Linear type global centers of linear systems with cubic homogeneous nonlinearities

A center p of a differential system in R2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {R}^2$$\end{document} is global if R2\{p}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {R}^2 \setminus \{p\}$$\end{document} is filled of periodic orbits. It is known that a polynomial differential system of degree 2 has no global centers. Here we characterize the global centers of the differential systems x˙=ax+by+P3(x,y),y˙=cx+dy+Q3(x,y),\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} {\dot{x}} = ax + b y +P_3(x,y), \quad {\dot{y}} = c x + d y + Q_3(x,y), \end{aligned}$$\end{document}with P3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$P_3$$\end{document} and Q3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Q_3$$\end{document} homogeneous polynomials of degree 3, and such that the center has purely imaginary eigenvalues, i.e. a linear type center.


Introduction and statement of the main results
The notion of center goes back to Poincaré and Dulac, see [6,10]. They defined a center for a vector field on the real plane as a singular point having a neighborhood filled of periodic orbits with the exception of the singular point. The problem of distinguishing when a monodromic singular point is a focus or a center, known as the focus-center problem started precisely with Poincaré and Dulac and is still active nowadays with many questions still unsolved. These B Johanna D. García last years also the centers are perturbed for studying the limit cycles bifurcating from their periodic solutions, see for instance [1,3]. If an analytic system has a center, then it is known that after an affine change of variables and a rescaling of the time variable, it can be written in one of the following three forms: x = −y + P(x, y),ẏ = x + Q(x, y), called linear type center, which has a pair of purely imaginary eigenvalues, x = y + P(x, y),ẏ = Q(x, y) called nilpotent centerẋ = P(x, y),ẏ = Q(x, y) called degenerated center, where P(x, y) and Q(x, y) are real analytic functions without constant and linear terms defined in a neighborhood of the origin.
We recall that a global center for a vector field on the plane is a singular point p having R 2 filled of periodic orbits with the exception of the singular point. The easiest global center is the linear centerẋ = −y,ẏ = x. It is known (see [2,11]) that quadratic polynomial differential systems have no global centers. The global degenerated centers of homogeneous or quasi-homogeneous polynomial differential systems were characterized in [4] and [8], respectively. However the characterization of the global centers in the cases that the center is nilpotent or of linear-type has not been done. This is the first paper in which such classification is done for the linear-type centers for the systems having a linear part at the origin with purely imaginary eigenvalues and cubic homogeneous nonlinearities. A polynomial differential system can be extended in a unique analytic way to infinity using the Poincaré compactification, for more details see Chapter 5 of [7].
We now state our main results. We first provide normal forms for the differential systems to be studied. Theorem 1 Any vector field having at the origin of coordinates a singular point with purely imaginary eigenvalues of the form linear plus cubic homogeneous terms and no infinite singular points in the Poincaré disc after a linear change of variables and a rescaling of the independent variable can be written aṡ where α = ±1 and c, d, ω, μ, p 1 , p 2 , p 3 ∈ R with c = 0, ω > 0 and μ > −1/3.
The proof of Theorem 1 is given in Sect. 2. A singular point p of a planar system is called hyperbolic if both eigenvalues of the Jacobian matrix at p have real part different from zero. It is called semi-hyperbolic if only one of the eigenvalues of the Jacobian matrix at p is zero, and if both eigenvalues of the Jacobian matrix at p are zero but this matrix is not identically zero it is called nilpotent. Finally, if the Jacobian matrix at p is identically zero then p is said to be linearly zero.
Let q be an infinite singular point and let h be a hyperbolic sector of q. We say that h is degenerated if its two separatrices are contained at infinity, that is, are contained in the boundary of the Poincaré disc.
It follows from Theorem 2.15 (for hyperbolic singular points), Theorem 2.19 (for semihyperbolic singular points) and Theorem 3.5 (for nilpotent singular points) in [7] that a singular point which is either hyperbolic, semihyperbolic or nilpotent cannot be formed by two degenerated hyperbolic sectors. So in order that an infinite singular point q can be formed by two degenerated hyperbolic sectors it must be linearly zero.
Theorem 2 Any vector field having at the origin of coordinates a singular point with purely imaginary eigenvalues of the form linear plus cubic homogeneous terms such that all infinite singular points in the Poincaré disc are linearly zero after a linear change of variables and a rescaling of the independent variable can be written as one of the following three systems: (2) (3) where α = ±1 and c, d, ω, The proof of Theorem 2 is given in Sect. 3.
The proof of Theorem 3 is given in Sect. 4. In order to state the following theorem we introduce the notation

c has a global center at the origin if and only if cα > 0.
Moreover, system (IV) with p 1 = −dα/c has a global center at the origin if and only if either cα > 0 and S 1 < 0, or cα > 0, S 1 > 0 and cdp 2 (e) A system (V) has a global center at the origin if and only if cα( p 2 + 4 p 2 1 + p 2 2 ) ≥ 0 and cα( p 2 − 4 p 2 1 + p 2 2 ) ≤ 0. The proof of Theorem 4 is given in Sect. 5.
The proof of Theorem 5 is given in Sect. 6.

