On the number of algebraically independent Poincaré–Liapunov constants

Abstract

In this paper, an upper bound for the number of algebraically independent Poincaré–Liapunov constants in a certain basis for planar polynomial differential systems is given. Finally, it is conjectured that an upper bound for the number of functionally independent Poincaré–Liapunov quantities would be m² + 3m − 7, where m is the degree of the polynomial differential system. Moreover, the computational problems which appear in the computation of the Poincaré–Liapunov constants and in the determination of the center cases are also discussed.

Introduction

One of the most interesting and difficult problems in the theory of planar differential equations is the control of the number of limit cycles that a differential equation or a family of differential equations can have. There exists different methods to produce limit cycles: the limit cycles which bifurcate from a singular point, from a center, from a homoclinic or heteroclinic orbit and finally the limit cycles which bifurcate from the infinity. To detect these different types of bifurcations of limit cycles there exists different methods: the first one is based on the study of the Poincaré return map, the Poincaré–Melnikov integral method, the Abelian integral method, the averaging method, and the last one uses the inverse integrating factor. When the first is used to detect the small limit cycles from a singular point is called Degenerate Hopf bifurcation, see [4], [21]. It can be also used to detect limit cycles which bifurcated from a center, see [6], [10]. The second and third methods are based in the study of small perturbations of Hamiltonian systems, and in fact in the plane are essentially equivalent, see [6], [17, Section 6, Chapter 4] and [3, Section 5, Chapter 6].

Other classical problem in the qualitative theory of planar analytic differential systems is to characterize the local phase portrait at an isolated singular point. This problem has been solved except if the singular point is of focus-center type, see [1], [2], [12]. Recall that a singular point is said to be of focus-center type if it is either a focus or a center. The problem of distinguishing between a center or a focus is called the center problem. Of course, if the linear part of the singular point in nondegenerate (i.e. its determinant does not vanish) the characterization is well known, see [18], [19]. If an analytic system has a nondegenerate singular point of focus type at the origin, then after a linear change of variables and a rescaling of the time variable it can be written into the form:

$$\begin{array}{cc}\hfill & \dot{x}=\lambda x-y+\stackrel{\sim }{X}(x\text{,}y)\text{,}\hfill \\ \hfill & \dot{y}=x+\lambda y+\stackrel{\sim }{Y}(x\text{,}y)\text{,}\hfill \end{array}$$

where $\stackrel{\sim }{X}(x\text{,}y)$ and $\stackrel{\sim }{Y}(x\text{,}y)$ are analytic functions without constant and linear terms defined in a certain neighborhood of the origin.

The aim of this paper consists on giving an upper bound for the number of algebraically independent Poincaré–Liapunov constants in a certain basis for planar polynomial differential systems. The final objective would be to find an upper bound for the number of functionally independent Poincaré–Liapunov constants for planar polynomial differential systems. If this upper bound existed, taking into account the works of Shi Songling, see [21], and under certain hypothesis about the Poincaré–Liapunov constants generators of the Dulac ideal, for instance the Dulac ideal to be radical, this upper bound would give an upper bound for the maximum number of small limit cycles which bifurcate from a nondegenerate singular point of focus type of a planar polynomial differential system.

We say that the analytic functions f₁ , … , f_m are functionally dependent in a neighborhood of the point p, where f_i(p) = 0 for i = 1 , … , m, if and only if there exist analytic functions g₁ , … , g_m in this neighborhood such that ${\sum }_{i=1}^m{f}_i{g}_i\equiv 0$ and $\sum \mid g_i{\mid }^2(p)\ne 0$. An ideal J of a ring A is radical if and only if rad(J) = J, where rad(J) = {f ∈ A : fⁿ ∈ J for some n > 0}. The equivalent definition states that if fⁿ ∈ J, then f ∈ J.

Therefore, we consider the two-dimensional autonomous differential system

$$\begin{array}{cc}\hfill & \dot{x}=\lambda x-y+X(x\text{,}y)\text{,}\hfill \\ \hfill & \dot{y}=x+\lambda y+Y(x\text{,}y)\text{,}\hfill \end{array}$$

where X(x, y) and Y(x, y) are polynomials of degree m, with m ⩾ 2. Hence, the nonlinear terms can be expressed as $X(x\text{,}y)={\sum }_{s=2}^m{X}_s(x\text{,}y)$ and $Y(x\text{,}y)={\sum }_{s=2}^m{Y}_s(x\text{,}y)$, where X_s(x, y) and Y_s(x, y) are homogeneous polynomials of degree s, that is, $X_s(x\text{,}y)={\sum }_{k=0}^s{a}_k{x}^k{y}^{s-k}$ and $Y_s(x\text{,}y)={\sum }_{k=0}^s{b}_k{x}^k{y}^{s-k}$, where a_k and b_k are arbitrary coefficients.

