Piecewise linear differential systems with only centers can create limit cycles?

In this article, we study the continuous and discontinuous planar piecewise differential systems formed only by linear centers separated by one or two parallel straight lines. When these piecewise differential systems are continuous, they have no limit cycles. Also if they are discontinuous separated by a unique straight line, they do not have limit cycles. But when the piecewise differential systems are discontinuous separated two parallel straight lines, we show that they can have at most one limit cycle, and that there exist such systems with one limit cycle.


Introduction and statement of the main result
A limit cycle is a periodic orbit of a differential system in R 2 isolated in the set of all periodic orbits of that system. The study of the limit cycles goes back essentially to Poincaré [25] at the end of the nineteenth century. The existence of limit cycles became important in the applications to the real world, because many phenomena are related with their existence, see for instance the Van der Pol oscillator [29,30], or the Belousov-Zhabotinskii reaction which is a classical reaction of non-equilibrium thermodynamics appearing in a nonlinear chemical oscillator [3,31].
The study of the continuous piecewise linear differential systems separated by one or two parallel straight lines appears in a natural way in the control theory, see for instance the books [2,10,12,13,19,24].
The easiest continuous piecewise linear differential systems are formed by two linear differential systems separated by a straight line. It is known that such systems have at most one limit cycle, see [7,15,21,22]. But it is also known that if both linear differential systems are linear centers, then the continuous piecewise linear differential system has no limit cycles.
Theorem 1 A continuous piecewise linear differential system separated by one straight line formed by two linear centers has no limit cycles.
Since we cannot find an explicit reference where it is proved Theorem 1 and its proof is short and easy, we will provide it in Sect. 2. Theorem 1 can be extended to continuous piecewise linear differential systems separated by two parallel straight lines formed by three linear centers.
Theorem 2 A continuous piecewise linear differential system separated by two parallel straight lines formed by three linear centers has no limit cycles.
The study of the discontinuous piecewise linear differential systems separated by straight lines goes back to Andronov et al. [1] and until nowadays they had special attention from the mathematicians, mainly because these systems appear in mechanics, electrical circuits, economy, etc, see for instance the books [6,27] and the surveys [23,28].
In the planar discontinuous piecewise linear differential systems, the limit cycles can be of two kinds: sliding limit cycles or crossing limit cycles; the first ones contain some segment of the lines of discontinuity, and the second ones did not contain any of such segments. In this paper, we only study the crossing limit cycles. In what follows, when we talk about limit cycles, we are talking about crossing limit cycles.
Again the easiest discontinuous piecewise linear differential systems are formed by two linear differential systems separated by a straight line. It is known that such systems can have three limit cycles, see [4,5,9,11,14,16]. It remains open to know if three is the maximum number of limit cycles that such systems can exhibit. In [18], we deal with discontinuous piecewise linear differential systems separated by a straight line such that each linear differential system has neither real nor virtual equilibria, and we show that such systems can have at most one limit cycle.
On the other hand, in [20] it is proved: Theorem 3 A discontinuous piecewise linear differential system separated by one straight line formed by two linear centers has no limit cycles.
Since Proof of Theorem 3 is short, we shall include it in Sect. 3 for completeness. We remark that the problem studied in Theorem 3 but changing the linear centers by linear foci becomes a difficult problem, already studied by several authors, see in particular the paper [8] and the references quoted therein.
Here we shall prove that Theorem 3 cannot be extended to discontinuous piecewise linear differential Fig. 1 The limit cycle of the discontinuous piecewise linear differential system (1) system separated by two parallel straight lines formed by three linear centers. Thus our main result is: Theorem 4 A discontinuous piecewise linear differential system separated by two parallel straight lines formed by three linear centers can have at most one limit cycle. Moreover there are systems in this class having one limit cycle.
Theorem 4 is proved in section 3. Consider the discontinuous piecewise linear differential system separated by the two parallel straight lines x = ± 1 defined bẏ (1) This system has the limit cycle of Fig. 1. This will be proved in Sect. 3.
We remark that it is not difficult to show that if a discontinuous piecewise differential system with two straight lines of discontinuity is symmetric with respect to the origin of coordinates, then when this system has a periodic solution it has a continuum of periodic solutions. So the lack of symmetry plays a role for the existence of limit cycles for the systems studied in Theorem 4.

