Limit cycles for m-piecewise discontinuous polynomial Liénard differential equations

We provide lower bounds for the maximum number of limit cycles for the m-piecewise discontinuous polynomial differential equations x˙=y+sgn(gm(x,y))F(x)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\dot{x} = y+{\rm sgn}(g_m(x, y))F(x)}$$\end{document}, y˙=-x\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\dot{y} = -x}$$\end{document}, where the zero set of the function sgn(gm(x, y)) with m = 2, 4, 6, . . . is the product of m/2 straight lines passing through the origin of coordinates dividing the plane into sectors of angle 2π/m, and sgn(z) denotes the sign function.


Introduction and statement of the main result
The discontinuous Lienard polynomial differential systems have many applications, see for instance the excellent paper of Makarenkov and Lamb [23]. As far as we know up to now, there are no papers studying their limit cycles. The main objective of this paper is to start this study.
Hilbert [14] in 1900 and in the second part of its 16th problem proposed to find an estimation of the uniform upper bound for the number of limit cycles of all polynomial differential systems of a given degree and also to study their distribution or configuration in the plane. Except for the Riemann hypothesis, the 16th problem seems to be the most elusive of Hilbert's problems. It has been one of the main problems in the qualitative theory of planar differential equations in the XX century. The contributions ofÉcalle [11] and Ilyashenko [15] proving that any polynomial differential system has finitely many limit cycles have been the best results in this area. But until now it is not proved the existence of a uniform upper bound. This problem remains open even for the quadratic polynomial differential systems. However, it is not difficult to see that any finite configuration of limit cycles is realizable for some polynomial differential system, see for details [21].
Thus, we have the finiteness of the number of limit cycles for every polynomial differential system of degree n, but we do not have uniform bounds for that number in the whole class of all polynomial differential systems of degree n. Following Smale [25], we consider an easier and special class of polynomial differential systems, the Liénard polynomial differential systems: where F (x) = a 0 + a 1 x + · · · + a n x n , and the dot denotes derivative with respect to the time t. For these systems, the existence of uniform bounds also remains unproved. But when the degree n of these systems The first author is partially supported by a MICINN/FEDER Grant MTM2008-03437, by a AGAUR Grant number 2009SGR-0410 and by ICREA Academia. The second author is partially supported by a FAPESP-BRAZIL Grant 2007/06896-5. All the authors are also supported by the joint project CAPES-MECD Grant PHB-2009-0025-PC. ZAMP is odd, Ilyashenko and Panov in [16] obtained a uniform upper bound for the number of limit cycles in a subclass of systems such that F is monic and its coefficients satisfy some estimations.
For the Liénard polynomial differential systems (1), Lins et al. [19] in 1977 conjectured that they have at most [(n − 1)/2] limit cycles if F (x) is a polynomial of degree n. Here, [z] denotes the integer part function of z. Moreover, he provided how to construct Liénard polynomial differential systems of degree n with [(n − 1)/2] limit cycles.
In 2007 Dumortier et al. [10] proved that for n = 7, there are 4 limit cycles when the conjecture stated at most 3. In fact as they comment in that paper, their arguments can be extended in order to show that for n ≥ 7 odd always there will be more limit cycles than the expected by the conjecture. Recently, De Maesschalck and Dumortier proved in [8] that the classical Liénard equation of degree n ≥ 6 can have [(n − 1)/2] + 2 limit cycles. In the last two papers, the discussions are based on singular perturbation theory, and the authors used relaxation oscillation solutions to study the number of limit cycles. Li and Llibre [18] proved in 2012 that the conjecture of Lins, de Melo and Pugh holds for the Liénard polynomial differential systems of degree 4. So, now only remains open the conjecture for degree 5.
The problem of determining the maximum number of limit cycles that a given differential system can have has become one of the main topics in the qualitative theory of differential systems. Our main concern is to bring this problem to a class of non-smooth dynamical systems. A good representative of this class is the mathematical model which is commonly found in many applications such as control theory (see [4] and [9]), relay systems (see [3]), economy (see [13]), impact systems (see [6]), mechanical systems [2], nonlinear oscillations [17] among others. And of course in these areas the detection of limit cycles is of fundamental importance. Thus, in this paper, we shall study the limit cycles of the m-piecewise discontinuous polynomial differential equationsẋ where the zero set of the function sgn(g m (x, y)) with m = 2, 4, 6, . . . is the union of m/2 different straight lines passing through the origin of coordinates dividing the plane into sectors of angle 2π/m. Here, sgn(z) denotes the sign function, i.e., Note that the differential system (3) is a particular case of the differential systems (2), which in some sense generalize the class of Liénard differential systems to the non-smooth differential systems.
We also consider the case m = 0 with g 0 (x, y) = 1. Therefore, the differential equation (3) for m = 0 coincides with the Liénard polynomial differential equation (1). For this reason, we shall call the m-piecewise discontinuous polynomial differential equations (3) for m = 2, 4, 6, . . . as the m-piecewise discontinuous Liénard polynomial differential equation of degree n if n is the degree of the polynomial F (x).
Piecewise discontinuous differential equations or more general non-smooth differential equations derived from ordinary differential equations when the non-uniqueness of some solutions is allowed. The theory of non-smooth differential equations has been developed very fast in these recent years due to various facts: its mathematical beauty, its strong relation with other branches of science and the challenge in establishing reasonable and consistent definitions and conventions. It has become certainly one of the common frontiers between Mathematics, Physics and Engineering. Also appears in a natural way in control systems, impact in mechanical systems and in nonlinear oscillations, in particular in electrical circuits. We understand that non-smooth systems are driven by applications, and they play an intrinsic Vol. 66 (2015) Discontinuous Liénard polynomial differential equations 53 role in a wide range of technological areas. See for more details on the non-smooth differential equations [26] and the references therein. Our main result on the limit cycles for m-piecewise discontinuous Liénard polynomial differential equations of degree n is the following theorem and conjecture. There and in the rest of the paper when we say that "the lower upper bound for the maximum number of limit cycles of a differential systems is M " this means that this system can have M limit cycles if the right-hand side of the system is chosen conveniently.