The analytic technique to set up the so-called nondegenerate centre problem was introduced by Poincaré [19]. More precisely, his solution consists in determining when a system of the form (1) has a local analytic first integral at the origin, and consequently a center at this point. Poincaré’s method consists on finding a formal power series of the form

$$H(x\text{,}y)=\sum _{n=2}^{\infty }{H}_n(x\text{,}y)\text{,}$$

where H₂(x, y) = (x² + y²)/2, and for each n, H_n(x, y) are homogeneous polynomials of degree n, so that $\dot{H}={\sum }_{k=2}^{\infty }{V}_{2k}(x^2+y^2{)}^k$, where V_2k are called the Poincaré–Liapunov constants. For the polynomial system (2) the Poincaré–Liapunov constants are polynomials whose variables are the coefficients of system (2).

In order to solve the problem of the stability at the origin of system (1), it is sufficient to consider the sign of the first Poincaré–Liapunov constant different from zero. If it is positive we have asymptotic stability for negative times, and if it is negative we have asymptotic stability for positive times. If all Poincaré–Liapunov constants are zero, then the origin is stable for all times, but there is no asymptotic stability for any time, see for instance [2]. In this last case, we have a center at the origin, i.e. there is an open neighborhood of the origin where all orbits are periodic, except of course the origin. The origin is said to be a fine focus of order k if V_2k+2 is the first non-zero Poincaré–Liapunov constant. In this case at most k limit cycles can bifurcate from this fine focus [5]; these limit cycles are called small-amplitude limit cycles. Therefore to obtain the maximum number of limit cycles which can bifurcate from the origin for a given system, one has to find the maximum possible order of a fine focus. It is known that this maximum number is three for quadratic system [4], see also [20] for the first part of the proof. Żoła¸dek gave an algebraic proof of the Bautin Theorem [23]. An analogous algebraic theorem holds for the space of vector fields the linear part of which are centers and the nonlinear part are homogeneous polynomials of degree 3. The result says that the maximum number for these systems is five, see [24]. It has been shown recently, by the same author, that it is greater or equal than eleven for general cubic systems using Abelian integrals [25].

The proofs of these theorems depend essentially on the knowledge of all centers for the studied system, which makes possible to establish the structure of the coefficients in the Taylor expansion of the Poincaré map. Żoła¸dek believes that probably a similar situation takes place for a fixed degree. Knowing sufficiently many cases of integrability we should be able to prove a corresponding algebraic theorem analog to the Bautin’s one, see [24]. Unfortunately, the knowledge of all centers is an open problem even for cubic systems. In this paper an upper bound for the number of algebraically independent Poincaré–Liapunov constants in a certain basis is given without the knowledge of all centers of the studied systems.

Let E be some space of planar differential systems. We say that a nondegenerate singular point of focus type p of system (1) has cyclicity k with respect to the space E if any perturbation in E of this system (1) has at most k limit cycles in a neighborhood of the point p and k is the minimal number with this property.

Shi Songling [22] proved that for polynomial systems we have uniqueness for the V_2k in the following sense. Let A be the ring of real polynomials whose variables are the coefficients of the polynomial differential system. Given a set of Poincaré–Liapunov constants V₁,V₂ , … , V_i, let J_k−1 be the ideal of A generated by V₁, V₂ , … , V_k−1. If $V_1^{\prime }\text{,}{V}_2^{\prime }\text{,}\dots \text{,}{V}_i^{\prime }$ is another set of Poincaré–Liapunov constants, then $V_k\equiv V_k^{\prime }\mathrm{mod}({\mathbf{J}}_{k-1})$. As it has been said above the origin is a center if and only if all the Poincaré–Liapunov constants are zero. Let J = 〈V₁, V₂, …〉 be the ideal of A generated by all the Poincaré–Liapunov constants. For polynomials systems, using the Hilbert’s basis theorem, J is finitely generated; i.e., there exist B₁, B₂ , … , B_q in J such that J = 〈B₁, B₂ , … , B_q〉. Such a set of generators is called a basis of J.

Notice that Hilbert’s basis theorem assures us the existence of a generators basis, but it does not provide us a constructive method to find it. The existent methods to solve this problem are based in the Buchberger’s algorithm to find a Gröebner basis, see [11]. Moreover, it is only applicable for very simple cases, see for instance [8], [9], [13], [14]. Therefore, it is a computational problem of algebraic nature due to the appearance, already for simple systems, of massive Poincaré–Liapunov constants that are polynomials with rational coefficients and efficient algorithms do not exist that allow to determine simple groups of generators. One of the main difficulties comes ultimately on the decomposition in prime numbers of a big integer number. Therefore the resolution of the computational problem goes to have efficient algorithms that work with big integers and in decomposition in primes numbers of big numbers, a classical problem in computational mathematics. On the other hand there are recursive methods for the determination of these Poincaré–Liapunov constants and the development of the algebraic manipulators has allowed to approach the calculation of the first constants, see for instance [15], [16].

Shi Songling also proved in [21] that for polynomials of degree m, under certain hypothesis about the constants of Poincaré–Liapunov generators of the ideal, the maximum number of limit cycles is given by M(m), being M(m) the minimum number of the ideal generators. An open problem nowadays is the determination of M(m), or in its default an upper bound of it.