Preliminaries and Proof of Theorems 1 and 2
The following lemma provides a normal form for an arbitrary linear differential system having a center.
Lemma 1 A linear differential system having a center can be written aṡ with a > 0 and ω > 0.
Proof Consider a general linear differential systeṁ in the R 2 and assume that it has a center. The eigenvalues of this system are Then this system has a center if Hence the lemma is proved.
Proof of Theorem 1 Assume that we have a continuous piecewise linear differential system separated by one straight line and formed by two centers. Without loss of generality, we can assume that the straight line of continuity is x = 0. Doing a linear change of coordinates, we can write the matrix of the linear center in x < 0 in its real Jordan normal form, so the system in and it has the first integral After this linear change of variables, the center in x > 0 has a general expression as the (2) given in the Lemma 1. Note that system (2) has the first integral Since we must have a continuous piecewise linear differential system both systems, the (2) and the (3) must coincide on x = 0, therefore Clearly if the continuous piecewise linear differential system has a periodic orbit candidate to be a limit cycle, because its two differential systems are linear centers and consequently their orbits are ellipses, such a periodic orbit must intersect the line x = 0 in exactly two points, namely (0, y 1 ) and (0, y 2 ) with y 1 < y 2 . Since H 1 and H 2 are two first integrals, we have that That is The solutions (y 1 , y 2 ) of this last system satisfying the necessary condition y 1 < y 2 are y 2 = −2β − y 1 with y 1 arbitrary. So the periodic orbits of the continuous piecewise linear differential system are in a continuum of periodic orbits, and consequently this differential system has no limit cycles. This completes the proof of the theorem.
Proof of Theorem 2 Suppose that we have a continuous piecewise linear differential system separated by two parallel straight lines formed by three linear centers. Doing a rotation and a scaling of the variables, we can assume without loss of generality that the two parallel straight lines of continuity are x = ±1. As in Proof of Theorem 1, we can assume that the linear center in the strip −1 < x < 1 is given by the system (3), that the linear center in the half-plane x > 1 is given by the system (2), and finally from Lemma 1 that the linear center in the half-plane x < −1 can be given by the systeṁ with A > 0, > 0 and having the first integral Since we have a continuous piecewise linear differential system, it follows that system (2)  By Theorem 1, if this continuous piecewise linear differential system has a limit cycle, this must intersect the three open regions x < − 1,− 1 < x < 1 and x > 1. Since the orbits in each one of these three regions are ellipses or a piece of one ellipse, a possible limit cycle must intersect each straight line x = ±1 in exactly two points, namely (1, y 1 ), (1, y 2 ), (−1, y 3 ) and (−1, y 4 ) with y 1 < y 2 and y 3 > y 4 . Hence the first integrals H 1 , H 2 and H 3 must satisfy or equivalently (y 1 − y 2 )(y 1 + y 2 + 2β)(4 + ω 2 ) = 0, y 2 2 − y 2 3 + 2β(y 2 − y 3 ) − 4γ = 0, (y 3 − y 4 )(y 3 + y 4 + 2β)(4 + 2 ) = 0, y 2 1 − y 2 4 + 2β(y 1 − y 4 ) − 4γ = 0, The solutions (y 1 , y 2 , y 3 , y 4 ) of this last system satisfying the necessary conditions y 1 < y 2 and y 3 > y 4 are − β − y 2 4 + 2βy 4 + β 2 + 4γ , with y 4 arbitrary. So the periodic orbits of the continuous piecewise linear differential system are in a continuum of periodic orbits, and consequently this differential system has no limit cycles. This completes the proof of the theorem.

Proof of Theorems and 4
Proof of Theorem 3 Assume that we have a discontinuous piecewise linear differential system separated by one straight line and formed by two centers. As in Proof of Theorem 1 we can suppose that the straight line of discontinuity is x = 0, and that the linear center in the half-plane x < 0 is given by the system (3), while the linear center in the half-plane x > 0 is given by the system (2). Therefore, if the piecewise linear differential system has a periodic orbit candidate to be a limit cycle, it must intersect the line x = 0 in exactly two points, namely (0, y 1 ) and (0, y 2 ) with y 1 < y 2 . Since H 1 and H 2 are two first integrals, they must satisfy the system (4), which now is equivalent to the system This system has either no solutions (y 1 , y 2 ) satisfying the necessary condition y 1 < y 2 , or it has a continuum of solutions. So the continuous piecewise linear differential system either does not have periodic solutions, or it has a continuum of periodic orbits, and consequently this differential system has no limit cycles. This completes the proof of the theorem.
Proof of Theorem 4 Assume that we have a discontinuous piecewise linear differential system separated by two parallel straight lines formed by three linear centers. As in Proof of Theorem 2, we can suppose that the two discontinuous parallel straight lines are x = ±1, and that the systems having a linear center are (2) in x > 1, (3) in −1 < x < 1, and (5) in x < −1.
Looking at system (7), we remark that if (y 1 , y 2 , y 3 , y 4 ) is a solution, then (y 2 , y 1 , y 4 , y 3 ) is also a solution, but due to the fact that y 1 < y 2 and y 3 > y 4 , at most one of these two solutions will be satisfactory.
Since K = 0 from the third equation of (7), we isolated y 3 and substituted it in the second equation of (7), and we get In a similar way, since k = 0 from the first equation of (7) we isolated y 1 and substituted it in the fourth equation of (7), and we obtain Now doing the Gröbner basis of the two left-hand polynomials in Eqs. (8) and (9) with respect the variables y 2 and y 4 , we obtain the two equations where Eventually system (10) could have a continuum of solutions (y 2 , y 4 ) if some coefficients of the polynomials that there appear are zero, but then the possible periodic solutions would not be limit cycles. So assume that system (10) has finitely many solutions. We recall that Bezout Theorem (see for instance [26]) states that if a polynomial differential system of equations has finitely many solutions, then the number of its solutions is at most the product of the degrees of the polynomials which appear in the system. Then by Bezout Theorem system (7) has at most two solutions. Finally from these at most two solutions (y 2 , y 4 ) of system (10) we get two solutions (y 1 , y 2 , y 3 , y 4 ) of system (7), but from the previous remark at most one of these two solution would satisfy y 1 < y 2 and y 3 > y 4 . In summary, we have proved that at most we can have one limit cycle. Now we shall prove that the discontinuous piecewise linear differential system (1) having a center in each of these three pieces has also one limit cycle. This will complete the Proof of Theorem 1.
The eigenvalues of the matrices of the three linear differential systems (1) are ±4i, ±2i and ±2i, respectively. So the three systems have a linear center.

Concluding section
We have studied the continuous and discontinuous planar piecewise differential systems formed only by linear centers separated by one or two parallel straight lines.
Such continuous piecewise differential systems appear in control theory, while the discontinuous ones appear in mechanics, electrical circuits, economy, etc. When these piecewise differential systems are continuous or discontinuous separated by a unique straight line, we prove that they have no limit cycles. But when the piecewise differential systems are discontinuous separated two parallel straight lines, we show that they can have at most one limit cycle, and that there exist such systems with one limit cycle.