On the 16th Hilbert Problem for Discontinuous Piecewise Polynomial Hamiltonian Systems

In this paper we study the maximum number of limit cycles of the discontinuous piecewise differential systems with two zones separated by the straight line y=0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$y=0$$\end{document}, in y≥0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$y\ge 0$$\end{document} there is a polynomial Hamiltonian system of degree m, and in y≤0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$y\le 0$$\end{document} there is a polynomial Hamiltonian system of degree n. First for this class of discontinuous piecewise polynomial Hamiltonian systems, which are perturbation of a linear center, we provide a sharp upper bound for the maximum number of the limit cycles that can bifurcate from the periodic orbits of the linear center using the averaging theory up to any order. After for the general discontinuous piecewise polynomial Hamiltonian systems we also give an upper bound for their maximum number of limit cycles in function of m and n. Moreover, this upper bound is reached for some degrees of m and n.


Introduction and Statement of Main Results
One of the most studied problems in the qualitatitve theory of the differential equations in the plane is to identify the maximum number of limit cycles that can exhibit a given class of differential systems. Thus a famous and challenging question is the Hilbert's 16th problem [22], which was proposed in 1900. In the second part of this question, Hilbert asked what is the maximum number of limit cycles that planar polynomial differential system of a given degree may have. Since 1900 many researchers are dedicated to study this problem and some excellent results were obtained, see for instance the survey paper [26]. But this question is far from being answered up to now, even for quadratic polynomial differential systems. Let H (n) be the maximum number of limit cycles that a planar polynomial differential system of degree n may have. Sometimes, H (n) is called Hilbert number. As far as we are concerned, the existing results showed H (2) ≥ 4, H (3) ≥ 13, H (4) ≥ 21, H (5) ≥ 33, H (6) ≥ 44, H (7) ≥ 65, H (8) ≥ 76, etc, see [15,17,19,20,27,33,37,39]. For n sufficiently large it is known that H (n) ≥ (n + 2) 2 ln(n + 2) 2 ln 2 , see [9,19,26]. In these last twenty years an increasing interest has appeared for studying the discontinuous piecewise smooth differential systems, stimulated by lots of nonsmooth or discontinuous phenomena that come from mechanical engineering with dry frictions, feedback control systems, electrical circuits with switches, neuron models, biology, see for example [1,6,7,11,32,[34][35][36] and the references therein. Furthermore, a particular interest is paid to the following discontinuous piecewise polynomial differential system where (x, y) ∈ R 2 and Z ± : R 2 → R 2 are two real polynomial vector fields. Here the x-axis is called the switching line or the discontinuity boundary. Throughout this paper we define the vector field on the switching line by Z + , or equivalently Z − , if Z + ≡ Z − on the entire switching line, so that the orbits of system (1) can be defined as in continuous systems. However, if Z + = Z − on the switching line, we adopt the so-called Filippov convention to define the orbits of the discontinuous piecewise differential system, see [12,25] for more details.
A crossing periodic orbit of a discontinuous piecewise polynomial differential systems (1) is a periodic orbit which intersect the swithching line at two crossing points. Here a point p in the switching line is said to be a crossing point if Z + 2 ( p)Z − 2 ( p) > 0, where Z ± 2 is the second coordinate of Z ± . A crossing periodic orbit isolated in the set all crossing periodic orbits is called a crossing limit cycle.
