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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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.
We describe an algebraic approach to the center problem for the ordinary differential equation(1.1)with coefficients ai from the Banach spaceof bounded measurable complex-valued functions on IT equipped with the supremum norm. Condition guarantees 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 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 we denote the set of centers of (1.1). The center problem is: given to determine whether It 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 (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 are closed. Passing to polar coordinates in (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.
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 It is defined as follows.
Let us consider X as a semigroup with the operations given for
Let be the vector space of sequences of complex numbers equipped with the product topology. For we denote a path in starting at 0. The one-to-one map sends the product to the product of paths that is, the path obtained by translating so that its beginning meets the end ofeb and then forming the composite path. Similarly,is
the path obtained by translating so that its end meets 0 and then taking it with the opposite orientation.
Forconsider the basic iterated integrals
By the Ree shuffle formula the linear space generated by all such functions on X is an algebra. For we write if all basic iterated integrals vanish at Then if and only
if for all basic iterated integrals, see . 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 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 on G(X) such that I· = The functions are 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 in which all basic iterated integralsare continuous. Then (G(X),)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 .
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
Let be the associative algebra with unit I of complex noncommutative polynomials in I and free noncommutative variables (i.e., there are no nontrivial relations between these variables).we denote the associative algebra of formal power series in t with coefficients from be the multiplicative semigroup generated by Consider a grading function determined by the conditions
This splits S in a disjoint union we denote the subalgebra of series f of the form(3.1)
We equip A with the weakest topology in which all coefficients in (3.1) considered as functions in 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 is continuous in this topology
By we denote the closed subset of elements f of form (3.1) with is a topological group. Its Lie algebra consists of elements f of form (3.1) with f0 = 0. is a homeomorphism.
Further, for an element consider the equation(3.2)
This can be solved by Picard iteration to obtain a solution whose coefficients in expansion in and t are Lipschitz functions on IT . We set (3.3)
By the definition we have (3.4)
Also, an explicit calculation leads to the formula (3.5)
From the last formula one obtains that there is a homomorphism such that that is,(3.6)
Formula (3.6) shows that is a continuous embedding. Moreover, one can determine a metric d1 on A compatible with topology such that is an
isometric embedding. Therefore is naturally extended to a continuous embedding (denoted also by) By definition, is an injective homomorphism of topological
groups and is the closure of in the topology of G.
In what follows we identify G(X) and Gf (X) with their images under
Recall that each element can be written as We say that g is a Lie element if each gn belongs to the free Lie algebra generated by In this case each gn has the form (4.1) with all(Here the term with
Let be the subspace of elements gn of form (4.1). Then (4.2)
where the sum is taken over all numbers that divide n, and is the Mobius function.
By we denote the subset of Lie elements of Then is a closed (in the topology of A) Lie subalgebra of The following result was proved in .
Theorem 4.1. The exponential map exp : maps homeomorphically onto
Thus can be regarded as the Lie algebra of
Let be the algebra of formal complex power series in z. By D, L : we denote the differentiation and the left translation operators defined on (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
Using a linearization of (5.2) we associate to this equation the following system of ODEs:
Solving (5.3) by Picard iteration we obtain a solution whose coefficients in the series expansion in D,L and t are Lipschitz functions on IT It was established
in  that (5.2) determines a center if and only if This implies the following result, see .
Theorem 5.1. We have
where is a complex polynomial of degree k defined by the formula
Let be the set of formal complex power series be such that coefficient in the series expansion of f. We equip with the weakest topology in which all di are continuous functions and consider the multiplication defined by the composition of series. Then s a separable topological group.
Moreover, it is contractible and residually torsion free nilpotent. By we denote the subgroup of power series locally convergent near 0 equipped with the induced topology. Next, we define the map (5.5)
Then Moreover, let be the Lipschitz solution of equation (5.2) with initial value v(0; r; a) = r. Clearly for every we have It is proved in  that is the first return map of (5.2)). In particular, we have (5.6)
Also, (5.5) implies that there is a continuous homomorphism such that is the quotient map). We extend it by continuity toretaining the same symbol for the extension. Then is a normal subgroup ofwe denote its closure in This group is called the group of formal centers C of equation (1.1).
Theorem 5.2 ().The Lie algebraconsists of elements
In particular, the map exp : is a homeomorphism