Theorem 1.
Lower upper bounds L(m, n) for the maximum number of limit cycles of the m-piecewise discontinuous polynomial Liénard differential equations (3) of degree n are Theorem 1 is proved in Sect. 3 using the averaging theory. In the "Appendix," we shall summarize the results on the averaging theory that we shall use in this paper. Of course the conjecture for L(0, n) was proved by Lins et al. [19], but here we shall present an easier and shorter proof using averaging theory.

Conjecture.
A lower upper bound L(m, n) for the maximum number of limit cycles of the m-piecewise discontinuous polynomial Liénard differential equations of degree n given by system (3) is In Sect. 3, we shall provide some analytical results providing evidence that the conjecture must hold for the cases L(m, n) with m = 6, 8, 10, . . ..
We must mention that our proof of Theorem 1 will work also if the slopes of the m/2 straight lines of g m (x, y) = 0 are tan(α + (2πj)/m) for an arbitrary α instead of tan((2πj)/m) for j = 1, 2, . . . , m.

Computations of L(0, n), L(2, n) and L(4, n)
As we shall see the proof of Theorem 1 is reduced to prove the next Propositions 2, 3 and 4.
First, we shall provide the proof of L(0, n). We recall the Descartes theorem about the number of zeros of a real polynomial. For a proof, see the pages 81-83 of the book [5], mainly the last lines of page 82 and the beginning of page 83. See also the Appendix A of [22] for checking that the maximum number of positive roots can be reached stated in the Descartes theorem can be reached. More details can be seen in "Appendix III". Proof. For proving that [(n − 1)/2] is a lower bound for the maximum number of limit cycles of the Liénard polynomial differential systems (1) of degree n, we shall prove that there are differential systems of the formẋ
System (9) in polar coordinates becomeṡ r = ε s δ (r cos θ) cos θ F (r cos θ), We must note that the Poincaré map of both systems (8) and (10) are smooth because in the first case it is composition of two smooth functions (one defined on x = 0 by the flow in x > 0, and the other also defined on x = 0 by the flow in x < 0), and in the second, it is smooth by the general results on smooth ordinary differential equations. Clearly, the limit of the Poincaré map of system (9) when δ → 0 tends to the Poincaré map of system (7). In fact, this convergence which is clear for the systems here studied has been proved in [20] for any discontinuous systems under convenient assumptions. On the other hand, if we do the Taylor expansion of the Poincaré map in the parameter ε, the averaged function f 0 of the "Appendix" is the coefficient of ε in such expansion, and for more details, see for instance the section 3 of [7]. Therefore, if f δ 0 (r) and f 0 (r) denotes the averaged function of systems (10) and (8), respectively, then the limit of f δ 0 (r) when δ → 0 is the function f 0 (r). Hence, by Theorem 10 the simple zeros of the function f 0 (r) provide limit cycles of the differential equation (8) and consequently of the discontinuous differential system (7). Now we shall compute the function f 0 (r). = εf (θ, r) + ε 2 g(θ, r, ε).
Taking θ as the new independent variable system, Eq. (13) can be written as Since the differential equation (14) is the limit of systems satisfying all the assumptions of Theorem 10 of the "Appendix," we shall apply directly Theorem 10 to system (14) for computing the averaged function f 0 (r). Thus, we have where cos θ F (r cos θ)dθ, cos θ F (r cos θ)dθ + 7π/4 5π/4 cos θ F (r cos θ)dθ.

