Centers and Uniform Isochronous Centers of Planar Polynomial Differential Systems

For planar polynomial vector fields of the form (-y+X(x,y))∂∂x+(x+Y(x,y))∂∂y,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} (-y+X(x,y))\dfrac{\partial }{\partial x}+(x+Y(x,y))\dfrac{\partial }{\partial y}, \end{aligned}$$\end{document}where X and Y start at least with terms of second order in the variables x and y, we determine necessary and sufficient conditions under which the origin is a center or a uniform isochronous centers.


Introduction and Statement of the Main Results
A real planar polynomial differential system is a system of the forṁ x = P(x, y),ẏ = Q(x, y), (1) where the dot denotes derivative with respect to an independent variable here called the time t, and P and Q are real coprime polynomials in R[x, y]. We say that the polynomial differential system (1)  The study of the centers of polynomial differential systems (1) has a long history. The first works are due to Dulac [4] and Poincaré [11]. Later on where developed by Lyapunov [9], Bendixson [2], Frommer [5] and many others.
Assume that the origin of the polynomial differential system (1) is a center. It is wellknown that, after a linear change of variables and a constant scaling of the time variable (if necessary), system (1) can be written in one of the next three forms: where F 1 (x, y) and F 2 (x, y) are polynomials without constant and linear terms defined in a neighborhood of the origin. Then the origin O of the polynomial differential system (1) is called linear type, nilpotent or degenerate if after a linear change of variables and a scaling of the time it can be written as the first, second and third system of (2), respectively.
A center O of system (1) is a uniform isochronous center if the equality xẏ − yẋ = a(x 2 + y 2 ) holds with a = 0; or equivalently if in polar coordinates (r, θ) defined by x = r cos θ, y = r sin θ , we have thatθ = a.
It is known that isochronous centers and in particular isochronous uniform centers are linear type centers. This is due to the fact that blowing up the origin where the center is located for the linear type centers the origin blows up to a periodic orbit, but for nilpotent and degenerate centers the origin blows up to a graphic and consequently the periodic orbits near the origin cannot have constant period. We recall that a graphic is formed by finitely many singular points and orbits connecting these points in such a way that in one of the two sides of the graphic a return Poincaré map is defined. To see examples of isochronous centers which are not uniform, see for instance [8].
The aim of this paper is to provide new results on the linear type centers and on the uniform isochronous centers. More precisely, the classical Poincaré-Lyapunov Theorem (see Theorem 1) provides information about the analytical first integral which exists in a neighborhood of a linear type center localized at the origin of coordinates of a polynomial differential system. While our Theorem 2 is a kind of inverse theorem of the Poincaré-Lyapunov Theorem, i.e. given an analytical function satisfying the conditions of the first integral of the Poincaré-Lyapunov Theorem we characterize all the polynomial differential systems having such analytical function as a first integral, and these polynomial differential system have a linear type center at the origin of coordinates.
On the other hand in our Theorems 4 and 7 we give two new and more precise characterizations of the polynomial differential systems having an isochronous center at the origin. Finally in our Corollary 8 and Theorem 10 we improve the previous two characterizations of the isochronous centers for the polynomial differential systems whose nonlinearities are homogeneous polynomials of the same degree.
The following necessary and sufficient condition in order that the origin O of the first polynomial differential of system (2) be a center was obtained by Poincaré, and it was extended to analytic differential systems by Lyapunov (see for more details [6,9]). Theorem 1 A planar polynomial differential systeṁ of degree m has a center at the origin if and only if it has an analytic first integral of the form where X j , Y j and H j are homogenous polynomials of degree j.
One of the main objectives of the present paper is to study the centers and the uniform isochronous centers using the inverse theory (see for instance [7,13]). More precisely, we want to determine the polynomials X j and Y j of system (3) in order that a function of the form (4) be a first integral of the polynomial differential system (3) in a neighborhood of the origin O.
As usual the Poisson bracket of the functions f (x, y) and g(x, y) is defined as Our first result provides the expression of the polynomial differential systems (3) in function of its first integral (4).

Theorem 2
Given an analytic function of the form (4) a polynomial differential system having such function as a first integral can be written aṡ where g 0 = 1 and g m+1− j is a homogenous polynomial of degree m + 1 − j ≥ 0 satisfying Theorem 2 is proved in Sect. 2.
We remark that Theorem 2 shows that all polynomial differential systems having a first integral of the form (4) have a center at the origin, because we have proved that such systems are of the form (3) and consequently they have a focus or a center at the origin, but the existence of the first integral forces that the origin is a center.
On the other hand, if a polynomial differential system has a linear type center at the origin by Theorem 1 it has an analytic first integral of the form (4), and the differential system has the structure described in Theorem 2. So this structure is necessary in order that a polynomial differential system has a center at the origin. Now we shall put our attention on the uniform isochronous centers by using Theorem 2.
The following result is well-known, see for instance [3].

Proposition 3 Assume that a planar polynomial differential system of degree m has a center at the origin of coordinates. Then this center is a uniform isochronous center if and only if doing a linear change of variables and a scaling of the time the system can be written aṡ
where ϕ n−1 = ϕ n−1 (x, y) is an arbitrary homogenous polynomial in x and y of degree n − 1.
This result can be improved as follows. Thus, using Theorem 2 we provide a first characterization of the uniform isochronous centers. (7) with

Theorem 4 A polynomial differential system (3) has a uniform isochronous center at the origin if and only if the system can be written as
where H 2 = (x 2 + y 2 )/2, g j for j = 1, . . . , m − 1 are arbitrary homogeneous polynomials of degree j, ϒ j for j ≥ 1 are homogenous polynomials of degree j such that and the system has the first integral in a neighborhood of the origin.
The following result is well-known (see for instance [10]) and give the linearization criterion for the isochronocity of a center.

