On the conditions of Kukles for the existence of a Centre

Kukles revisited: Advances in computing techniques

“…The origin is an isochronous centre for system (2) if and only if one of the following conditions holds:…”

“…In Section 4 we determine the necessary conditions for the origin to be an isochronous centre for (2). We perform all our computerised calculations on a Dual 3.2 GHz Xeon processor, with 2 GB of memory, running a Linux Gentoo operating system and make much use of the CA systems REDUCE 3.7 [18] and Fermat 3.8.7 [19].…”

“…A specific example of system (2) can be used to model predator-prey populations where intratrophic predation occurs [10,11]. We are interested in the conditions for the origin to be a centre for (1) and an isochronous centre for (2), when the coefficients a i are real. An isolated critical point is a centre if, within some neighbourhood, all orbits surrounding the point are closed.…”

Sufficient conditions for the existence of a center in polynomial systems of arbitrary degree

“…It is natural to think that this result will be valid for polynomial systems of an arbitrary degree n. In order to verify this hypothesis, we have carried out similar calculations as in Section 1, for polynomial systems up to degree 8 We recall that this conjectured theorem has been verified for 2 ≤ n ≤ 8. Let us remark that we have also considered all the particular cases by using the same method.…”

“…However, Jin and Wang showed in [8] that Kukles'conditions are only sufficient but not necessary, by giving explicitly another condition which is not covered by the Kukles'ones (see also [9]). C. J. Christopher and N. G. Lloyd showed in [10] that Kukles'conditions are necessary only in the case where a 7 = 0.…”