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ISSN: 1736-4337
Journal of Generalized Lie Theory and Applications
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An algebraic approach to the center problem for ODEs 1

Alexander BRUDNYI

Department of Mathematics and Statistics, University of Calgary,
2500 University Drive NW, T1N 1N4 Calgary, Canada

E-mail: [email protected]

Received date: December 14, 2007; Revised date:March 07, 2008

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Abstract

The classical Poincar´e Center-Focus problem asks about the characterization of planar polynomial vector fields such that all their integral trajectories are closed curves whose interiors contain a fixed point, a center. This problem is reduced to a center problem for certain ODE . We present an algebraic approach to the center problem based on the study of the group of paths determined by the coefficients of the ODE.

Introduction

We describe an algebraic approach to the center problem for the ordinary differential equationimage(1.1)with coefficients ai from the Banach spaceimageof bounded measurable complex-valued functions on IT equipped with the supremum norm. Condition imageguarantees that (1.1) has Lipschitz solutions on IT for all sufficiently small initial values. By X we denote the complex Fr´echet space of sequences image satisfying this condition. We say that equation (1.1) determines a center if every solution v of (1.1) with a sufficiently small initial value satisfies imagewe denote the set of centers of (1.1). The center problem is: given imageto determine whether imageIt arises naturally in the framework of the geometric theory of ordinary differential equations created by Poincar´e. In particular, there is a relation between the center problem for (1.1) and the classical Poincar´e Center-Focus problem for planar polynomial vector fields image(1.2)where F and G are polynomials of a given degree without constant and linear terms. This problem asks about conditions on F and G under which all trajectories of (1.2) situated in a small neighbourhood of imageare closed. Passing to polar coordinates imagein (1.2) and expanding the right-hand side of the resulting equation as a series in r (for F, G with sufficiently small coefficients) we obtain an equation of the form (1.1) whose coefficients are trigonometric polynomials depending polynomially on the coefficients of (1.2). This reduces the Center-Focus Problem for (1.2) to the center problem for (1.1) with coefficients depending polynomially on a parameter.

Group of paths

One of the main objects of our approach is a metrizable topological group G(X) determined by the coefficients of equations (1.1) (the, so-called, group of paths in image It is defined as follows.

Let us consider X as a semigroup with the operations given for imageimageimage

where for imageimage

Let image be the vector space of sequences of complex numbers imageequipped with the product topology. For imageimagewe denote a path in imagestarting at 0. The one-to-one map image sends the product image to the product of paths imagethat is, the path obtained by translating image so that its beginning meets the end ofeb and then forming the composite path. Similarly,IMAGEis
the path obtained by translating imageso that its end meets 0 and then taking it with the opposite orientation.

ForIMAGEconsider the basic iterated integrals

image(2.1)

By the Ree shuffle formula the linear space generated by all such functions on X is an algebra. For imagewe write imageif all basic iterated integrals vanish at imageThen image if and only
if imagefor all basic iterated integrals, see [1]. In particular, ~ is an equivalence relation on X. By G(X) we denote the set of equivalence classes. Then G(X) is a group with the product induced by the product * on X. By image we denote the map determined by the equivalence relation. By the definition each iterated integral I· is constant on fibres of π and therefore it determines a function IMAGEon G(X) such that I· = IMAGEimageThe functions IMAGEare referred to as iterated integrals on G(X). These functions separate the points on G(X).

Next, we equip G(X) with the weakest topology image in which all basic iterated integralsimageare continuous. Then (G(X),image)is a topological group. Moreover, G(X) is metrizable, contractible, residually torsion free nilpotent (i.e., finite dimensional unipotent representations of G(X) separate the points on G(X)) and is the union of an increasing sequence of compact subsets, see [2].