Theorem 6 Systems (3) have a linear type center at the origin and no more finite singular points with all infinite singular points formed by two degenerated hyperbolic sectors if and only if they can be written as one of the following systemṡ
with α = ±1, c, d ∈ R, c = 0, ω > 0 and cα > 0. These systems are Hamiltonian with The proof of Theorem 6 is given in Sect. 7.

Theorem 7 Systems (4) have a linear type center at the origin and no more finite singular points with all infinite singular points formed by two degenerated hyperbolic sectors if and
only if they can be written as one of the following systemṡ with α = ±1, c, d ∈ R, c = 0, ω > 0 and cα > 0. These systems are Hamiltonian with The proof of Theorem 7 is given in Sect. 8. An immediate consequence of Theorems 4, 5, 6 and 7 is the following.

Proof of Theorem 1
Doing a linear change of variables and a rescaling of the independent variable, planar cubic homogeneous differential systems can be classified into the following ten classes, see [4]: whose infinity in the Poincaré disc is formed by singular points; where α = ±1 and whose infinite singular points in the Poincaré disc are the real solutions of αx 4 = 0 at infinity.
whose infinite singular points in the Poincaré disc are the real solutions of 4x 3 y = 0 at infinity.
where α = ±1 and whose infinite singular points in the Poincaré disc are the real solutions of 6αx 2 y 2 = 0 at infinity.
In this theorem we are interested in systems without infinite singular points. The only classes of systems that have no infinite singular points are systems (ix) with μ > −1/3.
For studying the cubic planar polynomial vector fields having linear and cubic terms being the origin a singular point, it is sufficient to add to the above family in (ix) a linear part. This is due to the fact that the linear changes of variables that are done to obtain the classes (i)-(x) are not affine, they are strictly linear. So a linear plus a cubic vector field being the origin a singular point with no infinite singular points in the Poincaré disc can be written aṡ for some real constants a, b, c, d and with α = ±1, μ > −1/3. The eigenvalues of the linear part of system (6) at the origin are In order to have a linear type center at the origin, these eigenvalues must be ±ωi, for some ω > 0. So a = −d and d 2 + bc = −ω 2 . We see that bc = 0, otherwise the left hand side would be non-negative. Then we can solve for b and we get b = −(d 2 + ω 2 )/c. So system (6) becomesẋ This completes the proof of the theorem.

Proof of Theorem 2
The proof of Theorem 2 is the same as the proof of Theorem 1 taking into account that system (ii) has only the origin of the local chart U 2 as the infinite singular points which is linearly zero if and only if p 3 = 0. This gives system (2) in Theorem 2.
Moreover, system (iv) has only the origin of the local charts U 1 and U 2 as infinite singular points and they are both linearly zero if and only if p 1 = p 3 = 0. This gives system (3) in Theorem 2.
System (vii) has an infinite singular point which is the origin of the local chart U 1 and is linearly zero if and only if p 1 = 0. This gives system (4) in Theorem 2.
Any other of the systems (i)-(x) either the infinite is formed by singular points, or they do not provide infinite singular points in the Poincaré disc, or they provide at least one infinite singular point which is not linearly zero. After this, the proof is exactly the same as in the proof of Theorem 1 (that is, adding the linear part).

Proof of Theorem 3
We will use the following result, proved in [9].  First we write the linear part of system (1) into its real Jordan normal form doing the change of variables x = (ωY − d X)/c, y = X and then we do the rescaling τ = ωt. Doing so, the new system can be written as in (7) with Condition (a) in Theorem 8 yields p 1 = p 2 = p 3 = 0 and so we obtain system (I) in the statement of the theorem. Note that in this case system (I) is Hamiltonian with the Hamiltonian stated in the statement of the theorem.
For condition (c) of Theorem 8 with p 2 1 + p 2 2 + p 2 3 = 0 systems (1) have no linear type center at the origin. This completes the proof of the theorem.