As in polynomial differential systems, we can also consider the Hilbert's 16th problem for discontinuous piecewise polynomial differential systems (1), i.e. what is the maximum number of crossing limit cycles that systems (1) can exhibit in function of the degrees of the polynomial vector fields Z ± . Compared with polynomial differential systems, the determination of the Hilbert number of discontinuous piecewise polynomial systems (1) is more difficult, even for the discontinuous piecewise linear differential systems. It was conjectured in [21] that a discontinuous piecewise linear differential system has at most 2 crossing limit cycles. A negative answer to this conjecture was given in [23] by a numerical example with 3 crossing limit cycles, the first proof with 3 crossing limit cycles was given in [30]. Up to now, the best result is still 3 for discontinuous piecewise linear systems, see [5,13,29] for more systems with three crossing limit cycles. Recently, it was proved in [10] that discontinuous piecewise quadratic differential systems of form (1) can have 16 crossing limit cycles through perturbing a quadratic isochronous center. Regarding discontinuous piecewise cubic systems, 18 crossing limit cycles were obtained in [18], but this number was updated as 24 in [16] later on.
We point out that the averaging theory, which has been extended recently for discontinuous piecewise smooth differential systems [3,24,28,38], is an important tools to study crossing limit cycles bifurcating from a periodic annulus. For instance using the averaging theory it was proved in [8] that the cyclicity of a Hopf bifurcation for planar linear-quadratic discontinuous polynomial differential systems is at least 5. In [31] the maximum number of crossing limit cycles of discontinuous discontinuous piecewise quadratic and cubic polynomial perturbations of a linear center was computed using the averaging theory of order n for n = 1, 2, 3, 4, 5. In [10] the averaging theory of order two was applied to discontinuous piecewise quadratic perturbations of quadratic isochronous centers. More results about limit cycles bifurcating from the periodic annulus of a linear center can be found in [4].
It is well known that limit cycles cannot exist in polynomial Hamiltonian systems. But it is possible for discontinuous piecewise polynomial Hamiltonian systems to have crossing limit cycles as it was studied in [40], where a lower bound and an upper bound for the maximum number of small amplitude crossing limit cycles bifurcating from a non-smooth focus were provided. Motivated by the work [40], in this paper we further study the maximum number of crossing limit cycles of discontinuous piecewise polynomial Hamiltonian systems. This can be regarded as an extension of the Hilbert's 16th problem to discontinuous piecewise polynomial Hamiltonian systems.
Let H + (x, y) and H − (x, y) be two real polynomials of degree m + 1 and n + 1 given by First we consider a discontinuous piecewise polynomial Hamiltonian perturbation of the linear centerẋ = −y,ẏ = x, namely where ε ∈ R is a perturbation parameter with |ε| > 0 sufficiently small, the subscripts x and y denote the derivatives with respect to x and y respectively. Notice that system (3) is a discontinuous piecewise polynomial Hamiltonian system with the first integrals In this paper our first goal is to study the maximum number of crossing limit cycles of system (3), which can bifurcate from the period annulus of the linear centerẋ = −y,ẏ = x. Before stating the main theorem we give the following useful result.

