Centers: their integrability and relations with the divergence

This brief survey is mainly dedicated to the mathematicians which are interested in the centers of the analytic differential systems in the plane ℝ², and which are not specialists in this topic.

Let O be the origin of coordinates of ℝ² and let U be a neighborhood of O. We consider two real analytic functions X(x,y) and Y (x,y) in U which vanish at O. In this work we deal with the analytic differential systems of the form

x˙=X(x,y),y˙=Y(x,y),$$\begin{array}{} \displaystyle \dot x = X(x,y),\quad \dot y = Y(x,y), \end{array}$$(1)

where the dot denotes derivative with respect to an independent real variable t, usually called the time.

When all the orbits of system 1 in a punctured neighborhood of the singular point O are periodic, we say that the origin is a center. If the orbits of system 1 in a punctured neighborhood of O spiral to O when t → +∞ or when t → − ∞, then we say that the origin is a focus. In the first case (t → +∞), we say that it is stable and in the second case (t → −∞), we say that it is unstable. If the origin is either a focus or a center, we say that it is a monodromic singular point. The center–focus problem consists in distinguishing when a monodromic singular point is either a center or a focus. The center–focus problem started with Poincaré [22] and Dulac [6], and in the present days many questions remains open.

If a real analytic system 1 has a center at the origin, then after a linear change of variables and a rescaling of the time variable, it can be written in one of the following three forms:

x˙=−y+X2(x,y),y˙=x+Y2(x,y),$$\begin{array}{} \displaystyle \begin{array}{*{20}{l}} {\dot x} \hfill & = \hfill & { - y + {X_2}(x,y),} \hfill \\ {\dot y} \hfill & = \hfill & {x + {Y_2}(x,y),} \hfill \\ \end{array} \end{array}$$(2)

called a linear type center,

x˙=y+X2(x,y),y˙=Y2(x,y),$$\begin{array}{} \displaystyle \begin{array}{*{20}{l}} {\dot x = y + {X_2}(x,y),} \hfill \\ {\dot y = {Y_2}(x,y),} \hfill \\ \end{array} \end{array}$$(3)

called a nilpotent center,

x˙=X2(x,y),y˙=Y2(x,y),$$\begin{array}{} \displaystyle \begin{array}{*{20}{l}} {\dot x = {X_2}(x,y),}\\ {\dot y = {Y_2}(x,y),} \end{array} \end{array}$$(4)

called a degenerate center, where X₂(x,y) and Y₂(x,y) are real analytic functions without constant and linear terms, defined in a neighborhood of the origin.

The characterization of linear type centers using their first integrals is due to Poincaré [22] in the case of polynomial differential systems and to Liapunov [16] in the case of analytic differential systems, see also Moussu [20].

Using this theorem Poincaré and Liapunov [16, 22] constructed an algorithm which, when it can be applied because it usually needs huge computations, solves in principle the center problem for linear type centers.

We say that a non–zero formal power series H=∑i,j=0∞aijxiyj$\begin{array}{} \displaystyle H = \sum\limits_{i,j = 0}^\infty {{a_{ij}}} {x^i}{y^j} \end{array}$ is a formal first integral of system 3, if it formally satisfies that

∂H∂x(y+X2(x,y))+∂H∂yY2(x,y)≡0.$$\begin{array}{} \displaystyle \frac{{\partial H}}{{\partial x}}(y + {X_2}(x,y)) + \frac{{\partial H}}{{\partial y}}{Y_2}(x,y) \equiv 0. \end{array}$$

The integrability of the nilpotent centers was studied in [4].

We note that statement (b) provides a tool for detecting if a given analytic nilpotent center has or not a local analytic first integral.

Analytic systems having a nilpotent singular point at the origin were studied by Andreev [1] in order to obtain their local phase portraits. But Andreev’s results do not distinguish between a focus and a center. Takens [24] provided a normal form for nilpotent centers or foci. Moussu [19] found the C^∞ normal form for analytic nilpotent centers. Berthier and Moussu in [3] study the reversibility of the nilpotent centers. Teixeira and Yang [25] analyse the relationship between reversibility and the center–focus problem for systems 2 and 3, written in a convenient normal form. Strózyna and Zoladek [23] gave also a good normal form for nilpotent centers.

