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Centers for generalized quintic polynominal differential systems
https://ddd.uab.cat/record/182538
Giné, JaumeTue, 28 Nov 2017 07:46:29 GMThttps://ddd.uab.cat/record/1825382017Center problem for systems with two monomial nonlinearities
https://ddd.uab.cat/record/169455
We study the center problem for planar systems with a linear center at the origin that in complex coordinates have a nonlinearity formed by the sum of two monomials. Our first result lists several centers inside this family. To the best of our knowledge this list includes a new class of Darboux centers that are also persistent centers. The rest of the paper is dedicated to try to prove that the given list is exhaustive. We get several partial results that seem to indicate that this is the case. In particular, we solve the question for several general families with arbitrary high degree and for all cases of degree less or equal than 19. As a byproduct of our study we also obtain the highest known order for weak-foci of planar polynomial systems of some given degrees. Gasull i Embid, ArmengolMon, 23 Jan 2017 15:21:46 GMThttps://ddd.uab.cat/record/1694552016Analytic nilpotent centers as limits of nondegenerate centers revisited
https://ddd.uab.cat/record/169453
We prove that all the nilpotent centers of planar analytic differential systems are limit of centers with purely imaginary eigenvalues, and consequently the Poincaré-Liapunov method to detect centers with purely imaginary eigenvalues can be used to detect nilpotent centers. García, Isaac A.Mon, 23 Jan 2017 15:21:46 GMThttps://ddd.uab.cat/record/1694532016A method for characterizing nilpotent centers
https://ddd.uab.cat/record/150727
To characterize when a nilpotent singular point of an analytic differential system is a center is of particular interest, first for the problem of distinguishing between a focus and a center, and after for studying the bifurcation of limit cycles from it or from its period annulus. We give an effective algorithm in the search of necessary conditions for detecting nilpotent centers based in recent developments. Moreover we survey the last results on this problem and illustrate our approach by means of examples. Giné, JaumeFri, 06 May 2016 08:59:52 GMThttps://ddd.uab.cat/record/1507272014Centers for a class of generalized quintic polynomial differential systems
https://ddd.uab.cat/record/150687
We classify the centers of the polynomial differential systems in R2 of degree d ≥ 5 odd that in complex notation writes as z˙ = iz + (zz¯)d−5/2 (Az5 + Bz4z¯ + Cz3z¯2 + Dz2z¯3 + Ezz¯4 + Fz¯5), where A, B, C, D, E, F ∈ C and either A = Re(D) = 0, or A = Im(D) = 0, or Re(A) = D = 0, or Im(A) = D = 0. Giné, JaumeFri, 06 May 2016 08:59:49 GMThttps://ddd.uab.cat/record/1506872014Centers for the Kukles homogeneous systems with odd degree
https://ddd.uab.cat/record/145329
For the polynomial differential system x ̇ = −y, y ̇ = x Q n (x, y), where Q n (x, y) is a homogeneous polynomial of degree n there are the following two conjectures raised in 1999. (1) Is it true that the previous system for n 2 has a center at the origin if and only if its vector field is symmetric about one of the coordinate axes? (2) Is it true that the origin is an isochronous center of the previous system with the exception of the linear center only if the system has even degree? We prove both conjectures for all n odd. Giné, JaumeTue, 12 Jan 2016 16:38:09 GMThttps://ddd.uab.cat/record/1453292015Parallelization of the Lyapunov constants and cyclicity for centers of planar polynomial vector fields
https://ddd.uab.cat/record/145287
Christopher in 2006 proved that under some assumptions the linear parts of the Lyapunov constants with respect to the parameters give the cyclicity of an elementary center. This paper is devote to establish a new approach, namely parallelization, to compute the linear parts of the Lyapunov constants. More concretely, it is showed that parallelization computes these linear parts in a shorter quantity of time than other traditional mechanisms. To show the power of this approach, we study the cyclicity of the holomorphic center =iz z^2 z^3 z^n under general polynomial perturbations of degree n, for n 13. We also exhibit that, from the point of view of computation, among the Hamiltonian, time-reversible, and Darboux centers, the holomorphic center is the best candidate to obtain high cyclicity examples of any degree. For n=4,5, 13, we prove that the cyclicity of the holomorphic center is at least n^2 n-2. This result give the highest lower bound for M(6), M(7), M(13) among the existing results, where M(n) is the maximum number of limit cycles bifurcating from an elementary monodromic singularity of polynomial systems of degree n. As a direct corollary we also obtain the highest lower bound for the Hilbert numbers H(6) 40, H(8) 70, and H(10) 108, because until now the best result was H(6) 39, H(8) 67, and H(10) 100. Liang, HaihuaTue, 12 Jan 2016 16:38:07 GMThttps://ddd.uab.cat/record/1452872015