Proposition 1
For |ε| > 0 sufficiently small the number of crossing periodic orbits for the discontinuous piecewise polynomial Hamiltonian system (3) and for the discontinuous piecewise polynomial Hamiltonian system are the same.
According to Proposition 1, it is enough to consider system (4) in order to obtain the maximum number of crossing limit cycles that system (3) can have for |ε| > 0 sufficiently small. Applying the averaging method, up to any order, to system (4) we obtain the next result.
Theorem 2 For |ε| > 0 sufficiently small the maximum number of crossing limit cycles, bifurcating from the periodic orbits of the linear differential centerẋ = −y,ẏ = x, which can be obtained with the averaging theory for the discontinuous piecewise polynomial Hamiltonian system (4) with m ≤ n is at most where [z] denote the integer part function of z ∈ R. Moreover this maximum is achievable.
Our second goal deals with the maximum number of crossing limit cycles for the general discontinuous piecewise polynomial Hamiltonian system where H ± (x, y) are given in (2). The following theorem provides an upper bound for a such maximum.

Theorem 3
Using the averaging theory the discontinuous piecewise polynomial Hamiltonian system (5) with m ≤ n has at most mn 2 resp. n(n − 1) 2 It is worth mentioning that Yang, Han and Huang in the proof of [40, Theorem 1.2] also observed that [nm/2] is an upper bound for the maximum number of crossing limit cycles that discontinuous piecewise polynomial Hamiltonian system (5) can have. However, we provide a better upper bound for the case of m = n in Theorem 3, because [n(n − 1) Regarding the lower bound, it follows from [40, Theorem 1.2] that [(n − 1)/2] + [m/2] small amplitude crossing limit cycles can bifurcate from a non-smooth focus if m ≤ n, so that [(n − 1)/2] + [m/2] is a lower bound for the maximum number of crossing limit cycles that system (5) with m ≤ n can have. According to Theorem 2, we can update the lower bound as Although the lower bounds given in [40, Theorem 1.2] and Theorem 2 are the same for m ≥ 2, the methods used to obtain the bounds are different. Our method, based in the application of the averaging theory to discontinuous piecewise polynomial Hamiltonian perturbations of a linear center, allows to obtain large amplitude crossing limit cycles. Thus, in terms of amplitude, we give a new lower bound for the maximum number of large amplitude crossing limit cycles. More importantly, combining Theorems 2 and 3 we directly get the exact maximum number of crossing limit cycles for some classes of discontinuous piecewise polynomial Hamiltonian systems (5) as it is stated in the following result.
Theorem 4 Consider the discontinuous piecewise polynomial Hamiltonian system (5) with m ≤ n. Using the averagin theory the following statements hold.
(i) If m = 1, the maximum number of crossing limit cycles that system (5) can have is exactly [n/2].
(ii) If m = n = 2, the maximum number of crossing limit cycles that system (5) can have is exactly 1.
This paper is organized as follows. Section 2 contains a brief review for the averaging theory of any order. In Sect. 3 we study the perturbation problem (3) and give the proofs of Proposition 1 and Theorem 2. Section 4 is devoted to the proof of Theorem 3.

Preliminaries
In this section we recall the averaging theory and some important theorems in order to prove our main results.
As a main tool studying limit cycle bifurcations the averaging theory has been generalized for discontinuous piecewise smooth differential systems, see [24,28,38]. In what follows we shall state the averaging method of any order by considering the perturbed discontinuous piecewise smooth differential system where f ± , g ± : R 2 → R are real polynomials, and ε ∈ R is a perturbation parameter. Using the polar coordinates (x, y) = (r cos θ, r sin θ) and taking θ as the new independent variable, for any given N ∈ N + we transform system (6) into where .., N , are analytical functions of period 2π in the variable θ and are given by According to the results of [24] we get that the averaged function F i (r ) : (0, +∞) → R of order i = 1, 2, ..., N is The functions y ± i : [0, 2π] × (0, +∞) → R are defined recurrently as where S l is the set of all l-tuples of non-negative integers (b 1 , b 2 , ..., b l ) satisfying b 1 +2b 2 + · · · + lb l = l, and L = b 1 + b 2 + · · · + b l . Moreover we are assuming that F ± 0 = 0 in (9) for convenience.
The next theorem proved in [24] implies that we can study the zeros of these averaged functions in order to obtain crossing limit systems of system (6) bifurcating from the periodic annulus of the linear centerẋ = −y,ẏ = x.
The following two theorems will be used to prove our main results, see [2] and [14] for their proofs.

Proofs of Proposition 1 and Theorem 2
In this section we study the maximum number of crossing limit cycles bifurcating from the period annulus of the linear centerẋ = −y,ẏ = x via discontinuous piecewise polynomial Hamiltonian perturbations. We start with the proof of Proposition 1.
As it was indicated in Proposition 1 it is enough to consider system (4) in order to obtain the maximum number of crossing limit cycles that system (3) can have for |ε| > 0 sufficiently small. Next we will apply the averaging method of any order to system (4) in subsections 3.1, 3.2, 3.3.
and a ± i = a ± i0 . More precisely, we write where if m is even, if n is even.

First Order Averaging Method
Applying the first order averaging method to system (4) we have the following.