The next proposition provides, probably, the easiest example of an analytic nilpotent center without a local analytic first integral, for a proof see Proposition 7 of [4].

Assume that we have the analytic differential system

x˙=Xn(x,y)+On+1(x,y),y˙=Ym(x,y)+Om+1(x,y),$$\begin{array}{} \displaystyle \begin{array}{*{20}{l}} {\dot x = {X_n}(x,y) + {O_{n + 1}}(x,y),}\\ {\dot y = {Y_m}(x,y) + {O_{m + 1}}(x,y),} \end{array} \end{array}$$(5)

where n ≥ 1 and m ≥ 1 are integers and X_n(x,y) and Y_m(x,y) are non–zero homogeneous polynomials of degrees n and m respectively, formed by the lowest order terms of X(x,y) and Y(x,y) of system 1, respectively.

Define the homogeneous polynomial

Δ(x,y)={yXn(x,y)−xYm(x,y) if n=m,yXn(x,y) if n<m,−xYm(x,y) if n>m.$$\begin{array}{} \displaystyle \Delta (x,y) = \left\{ \begin{array}{*{20}{l}} {y{X_n}(x,y) - x{Y_m}(x,y)} \hfill & {ifn = m,} \hfill \\ {y{X_n}(x,y)} \hfill & {ifn \lt m,} \hfill \\ { - x{Y_m}(x,y)} \hfill & {ifn \gt m.} \hfill \\ \end{array}\right. \end{array}$$(6)

The real linear factors of the polynomial ∆(x,y) provide the possible directions that the trajectories can reach or exit the singular point located at the origin of coordinates of system 5. Such directions are called characteristic directions.

A sufficient condition in order that system 5 has a monodromic singular point at the origin is that ∆(x,y) = 0 only if (x,y) = (0,0). In this case the origin has no characteristic directions.

A necessary condition in order that system 5 has a monodromic singular point at the origin is that ∆(x,y) ≥ 0 or ∆(x,y) ≤ 0 for all (x,y) ∈ ℝ². For more details on characteristic directions see for instance [2].

Suppose that the analytic differential system 1 has a monodromic singular point at the origin O. Let Σ be an analytic transversal section at O, that is, an analytic arc transverse to the flow of the system such that O ∈ ∂ Σ, the boundary of Σ. We consider a parameter ρ of Σ such that ρ = 0 corresponds to the origin of coordinates and Σ is parameterized by the interval (0,ρ*) with ρ* > 0. Given a point ρ in Σ we consider the orbit of system 1 with ρ as initial condition. Due to the fact that the origin is monodromic, if ρ is close enough to O and we follow the orbit for positive values of the time t, it will cut Σ again at some point. We define the Poincaré map 𝒫 : Σ → Σ being 𝒫(r) the point in Σ corresponding to the first cut with Σ of the orbit through ρ in positive time.

It is clear that the origin of the analytic differential system 1 is a center if and only if the Poincaré map is the identity. For systems with the linear type form 2 and for systems with the nilpotent form 3, it is possible to find a parametrization of Σ such that the Poincaré map is analytic in ρ = 0 and writes as

P(ρ)=ρ+∑i=3∞αiρi,$$\begin{array}{} \displaystyle {\scr P}(\rho ){\mkern 1mu} = {\mkern 1mu} \rho {\mkern 1mu} + {\mkern 1mu} \sum\limits_{i = 3}^\infty {{\alpha _i}} {\mkern 1mu} {\rho ^i}, \end{array}$$

where α_i are algebraic expressions in the coefficients of X₂ and Y₂. There are systems with the degenerate form 4 for which such a parametrization is also possible, for instance the ones which do not have characteristic directions, see for instance [9].

The stability of the origin is clearly given by the sign of the first non–zero α_i (that is, it is unstable if α_i > 0, and stable if α_i < 0), and if all the α_i are zero then the origin is a center. Moreover, the even–indexed terms α_2k are algebraic expressions of the previous α_i. Therefore the interesting expressions are the ones with odd index, i.e. the α_2k+1’s. Moreover we define the (k + 1)−th Poincaré–Liapunov constant as the expression α_2k+1 modulus the vanishing of all the previous ones. In general the computation of the Poincaré–Liapunov constants is a very difficult computational problem. See for more details [5].