Some analytic results and numerical computations
For m = 2, 4, 6, . . ., let g m (x, y) be the function which appears in system (3). The set of points (x, y) satisfying g m (x, y) = 0 divides the plane into m sectors. We can assume that the slopes of the m/2 straight lines of g m (x, y) = 0 are tan(α + (2πj)/m) for j = 0, 1, . . . , m/2 − 1. Then, by the arguments of the proof of Propositions 3 and 4 for studying the limit cycles of the m-piecewise discontinuous Liénard polynomial differential equation of degree n via de averaging method, we must study the simple zeros of the averaged function If as usual F (x) = a 0 + a 1 x + · · · + a n x n with a n = 0, then Consequently f 0 (r) is always a polynomial of degree at most n. Note that we have taken (−1) j in the expression of the function f 0 (r), but it could also be (−1) j+1 , this depends on the explicit expression of the function g m (x, y). But this change of sign in the function f 0 (r) does not affect its zeros.
Remark. From (15), for studying the simple zeros of the polynomial f 0 (r), we must know if the constants d i which depend on m are zero or not. The problem of solving the conjecture is reduced to know which d i are zero for a given m and to apply the Descartes theorem.
We recall that a function g : R → R is called odd if g(−r) = −g(r), and it is called even if g(−r) = g(r). If g is an odd polynomial, then g only has monomials of odd degree. If g is an even polynomial, then g only has monomials of even degree.  (x, y)) in an open sector defined by g m (x, y) = 0 and in its symmetric with respect to the origin of coordinates are equal. But in one of these sectors, cos i+1 θ is positive and in the other, it is negative because i is even. So the addition of the two integrals of d i over these two symmetric sectors is zero, and consequently d i holds. Therefore, from (15) if follows that f 0 (r) is an odd polynomial. Proof. If m is not a multiple of 4, the signs (−1) j and (−1) j+m/2 in an open sector and in its symmetric with respect to the origin of coordinates are different. Since in these two sectors, cos i+1 θ is positive because i is odd; again the addition of the two integrals of d i over these two symmetric sectors is zero, and consequently d i holds. Therefore, from (15) if follows that f 0 (r) is an even polynomial.  Note that for proving the conjecture for m = 6, from Proposition 7, we only need to show that d 2j = 0 for j = 1, 2, 3, . . . and to apply the arguments of the end of the proof of Proposition 3.
With the help of the program mathematica, we obtain for m = 6 and α = π/2 that ) is an incomplete beta function and Γ(z) is the Euler gamma function, for more details see [1]. Then, again we can verify that d 0 = 0, and we must compute d 2j for many j ∈ {1, 2, 3, . . .} and check that for those d 2j = 0. But we do not know how to prove that d 2j = 0 for all j = 1, 2, 3, . . .. If this can be proved, then the conjecture is proved for m = 6. We must prove that d 1 = 0. We have that An easy computation shows that Again note that for proving the conjecture for m = 8, from Proposition 8, we only need to show that d 2j+1 = 0 for j = 1, 2, 3, . . . and to apply the arguments of the end of the proof of Proposition 4.
Again using the program mathematica, we obtain for m = 8 and α = π/8 that Then, we can verify that d 1 = 0, and we can compute d 2j+1 for many j ∈ {1, 2, 3, . . .} and check that for those d 2j+1 = 0. But again we do not know how to prove that d 2j+1 = 0 for all j = 1, 2, 3, . . .. If this can be proved, then the conjecture is proved for m = 8. We must prove that d 0 = d 2 = 0. We have that Note that for proving the conjecture for m = 6, from Proposition 7, we only need to show that d 2j = 0 for j = 1, 2, 3, . . . and to apply the arguments of the end of the proof of Proposition 3.
With the program mathematica, we obtain for m = 10 and α = π/2 that Then, again we can verify that d 0 = d 2 = 0, and we can compute d 2j for many j ∈ {2, 3, . . .} and check that for those d 2j = 0. But we do not know how to prove that d 2j = 0 for all j = 2, 3, . . .. If this can be proved, then the conjecture is proved for m = 10 and so on.