By Gf (X) we denote the completion of G(X) with respect to the metric d. Then Gf (X)is a topological group which is called the group of formal paths in image

Representation of paths by noncommutative power series

Let imagebe the associative algebra with unit I of complex noncommutative polynomials in I and free noncommutative variables image(i.e., there are no nontrivial relations between these variables).imagewe denote the associative algebra of formal power series in t with coefficients from imageimagebe the multiplicative semigroup generated by imageConsider a grading function imagedetermined by the conditions image

This splits S in a disjoint union imageimagewe denote the subalgebra of series f of the formimage(3.1)

We equip A with the weakest topology in which all coefficients in (3.1) considered as functions in image are continuous. Since the set of these functions is countable, A is metrizable. Moreover, if d is a metric on A compatible with the topology, then (A, d) is a complete metric space. Also, by the definition the multiplication imageis continuous in this topology

By imagewe denote the closed subset of elements f of form (3.1) with imageis a topological group. Its Lie algebra imageconsists of elements f of form (3.1) with f0 = 0. image is a homeomorphism.

Further, for an element image consider the equationimage(3.2)

This can be solved by Picard iteration to obtain a solution imagewhose coefficients in expansion in imageand t are Lipschitz functions on IT . We set image(3.3)

By the definition we have image(3.4)

Also, an explicit calculation leads to the formula image(3.5)

From the last formula one obtains that there is a homomorphism image such thatimage that is,image(3.6)

Formula (3.6) shows that iamgeis a continuous embedding. Moreover, one can determine a metric d1 on A compatible with topology such that imageis an
isometric embedding. Therefore imageis naturally extended to a continuous embedding image(denoted also byimage) By definition,image is an injective homomorphism of topological
groups and image is the closure ofimage in the topology of G.

In what follows we identify G(X) and Gf (X) with their images under image

Lie algebra of the group of formal paths

Recall that each element image can be written asimage We say that g is a Lie element if each gn belongs to the free Lie algebra generated by imageIn this case each gn has the form image(4.1) with allimage(Here the term with image

Let imagebe the subspace of elements gn of form (4.1). Then image(4.2)

where the sum is taken over all numbers imagethat divide n, and imageis the Mobius function.

By imagewe denote the subset of Lie elements of imageThen image is a closed (in the topology of A) Lie subalgebra of image The following result was proved in [3].

Theorem 4.1. The exponential map exp :image maps image homeomorphically ontoimage

Thus image can be regarded as the Lie algebra of image

Center Problem for ODEs

Let iamge be the algebra of formal complex power series in z. By D, L :image we denote the differentiation and the left translation operators defined on imageimage(5.1)

Let A(D,L) be the associative algebra with unit I of complex polynomials in I, D and L. By A(D,L)[[t]] we denote the associative algebra of formal power series in t with coefficients from A(D,L). Also, by G0(D,L)[[t]] we denote the group of invertible elements of A(D,L)[[t]] consisting of elements whose expansions in t begin with I.

Further, consider equation (1.1) corresponding to an image

image(5.2)

Using a linearization of (5.2) we associate to this equation the following system of ODEs:

image(5.3)

Solving (5.3) by Picard iteration we obtain a solution image whose coefficients in the series expansion in D,L and t are Lipschitz functions on IT It was established
in [1] that (5.2) determines a center image if and only ifimage This implies the following result, see [3].

Theorem 5.1. We have

image(5.4)

where image is a complex polynomial of degree k defined by the formula

image

Let image be the set of formal complex power seriesimage be such thatimage coefficient in the series expansion of f. We equip imagewith the weakest topology in which all di are continuous functions and consider the multiplication image defined by the composition of series. Then images a separable topological group.
Moreover, it is contractible and residually torsion free nilpotent. By imagewe denote the subgroup of power series locally convergent near 0 equipped with the induced topology. Next, we define the map imageimage(5.5)

Then imageMoreover, let image be the Lipschitz solution of equation (5.2) with initial value v(0; r; a) = r. Clearly for every imagewe have image It is proved in [1] that image is the first return map of (5.2)). In particular, we have image(5.6)

Also, (5.5) implies that there is a continuous homomorphism image such that image is the quotient map). We extend it by continuity toimageretaining the same symbol for the extension. Then image is a normal subgroup ofimagewe denote its closure in imageThis group is called the group of formal centers C of equation (1.1).

Theorem 5.2 ([3]).The Lie algebraIMAGEconsists of elements

IMAGE

such that image

In particular, the map exp : image is a homeomorphism

For further results and open problems we refer to papers [1]–[3] and references therein.

References

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