Theorem 5 A center of an analytic system is isochronous if and only if there exits an analytic change of coordinates X
Corollary 6 (Linearization of uniform isochronous center) Under the assumptions of Theorem 4 we get that by the analytic change of coordinates Theorem 4 and Corollary 6 are proved in Sect. 3. In the next result we provide a second characterization of the uniform isochronous centers.

Theorem 7
The origin of the polynomial differential system (3) is a uniform isochronous center if and only if this system can be written as (7) and is a periodic solution with the initial conditions (ε, 0) where ε is a small parameter, and s j (t) is a convenient 2π-periodic function such that s 1 (t) = 1, and s j (0) = s j (2π) = 0 for j > 1.
Theorem 7 is proved in Sect. 4. The next corollary provides a characterization of the linear type centers of polynomial differential systems when their nonlinearities are homogenous polynomials.

Corollary 8
Under the assumptions of Theorem 2 the planar polynomial differential system of degree mẋ where X m and Y m are homogenous polynomials of degree m, has a center at the origin if and only if it can be written aṡ where and g m−1 = g m−1 (x, y) is a homogenous polynomial of degree m − 1 satisfying Corollary 8 is proved in Sect. 5.
In [3] Conti proved the following result.
Theorem 9 The polynomial differential system (9) with

has a uniform isochronous center at the origin if either m is even, or m is odd and
Conti's result can be improved as follows.
Theorem 10 The polynomial differential system (9) has a uniform isochronous center at the origin if and only if this system can be written aṡ where H 2 = (x 2 + y 2 )/2 and g m−1 = g m−1 (x, y) is an arbitrary homogenous polynomial of degree m − 1 satisfying (11). Moreover this system has the rational first integral Theorem 10 is proved in Sect. 5. Theorem 10 characterizes the form of the polynomial uniform isochronous centers with homogeneous nonlinearities.
Note that, under the assumptions of Theorem 10 and using the notation of Theorem 4, the first integral F 1/(m−1) has the following development as the origin i.e. system (12) has a local first integral at the origin having H 2 as a factor, where ϒ j is a convenient homogenous polynomial of degree j.

Corollary 11 (Linearization of uniform isochronous centers with homogenous nonlinearity)
Under the assumptions of Theorem 10 we get that by the analytic change of coordinates in a neighborhood of the origin 1/(m−1) , Corollary 11 is proved in Sect. 5.

Proof of Theorem 2
Proof of Theorem 2 Consider a general polynomial vector field of degree m that we write as where X j and Y j for j = 0, 1, . . . , m are homogenous polynomials of degree j. Since the analytic first integral H starts with H 2 = (x 2 + y 2 )/2, without loss of generality this implies that X 0 (x, y) = Y 0 (x, y) = 0, X 1 (x, y) = −y and Y 1 (x, y) = x. Hence the following infinite number of equations follow The first equation can be rewritten as by solving with respect to X 2 and Y 2 we obtain the following polynomial solutions where g 1 = g 1 (x, y) is an arbitrary homogenous polynomial of degree one. By substituting these polynomials into the second equation of (13) we get By solving this equation with respect to X 3 and Y 3 we have where g 2 = g 2 (x, y) is an arbitrary homogenous polynomial of degree two. By continuing this process recursively we obtain X 4 , Y 4 , . . . , X m , Y m . By substituting X j and Y j for j = 1, . . . , m into the rest of equations (13) we deduce the partial differential equations (6).
Introducing the respectively notations we get with g 0 = 1. Thus the proof of the theorem follows.
Remark 12 A polynomial differential system having a linear type center at the origin can written as system (5), consequently this system gives a necessary condition in order to have a linear type center, but this condition is not sufficient. Indeed it is known (see [1]) that a quadratic differential system with a center at the origin can be written aṡ or equivalently (see (10) for m = 2) It is well known that there are values of the parameters λ 2 , λ 3 , λ 4 , λ 5 and λ 6 for which the origin is a focus. From (6) and (14) it follow (15).
The following result is due to Liapunov, see Theorem 1, page 276 of [9].
Theorem 14 Let U = U (x, y) be a homogenous polynomial of degree k. The linear partial differential equation has a unique homogenous polynomial solution V of degree k if k is odd; and if V is a homogenous polynomial solution when k is even then V + c(x 2 + y 2 ) k/2 where c ∈ R are all the homogenous polynomial solutions.

Proof of Theorem 4 and Corollary 6
First we prove the following lemma.

Lemma 16 Under the assumptions of Theorem 2 and assuming that
for n > 1. Then Proof Substituting the polynomial H n+1 into X n (x) we obtain Here we use the property of the homogenous polynomial jg j = x ∂g j ∂ y + y ∂g j ∂ y . In a similar way we obtain the expression for X n (y).

Proof of Theorem 4
If the origin is a linear type center of a polynomial differential system then in view of Theorem 2 we obtain that this system can be written as (5), or equivalently, This center is uniform isochronous center if and only if xẏ − yẋ = x 2 + y 2 , consequently the following relation holds xX (y) − yX (x) = 0, i.e. x In view of Corollary 16 X (x) and X (y) can be written as Substituting X (x) and X (y) into (5) we geṫ To finish the proof of the theorem we prove that conditions (6) hold. Indeed in view of the relations