Centers for a class of generalized quintic polynomial differential systems

Abstract

Abstract. We classify the centers of the polynomial differential systems in R 2 of degree d ≥ 5 odd that in complex notation writes asż

On the Mechanisms for Producing Linear Type Centers in Polynomial Differential Systems

“…These two mechanisms for producing centers, the Liouvillian integrability and the algebraic reducibility, explain all the known linear type centers of the polynomial differential systems studied up to now, see for instance [2,8,9,11,16,24,25,29,30,37,38,39,40,41,42,47,48] and the references therein. As far as we know does not exist a known linear type center produced by a different mechanism than these two explained mechanisms.…”

A hybrid symbolic-numerical approach to the center-focus problem

“…From the computational point of view, the biggest obstacle in solving the center-focus problem for a specific system is the determination of the irreducible components of the variety (i.e., solution set) defined by a certain number of focus quantities. The most common approach [2,32,24] is the application of computer algebra algorithms for computing the primary decomposition of the ideal generated by the focus quantities such as Gianni-Trager-Zacharias (GTZ) [31] or Shimoyama-Yokoyama (SY) [58], which have been implemented in various symbolic packages (e.g. Singular [35], or Macaulay [34]).…”

Integrability conditions of a resonant saddle perturbed with homogeneous quintic nonlinearities

“…Doing the change of variable (10) withξ = ( i i + b − i √3b satisfies among others A = i, E = Re(C) = B + 3D = 0. Thus this equation was studied in[8] and corresponds to case (h) of Theorem 3 where it is proved that for this case there exist a Darboux integrating factor. Hence when system (19) corresponds to a real center, this center is Darboux integrable.Form the results presented in this work we have the following conjecture: The conditions of Theorem 1, Theorem 2 (with the corrections given by Lemma 1 for the case (C 2 )(ζ) and the conditions in (13) for the case (C 2 )(η)), Theorem 3 and Theorem 4 are the necessary and sufficient conditions for integrability of system (4) under the conditions (C 1 ), (C 2 ), (C 3 ) and (C 4 ) respectively.…”