Proof of Theorem 4
We will study each of the systems (I)-(V) separately.
First note that in case of system (I) we get that the system is Hamiltonian. It was proved in [5] that in this case there unique condition so that the unique finite singular point is the origin is cα > 0. Hence statement (a) is proved.
It was proved in [5] that in this case the unique infinite singular point which is the origin of the local chart U 2 is formed by two degenerated hyperbolic sectors if and only if cα > 0 and that in this case there are no more finite singular points besides the origin. Condition (b) in Theorem 8 with p 2 1 + p 2 2 + p 2 3 = 0 yields the conditions (b.1) p 2 = p 1 (d 2 + ω 2 )/(cd) with cd = 0 and p 1 = 0 because otherwise this system becomes a system (VI); (b.2) p 1 = 0, d = 0 and p 2 = 0 because otherwise this system becomes a system (VI); System (2) with the conditions (b.1) becomes The singular points of system (10) are In order that all the candidates to be singular points different from the origin do not exist, and taking into account that condition cα > 0, dp 1 > 0 and cd(d 3 α + (cp 1 + dα)ω 2 ) < 0 never hold, we must have cα > 0 and dp 1 < 0. Now we study the infinite singular points. We already know that the unique infinite singular point is the origin of the local chart U 2 that is linearly zero. On the local chart U 2 we havė Doing two horizontal blow ups and one vertical blow up and taking into account that cα > 0 and dp 1 < 0 we conclude that the origin of the local chart U 2 is formed by two degenerated hyperbolic sectors. So we obtain system (VII). System (2) with the conditions (b.2) becomes The singular points of system (11) are In order that all the candidates to be singular points different from the origin do not exist, and taking into account that condition cα > 0, cp 2 > 0 and c(c 2 p 2 + αω 2 ) < 0 is null, we must have cα > 0 and cp 2 < 0. Now we study the infinite singular points. We already know that the unique infinite singular point is the origin of the local chart U 2 that is linearly zero. On the local chart U 2 we havė Doing one horizontal and one vertical blow up, taking into account that cα > 0 and cp 2 < 0 we conclude that the origin of the local chart U 2 is formed by two degenerated hyperbolic sectors. So we obtain system (VIII). For condition (c) of Theorem 8 with p 2 1 + p 2 2 + p 2 3 = 0 systems (2) have no linear type center at the origin. This completes the proof of Theorem 5.

Proof of Theorem 6
First we write the linear part of system (3) into its real Jordan normal form doing the change of variables x = (ωY − d X)/c, y = X and then we do the rescaling τ = ωt. Doing so, the new system can be written as in (7) with p 2 + 3α = 0 and cα > 0, then the origin of the local chart U 1 is formed by two degenerated hyperbolic sectors. Otherwise there are parabolic sectors arriving at the origin of the local chart U 1 and so it is not formed by two degenerated hyperbolic sectors. These conditions are equivalent to: either p 2 + 3α > 0, c > 0 and α = 1; or p 2 + 3α < 0, c < 0 and α = −1; or p 2 + 3α = 0 and cα > 0. On the local chart U 2 system becomeṡ Doing one horizontal and one vertical blow up, we get that if c( p 2 −3α) < 0 and α( p 2 −3α) < 0, or p 2 − 3α = 0 and cα > 0, then the origin of the local chart U 2 is formed by two degenerated hyperbolic sectors. Otherwise, there are parabolic sectors arriving to the origin of the local chart U 2 and so it is not formed by two degenerated hyperbolic sectors. These conditions are equivalent to: either p 2 − 3α > 0, c < 0 and α = −1; or p 2 − 3α < 0, c > 0 and α = 1; or p 2 − 3α = 0 and cα > 0. In short, for having the origin as the unique finite singular point and that both the origins of the local charts U 1 and U 2 are formed by two degenerated hyperbolic sectors we must have cα > 0 and p 2 ∈ [−3, 3]. Hence we obtain system (X).
For condition (c) of Theorem 8 with p 2 = 0 systems (3) have no linear type center at the origin. This completes the proof of Theorem 6.
Condition (a) in Theorem 8 yields p 2 = p 3 = 0 and so we obtain system (XI). Note that system (XI) is Hamiltonian with Hamiltonian It was proved in [5] that in this case the unique infinite singular point is the origin of the local chart U 1 which is formed by two degenerated hyperbolic sectors if and only if cα > 0 and that in this case there are no more finite singular points besides the origin. Condition (b) in Theorem 8 with p 2 2 + p 2 3 = 0 and α = 0 yields the condition p 3 = d = 0. So we get the systeṁ whose finite singular points are , ±i c p 2 + 3α .
Note that if p 2 = ±3α the last four points do not exist. So in order that they do not exist we must have cα > 0 and either c( p 2 + 3α) ≥ 0, or c( p 2 + 3α) < 0 and ( p 2 − 3α)(( p 2 + 3α)ω 2 − c 2 α) > 0. Now we study the infinite singular points of system (13). We already know that it is the origin of the local chart U 1 . On the local chart U 1 system (13) becomeṡ Doing one horizontal and one vertical blow up and taking into account that cα > 0, we get that if c( p 2 + 3α) ≥ 0 and α( p 2 + 3α) ≥ 0, then after the blow-down the origin of the local chart U 1 is formed by two degenerated hyperbolic sectors. On the other hand, if the above conditions do not hold, then the origin of the local chart U 1 is not formed by two degenerated hyperbolic sectors. In short, in order that system (13) has no more finite singular points besides the origin and the origin of the local chart U 1 is formed by two degenerated hyperbolic sectors we must have c( p 2 + 3α) ≥ 0 and cα > 0. Hence we get system (XII). For condition (c) of Theorem 8 with p 2 2 + p 2 3 = 0 systems (4) have no linear type center at the origin. This completes the proof of Theorem 7.