Limit Cycles of a Class of Generalized Liénard Polynomial Equations

We prove that the generalized Liénard polynomial differential system 1ẋ=y2p−1,ẏ=−x2q−1−εf(x)y2n−1,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \dot x = y^{2p-1}, \qquad \dot y= -x^{2q-1}-\varepsilon f(x)\, y^{2n-1}, $$\end{document}where p, q, and n are positive integers; ε is a small parameter; and f(x) is a polynomial of degree m which can have [m/2] limit cycles, where [x] is the integer part function of x.


Introduction and Statement of the Main Results
In 1900, Hilbert [2] in the second part of his 16-th problem, proposed to find a 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. The 16-th problem, except the one related with the Riemann hypothesis, 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 during the XX century. Until now, it has not proved the existence of such a uniform upper bound. This problem remains open even for the polynomial differential systems of degree 2. However, it is not difficult to see that any finite configuration of limit cycles is realizable for some polynomial differential system (see for details [6]).
Following Smale [8], many authors consider an easier and special class of polynomial differential systems, the Liénard polynomial differential equation: where and the dot denotes derivative with respect to the time t. The Liénard polynomial (2) of second order can be written as the Liénard differential system of first ordeṙ For these systems, the existence of uniform bounds also remain unproved.
Here, we want to study the number of limit cycles of the following generalized Liénard polynomial differential system of first ordeṙ where p, q, and n are positive integers; ε is a small parameter; and f (x) is the polynomial (3). In fact, system (4) with ε = 0 is a Hamiltonian system with Hamiltonian This system has a global center at the origin of coordinates, i.e., the periodic orbits surrounding the origin filled the whole plane R 2 , and we want to study how many periodic orbits persist after perturbing the periodic orbits of this center as in the system (4) for ε = 0 sufficiently small. Let [x] denotes the integer part function of x ∈ R. Our main result is the following one.
Theorem 1 For ε = 0 sufficiently small, the polynomial differential system (4) can have at least [m/2] limit cycles if m is the degree of the polynomial f (x).
Theorem 1 is proved in Section 2 using averaging theory of first order. See the Appendix for a summary of the results on averaging theory used here. Note that the maximum number of limit cycles obtained using the averaging theory of first order only depends on m, the degree of the polynomial f (x), and it is independent of p, q, and n.
System (4) with p = q = n = 1 was studied by Lins et al. [5] in 1977, and for p = n = 1 and q, arbitrary has been studied by Urbino et al. [9] in 1993. Other generalizations of Liénard systems have also been studied (see for instance [3,7]).

Proof of Theorem 1
The key point in the proof of many results on differential systems is the election of good coordinates for studying the systems.
Following Liapunov [4], we introduce the (p, q)-trigonometric functions z(θ) = Csθ and w(θ) = Snθ as the solution of the following initial value probleṁ It is easy to check that the functions Csθ and Snθ satisfy the equality p Cs 2q θ + q Sn 2p θ = 1.
For p = q = 1, we have that Csθ = cosθ and Snθ = sin θ , i.e., the (1, 1)-trigonometric functions are the classical ones. It is known that Csθ and Snθ are T -periodic functions with where (x) is the gamma function. The (p, q)-polar coordinates (r, θ) are defined as x = r p Csθ, y = r q Snθ.
Taking as independent variable the angular variable θ , the differential system (6) writes Now, we shall apply the Theorem 1 of the Appendix. Using the notation defined there, we have Then, using (3), we have that From [4], it follows that b k = 0 if k is odd, and clearly b k > 0 if k is even. So we have We recall the Descartes Theorem about the number of zeros of a real polynomial (for a proof see for instance [1]). Descartes Theorem. Consider the real polynomial p(x) = a i 1 x i 1 + a i 2 x i 2 + · · · + a i k x i k with 0 ≤ i 1 < i 2 < · · · < i k and a i j = 0 real constants for j ∈ {1, 2, · · · , k}. When a i j a i j +1 < 0, we say that a i j and a i j +1 have a variation of sign. If the number of variations of signs is m, then p(x) has at most m positive real roots. Moreover, it is always possible to choose the coefficients of p(x) in such a way that p(x) has exactly k − 1 positive real roots.
By Descartes Theorem, the polynomial g(r) has at most [m/2] positive roots r. Then, by Theorem 2, it follows that the differential (7) can have [m/2] periodic solutions, and consequently the differential systems (6) or equivalently (4) can have [m/2] periodic solutions which can be chosen alternatively stable and unstable. In short, the differential system (4) can have [m/2] limit cycles. This completes the proof of Theorem 1.