > 1), and u ∈ (u d , 1) if it is an ellipse (0 < d < 1), with u d = (1 − d)/(1 + d). With this change, we get Notice that both factors inside the square root are positive for all values of d, but restricted to the respective intervals of definition. When d = 1 this can be easily seen by direct substitution. Otherwise, this becomes apparent by writing In this new variable, the Melnikov function M C 2 (u) . = M C 2 (Φ(u)) writes as From these functions we can compute the ordered list of Wronskians, , .
Thus, the proof in this case (K = 1 − d 2 ) finishes because these Wronskians are non-vanishing and, by Proposition 4, the above 4 functions form an ECT-system in the respective intervals of definition. For the third case, K > 1, we write K = 1 + d 2 , for d > 0. Then we choose C 2 to be a piece of the quadratic curve which is always an ellipse. In this situation and here, to simplify the computations, it is convenient to introduce the new variable v as, , also to avoid the inner square root in f 1 . Doing similar computations, that we omit for the safe of brevity, we also prove that the corresponding set with 4 functions is an ECT in (0, 2d).
Hence, in particular, for all K ∈ R it is possible to find a Melnikov function M C 2 (r) of the form (6) with 3 simple zeros in any given interval in the domain of definition of the function. These zeros give rise to the 3 limit cycles stated in the theorem and prove that L 2 ≥ 3. This lower bound will be improved in the proof of item (b).
Proof of Theorem 3 (b) and (c). We will study in terms of C n , and for δ > 0 small enough, whether the first 5 or all the 6 functions that define (5) form an ECT-system for r ∈ (0, δ).
We will choose C 3 to be a piece of the cubic curve that passes trough the origin. Notice that when a = 0, it is either a parabola if d = 1, or an ellipse if 0 < d < 1, or a hyperbola if d > 1. When a = 0 it is a proper cubic. This curve always has at the origin a horizontal tangent point, it is not symmetric with respect to the y-axis and it is locally convex. As in the proof of item (a) of the theorem, the functions f k (r), k = 1, 2, that appear in the Melnikov function given in Proposition 5, can be obtained by substituting (x, y) = (rf, r 1 − f 2 ) in (8). Then, they are the two solutions, f = f k (r) for k = 1, 2, of equation that satisfy f 1 (0) = 1 and f 2 (0) = −1, respectively. The series expansion of f 1 writes as and it can be seen that f 2 (r) = −f 1 (−r). Then we can obtain the corresponding series expansions of the six functions defined in (5), g 0 (r) =1, Notice that since the coefficients of r −1 and r −3 in W 3 have no common roots as polynomials in d, the Wronskian W 3 is clearly non-vanishing in a neighborhood of the origin. Then, when a = 0, for all d, the first 5 Wronskians do not vanish in an open neighborhood of the origin. Moreover, by item (b) of Lemma 6, since F 2 has the monomial xy, we already knew that g 5 is a linear combination of the previous 4 functions and 5 is the maximum number of Wronskians that can be non-zero.
On the other hand, when a = 0 and for most values of d (it suffices that the huge polynomial that gives the first order terms of W 5 (r) does not vanish) we know that the six functions g k , k = 0, 1, . . . , 5, form an ECT-system in a small enough neighborhood of r = 0, as we wanted to prove.
In particular, when a = 0 (respectively, a = 0) it is possible for all d (respectively, for most d) to find a Melnikov function M Cn (r) of the form (5) with 4 (respectively, 5) simple zeros in any small enough given interval (0, ρ). Moreover, the functions g i and W i , i = 0, . . . , 5 have a convergent series in a small enough neighborhood of the origin and the size of such interval is completely independent of the interval [r 0 , β] in Proposition 5. Hence, we can choose adequately the value of r 0 in such a way that the desired zeros of M Cn (r) are in (r 0 , ρ). Consequently, the simple zeros of M Cn (r) give rise to the limit cycles stated in the theorem.