Proposition 8 For |ε| > 0 sufficiently small the first order averaging method predicts at most [n/2] crossing limit cycles for the discontinuous piecewise polynomial Hamiltonian system (4) with m ≤ n. Moreover this number is reachable.
Proof Writing the discontinuous piecewise polynomial Hamiltonian system (4) in the form of system (7) and using the expression (11), we get . Then, according to (8) and (9), the first order averaged function is Here the last equality is due to k + ≤ k − under the assumption m ≤ n. Clearly F 1 (r ) is a polynomial of degree 2k − , and all odd order terms vanish. This means that F 1 (r ) has at most k − positive real simple zeros. Since all coefficients are free, we can choose them such that this number is reachable. By Theorem 5 this proposition is proved because k − = [n/2] as defined in (12).

Second Order Averaging Method
In order to apply the second order averaging method to system (4), we have to take F 1 = 0. From (13) this is equivalent to take for 0 = 1, 2, ..., k + , and In particular we have the following.

Proposition 9 For |ε| > 0 sufficiently small the second order averaging method predicts at most [(n − 1)/2] + [m/2] crossing limit cycles for the discontinuous piecewise polynomial Hamiltonian system (4) with m ≤ n. Moreover this number is reachable.
Proof Writing the discontinuous piecewise polynomial Hamiltonian system (4) into the form of system (7) and using (11) and (14), we have Then and Therefore from (15) and (16) it follows that and similarly, Associate with (8) and (9), the above calculations yield the second order averaged function where Here l + ≤ l − because we are assuming that m ≤ n. Observe that F 21 (r ) (resp. F 22 (r )) has at most k + (resp. l − ) variations of signs. Hence, it follows from Theorem 6 that F 21 (r ) (resp. F 22 (r )) has at most k + (resp. l − ) positive real zeros. Furthermore, since all coefficients of F 21 (r ) and F 22 (r ) can be chosen arbitrarily, by Theorem 6 again we can choose them in such a way that F 21 (r ) (resp. F 22 (r )) has exactly k + (resp. l − ) positive real simple zeros and these zeros of F 21 (r ) are different from the ones of F 22 (r ). Consequently, F 2 (r ) has at most k + +l − positive real simple zeros from (17), and this number is reachable. Using Theorem 5, we conclude the proof of Proposition 9 because l − = [(n − 1)/2] and k + = [m/2] as defined in (12).

Higher Order Averaging Method
Now we apply the higher order averaging method to system (4). To do this, in the next lemma we explore the values of the coefficients for which the first and second order averaged functions are identically zeros, i.e. F 1 = F 2 = 0.

Lemma 10
Consider the first and second order averaged functions F 1 and F 2 given in (13) and (17) respectively. Then F 1 = F 2 = 0 if and only if condition (14) and one of the following conditions holds: Proof From the expression of F 1 given in (13) we directly obtain that F 1 = 0 if and only if condition (14) holds. On the other hand, it follows from (18) (17). This ends the proof the lemma.
In order to calculate the higher order averaged functions for system (4) we need some technical lemmas.
Proof From Lemma 11 we know thatF ± 1 (s, r ) are odd functions with respect to s, i.e. F ± 1 (−s, r ) = −F ± 1 (s, r ). Then using the transformation τ → −τ we havẽ where the second and fifth equalities are due to the oddness ofF ± 1 (s, r ). Henceỹ ± 1 (s, r ) are even functions with respect to s. Again by Lemma 11 we get that are odd functions with respect to s, where the assumptionF ± 0 = 0 is used. Since the integral of an odd function is always an even function,ỹ ± 2 (s, r ) are even functions with respect to s by (19) and the definition ofỹ ± 2 (s, r ). By the method of induction we finally obtain Lemma 12.
Having these lemmas we can prove the following proposition.

Proposition 13
If the discontinuous piecewise polynomial Hamiltonian system (4) satisfies (14) and the condition (i) of Lemma 10, then the averaged function of any order is identically zero.
Thus system (25), or equivalently (24), has at most n(n − 1) solutions by Theorem 7 again. Finally, using the symmetry of the solutions of (24), we get that system (5) has at most [n(n − 1)/2] crossing limit cycles in the case of m = n. From these last two paragraphs the proof of Theorem 3 follows.