The classification of reversible cubic systems with center

Classification of the centers, their cyclicity and isochronicity for a class of polynomial differential systems generalizing the linear systems with cubic homogeneous nonlinearities

“…In any case some interesting results on some subclasses of cubic systems are the ones of Rousseau and Schlomiuk [21], and the ones ofŻoładek [29,30].…”

Reversibility and quasi-homogeneous normal forms of vector fields

“…Let F be the vector field given by (4.11). F is a sum of three quasi-homogeneous vector field of type t = (3, 8,10) and degree 5, 6 and 7, respectively. The first quasi-homogeneous term, F 5 := (y, xz, x 5 ) T is already in a desired simplified form (it is R x -reversible) and this is the unique zero degree involution which carry the first term of the vector field into an R x -reversible form.…”

The problem of distinguishing between a center and a focus for nilpotent and degenerate analytic systems

“…We remark that all the nilpotent centers that we know are time-reversible or have an analytic first integral at the origin. The notion of time-reversibility has been generalized in [34] with the notion of rational reversibility, see also [33].…”

“…But it has neither a local analytic first integral, nor a formal first integral defined at the origin, see the proof in [10]. Consider now the following perturbation of system (34).…”