Center conditions for a simple class of quintic systems

We obtain center conditions for a $O$-symmetric system of degree 5 for which the origin is a uniformly isochronous singular point. In the revised paper some misprints are corrected in the reference list.

1. Let us consider a planar differential systeṁ x = y + xR n−1 (x, y), y = −x + yR n−1 (x, y), where R n−1 (x, y) is a polynomial in x, y of degree n − 1. System (1) has a unique singular point O(0, 0) whose linear part of center type.
Orbits of system (1) move around the origin with a constant angular velocity and the origin is a uniformly isochronous singular point.
In [1] the following problem was proposed: Problem 19.1. Identify systems (1) of odd degree which are O-symmetric (not necessarily quasi-homogeneous) having O as a (uniformly isochronous) center.
(2) * The work is supported by Russian Foundation for Basic Research Theorem. The origin is a center of system (2) if and only if one of the following sets of conditions is satisfied: Proof.

Necessity:
To describe the behaviour of trajectories of (2) near the origin we construct the comparison function [2] F (x, y) = ( where f k is a homogeneous polynomial of degree k whose derivative is The number of the first coefficient D i other than zero defines the multiplicity of a complex focus and the sign of this coefficient defines stability of a focus; if D i = 0 for all i the origin is a center of (2). We refer to coefficients D i as the Poincaré-Lyapunov constants.
To find the Poincaré-Lyapunov constants of a systemẋ = p(x, y),ẏ = q(x, y) with a linear center we used computer algebra and wrote a Mathematica code that rests on the Poincaré algorithm in [2]; see [3] for more details. The procedure PLconst[n] returns a list {D 1 , . . . , D n } of the Poincaré-Lyapunov constants if we define the coefficients p ij , q ij (2 ≤ i + j ≤ 2n + 1) in the Taylor series expansion of functions p(x, y), q(x, y) beforehand.
Using this procedure, we found the first four Poincaré-Lyapunov constants of (2).
It is easy to verify that the equalities D i = 0, i = 1, 2, 3, 4 are equivalent to the following relations If a = 0 then our simultaneous polynomial equations have two sets of solutions indicated in (i) and (ii). If a = 0 then, in view of the condition c = −a, we see that the other three equations constitute a non degenerate linear system for determining the variables f, g, h. The solution is given by (iii).
The necessity part of the theorem is proved. Sufficiency: Case (i). System (2) now takes the forṁ This is a quasi-homogeneous system of degree 5 whose coefficients satisfy the equality f = −3(d + h) which is the necessary and sufficient center condition in the case we study [4].
Case (ii). System (2) now takes the forṁ The planar differential systeṁ is said to be reversible (in the sense ofŻoladek) if its orbits are symmetric with respect to a line passing through the origin. System (6) is reversible if there is a linear transformation S : R 2 → R 2 , sending a point (x, y) to the point (x ′ , y ′ ) symmetric to (x, y) with respect to the line αx + βy = 0 and satisfying the condition S(p(x, y), q(x, y)) = −(p(S(x, y)), q(S(x, y))).
A more general condition of reversibility is as follows It is well known that if system (6) is reversible and has a linear center at the origin then the origin is a center of this system (see [2], for example).
Obviously, system (5) is reversible because its trajectories are symmetric with respect to both coordinate axes. So, the origin is a center for system (5).
Case (iii). System (2) now takes the form It turns out that system (7) is reversible. Its trajectories are symmetric with respect to each of the two perpendicular lines defined by the equation ax 2 + bxy − ay 2 = 0. The appropriate linear transformation S is given by each of the two matrices This fact is confirmed by the straight calculations. We used Mathematica here.
Hence the origin is a center for system (2) in this case once again. The theorem is proved.

2.
It is known that isochronism of a center of a planar polynomial system is equivalent to existence of an analytic transversal system commuting with a given system in a neighbourhood of a center [5]; observe that an arbitrary polynomial system with isochronous center not necessarily commutes with a polynomial system [6,7].
It is proved in [8] that if the systemṡ commute then µ(x, y) = 1/(p(x, y)s(x, y)−q(x, y)r(x, y)) is an integrating factor of both systems. Thereby if both commuting systems are polynomial then we can find the integrating Darboux factor for the given system and integrate the latter (about the method of Darboux and the relevant definitions see [9], for example).
We now state the following fact which will be useful later. Considering (8), assume that where R(x, y), Q(x, y) are polynomials in x, y. Then the algebraic curves x 2 + y 2 = 0, Q(x, y) = 0 are invariants for each of these systems. Indeed, it is immediately obvious that x 2 +y 2 = 0 is an invariant of both systems with the cofactor 2R(x, y) and 2Q(x, y) respectively. The curve Q(x, y) = 0 is an invariant of the second system with the cofactor xQ x (x, y) + yQ y (x, y).
We see that the curve Q(x, y) = 0 is an invariant with the cofactor xR x (x, y)+ yR y (x, y).
3. In each of the three cases we have found a non trivial polynomial system commuting with the respective system.
In case (i) such a system iṡ q 4 (x, y)).
A straight analysis of this expression allows us to conclude that in our case a center may be of type B 2 or B 4 only.
If b = 0 system (5) is a system of the form (4) for which the condition f = −3(d + h) is obviously fulfilled. Then its first integral is If b = 0 then we may suppose that b = 1. The general case reduces to this by the change of variables Then our system takes the forṁ The functions are invariants for (11) with the cofactors L 1 = 2xy(1 + ex 2 + gy 2 ), L 2 = 2xy(1 + 2(ex 2 + gy 2 )).
Moreover, if e = g the function is invariant with the cofactor L 3 = 2xy.
We have 2L 1 − L 2 + 1 e L 3 = 0. Then the function is the first Darboux integral of (11) for e = g. Since (11) has a unique finite singular point at the origin, the phase portraits are obtained by studying the points at infinity. A standard inspection of the location and types of such points on the equator of the Poincaré sphere allows us to conclude that (11) has phase portraits of two types only: a center is of type B 2 when eg ≥ 0 or of type B 4 when eg < 0.
In case (iii) a commuting system and first integral may be found on considering that system (7) is equivalent to (11).
Summarizing we conclude that the system under consideration has phase portraits of two types only: a center is of type B 2 or of type B 4 .