We consider the Abel equation
formula here
where p and q are polynomials, and a is a fixed constant. We denote the solution
of (1) by y(x, c), where
y(a, c) = c. Standard existence theorems ensure
that y(x, c) is well-defined and analytic in both its arguments,
for c sufficiently small. If y(b, c) = c,
then we call y(x, c) a periodic solution. Likewise, if
y(b, c) ≡ c for all c close to 0,
then we say that the system has a centre between a and b.
The numbers a and b are
not important; by a simple transformation, we can always choose a = 0 and b = 1,
and we shall usually do so from now on.
Abel equations arise in several circumstances, but perhaps the main reason for
their recent study is connected to the family of systems
formula here
where M and N are homogeneous polynomials of
the same degree n. A transformation
due to Cerkas allows us to bring these systems to the form (1), where p and
q are now trigonometric polynomials. It is not hard to show that, setting a = 0
and b = 2π, the definitions of periodic solution and centre for (1) coincide with
their usual definitions in the planar system (2). There are also transformations to
Abel-type equations for more general systems; see [8, 10].
This trigonometric Abel equation has been used in a large number of works in
order to estimate the number of limit cycles or obtain centre conditions, as well as in
more general investigations relating the derivatives of the return map with iterated
integrals. However, studying system (2) in whatever form is by no means easy, and
a natural question is to ask whether we can still capture the essence of this problem
if we take p and q to be polynomials, in the hope that the calculations will become
easier. The recent series of investigations by Briskin,
Françoise and Yomdin [3, 4, 5]
seems to indicate that this could be the case.
Our interest here is to see what conditions the existence of a centre in (1)
imposes on the defining equations. For ease of reference, we shall always denote the
antiderivative of the polynomials p and q as P and Q; that is,
formula here