The technique of the Poincaré–Liapunov constants for solving the focus–center problem for linear type centers and of the centers without characteristic directions can be extended to nilpotent centers, see [8, 11, 12].

In short, now we have algorithms based in the computation of Poincaré–Liapunov constants for solving the focus–center problem for linear type centers, nilpotent centers and centers without characteristic directions. Moreover, for such centers the Poincaré–Liapunov constants are algebraic polynomials in function of the coefficients of the polynomial differential systems.

The conditions on the coefficients of a polynomial differential system which determine when such a system has a center, which is neither of linear type, nor nilpotent and with characteristic directions, can be non–algebraic as it was proved by Ilyashenko [14], see also [17].

As usual we define the divergence of system1, and we denote it by div(x,y), as the function

div(x,y)=∂X∂x(x,y)+∂Y∂y(x,y).$$\begin{array}{} \displaystyle {\rm div} (x,y) \, = \, \frac{\partial X}{\partial x}(x,y) \, + \, \frac{\partial Y}{\partial y}(x,y). \end{array}$$

System 1 is said to be Hamiltonian if div(x,y) ≡ 0. In such a case there exists a neighborhood of the origin Uand an analytic function H : U ⊆ ℝ² → ℝ, called the Hamiltonian, such that

X(x,y)=−∂H∂y andY(x,y)=∂H∂y.$$\begin{array}{} \displaystyle X(x,y) = - \frac{{\partial H}}{{\partial y}}{\rm{ and}}\quad Y(x,y) = \frac{{\partial H}}{{\partial y}}. \end{array}$$

We note that the level curves of H are formed by orbits of system 1. A Hamiltonian system 1 with a monodromic singular point at O necessarily has a center at the origin because the function H is a first integral defined at the origin.

Our aim is to illustrate that the divergence function also can help us to solve the focus–center problem in other non–Hamiltonian differential systems.

Given a real analytic function f : U ⊆ ℝ² → ℝ, where U is a neighborhood of the origin O = (0,0), we consider its Taylor expansion at O, i.e.

f(x,y)=fd(x,y)+Od+1(x,y),$$\begin{array}{} \displaystyle f(x,y) \, = \, f_d(x,y) \, + \, \mathscr{O}_{d+1}(x,y), \end{array}$$

where d ≥ 0 is an integer and f_d(x,y) is a non–zero homogeneous polynomial of degree d. We say that f is positive sign definite if f_d(x,y) > 0, or negative sign definite f_d(x,y) < 0 for all (x,y) ∈ ℝ² \ {(0,0)}. It is clear that a necessary condition for f (x,y) to be of sign definite is that d is even.

The next result is Proposition 1 of [13].

The next result is Theorem 2 of [13].

The following result is Theorem 4 of [13].

Our next main result deals with a system of linear type 2 or in degenerate form 4 with a monodromic singular point at the origin.

We assume that the origin has no characteristic directions. As we have already stated, this means that the polynomial ∆(x,y) defined in 6 is such that ∆(x,y) = 0 only if (x,y) = (0,0). We remark that in this case the degree of the lowest order terms of ẋ and ẏ must coincide, that is, X(x,y) = X_n(x,y) + O_n+1(x,y) and Y (x,y) = Y_n(x,y) + O_n+1(x,y) where X_n(x,y) and Y_n(x,y) are nonzero homogeneous polynomials of degree n formed by all the terms of this degree in X(x,y) and Y (x,y). We define

ν(θ)=exp[∫0θcosσPn(cosσ,sinσ)+sinσQn(cosσ,sinσ)cosσQn(cosσ,sinσ)−sinσPn(cosσ,sinσ)dσ].$$\begin{array}{} \displaystyle \nu(\theta) \, = \, \exp\left[ \int_{0}^{\theta} \frac{\cos \sigma P_n(\cos \sigma, \sin \sigma)+\sin \sigma Q_n(\cos \sigma, \sin \sigma)}{\cos \sigma Q_n(\cos \sigma,\sin \sigma) - \sin \sigma P_n(\cos \sigma,\sin \sigma)} \, d\sigma \right]. \end{array}$$

The following result is Theorem 11 of [13].