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Made available in DSpace on 2021-06-25T12:30:34Z (GMT). No. of bitstreams: 0 Previous issue date: 2021-02-25 Catalan AGAUR Spanish Ministerio de Ciencia, Innovacion y Universidades -Agencia estatal de investigacion European Community Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq) Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP) In this paper, we are interested in how the local cyclicity of a family of centers depends on the parameters. This fact was pointed out in [21], to prove that there exists a family of cubic centers, labeled by C D-31(12) in [25], with more local cyclicity than expected. In this family, there is a special center such that at least twelve limit cycles of small amplitude bifurcate from the origin when we perturb it in the cubic polynomial general class. The original proof has some crucial missing points in the arguments that we correct here. We take advantage of a better understanding of the bifurcation phenomenon in nongeneric cases to show two new cubic systems exhibiting 11 limit cycles and another exhibiting 12. Finally, using the same techniques, we study the local cyclicity of holomorphic quartic centers, proving that 21 limit cycles of small amplitude bifurcate from the origin, when we perturb in the class of quartic polynomial vector fields. (C) 2020 Elsevier Inc. All rights reserved. Univ Lleida, Dept Matemat, Avda Jaume II 69, Lleida 6925001, Catalonia, Spain Univ Autonoma Barcelona, Dept Matemat, Barcelona 08193, Catalonia, Spain Univ Estadual Paulista, Dept Matemat, BR-15054000 Sao Jose Do Rio Preto, Brazil Ctr Recerca Matemat, Campus Bellaterra, Barcelona 08193, Catalonia, Spain Univ Estadual Paulista, Dept Matemat, BR-15054000 Sao Jose Do Rio Preto, Brazil Catalan AGAUR: 2017SGR1617 Catalan AGAUR: 2017SGR127 Spanish Ministerio de Ciencia, Innovacion y Universidades -Agencia estatal de investigacion: MTM2017-84383-P Spanish Ministerio de Ciencia, Innovacion y Universidades -Agencia estatal de investigacion: PID2019-104658GB-I00 European Community: H2020-MSCA-RISE-2017-777911 CNPq: 200484/2015-0 FAPESP: 2020/04717-0

Made available in DSpace on 2021-06-25T12:27:56Z (GMT). No. of bitstreams: 0 Previous issue date: 2021-01-15 Catalan AGAUR Spanish Ministerio de Ciencia, Innovacion y Universidades - Agencia estatal de investigacion European Community Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq) Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP) We are interested in small-amplitude isolated periodic orbits, so-called limit cycles, surrounding only one equilibrium point, that we locate at the origin. We develop a parallelization technique to study higher order developments, with respect to the parameters, of the return map near the origin. This technique is useful to study lower bounds for the local cyclicity of centers. We denote by M(n) the maximum number of limit cycles bifurcating from the origin via a degenerate Hopf bifurcation for a polynomial vector field of degree n. We get lower bounds for the local cyclicity of some known cubic centers and we prove that M(4) >= 20, M(5) >= 33, M(7) >= 61, M(8) >= 76, and M(9) >= 88. (C) 2020 Elsevier Inc. All rights reserved. Univ Autonoma Barcelona, Dept Matemat, Barcelona 08193, Catalonia, Spain Univ Estadual Paulista, Dept Matemat, BR-15054000 Sao Jose Do Rio Preto, Brazil Univ Estadual Paulista, Dept Matemat, BR-15054000 Sao Jose Do Rio Preto, Brazil Catalan AGAUR: 2017SGR1617 Spanish Ministerio de Ciencia, Innovacion y Universidades - Agencia estatal de investigacion: MTM201677278-P European Community: H2020-MSCA-RISE-2017-777911 CNPq: 200484/2015-0 FAPESP: 2020/04717-0

In this paper, limit cycles bifurcating from a third-order nilpotent critical point in a class of quartic planar systems are studied. With the aid of computer algebra system MAPLE, the first 12 Lyapunov constants are deduced by the normal form method. As a result, sufficient and necessary center conditions are derived, and the fact that there exist 12 or 13 limit cycles bifurcating from the nilpotent critical point is proved by different perturbations. The result in [Qiu et al. in Adv. Differ. Equ. 2015(1):1, 2015] is improved.

We classify the centers of polynomial differential systems in $R^2$ of odd degree $d \ge 5$, in complex notation, as $\dot{z} = iz + (z \bar z)^(d-5)/2(A z^5 + B z^4 \bar z + C z^3 \bar z^2 + D z^2 \bar z^3 + E z \bar z^4 + F \bar z^5)$, where $A,B,C,D,E, F \in mathbb{C}$ and either $A = Re(D) = 0$, $A = Im(D) = 0$, $Re(A) = D = 0$ or $Im(A) = D = 0$. We thank to Professor Colin Christopher for his help in the proof of statement (g) of Theorem 3. The first author is partially supported by a MINECO/ FEDER grant number MTM2014-53703-P and an AGAUR (Generalitat de Catalunya) grant number 2014SGR 1204. The second author is partially supported by a MINECO/ FEDER grant number MTM2008-03437, an AGAUR grant number 2009SGR 410, ICREA Academia, two FP7-PEOPLE-2012-IRSES numbers 316338 and 318999, and FEDERUNAB10-4E-378. The third author is partially supported by FCT/Portugal through the project UID/MAT/04459/2013.

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. The first author is partially supported by a MINECO/ FEDER grant number MTM2011-22877 and an AGAUR (Generalitat de Catalunya) grant number 2009SGR 381. The second author is partially supported by a MINECO/ FEDER grant number MTM2008-03437, an AGAUR grant number 2009SGR 410, ICREA Academia and two FP7-PEOPLE-2012-IRSES numbers 316338 and 318999.

For the polynomial differential system x˙ = −y, y˙ = x+Qn(x; y), where Qn(x; y) is a homogeneous polynomial of degree n there are the following two conjectures done 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. The first author is partially supported by a MINECO/FEDER grant number MTM2011-22877 and an AGAUR (Generalitat de Catalunya) grant number 2014SGR 1204. The second author is partially supported by a MINECO/FEDER grant MTM2008-03437 and MTM2013-40998-P, an AGAUR grant number 2014 SGR568, an ICREA Academia, the grants FP7-PEOPLE-2012-IRSES 318999 and 316338, FEDER-UNAB-10-4E-378. The third author is supported by Portuguese National Funds through FCT – Fundação para a Ciência e a Tecnologia within the project PTDC/MAT/117106/2010 and by CAMGSD

In this paper, we provide necessary and sufficient conditions for the existence of local analytic first integrals for a seventh-parameter family of complex cubic systems called the complex Kukles systems. The first author is partially supported by a MINECO/ FEDER [grant number MTM2014-53703-P]; an AGAUR (Generalitat de Catalunya) [grant number 2014SGR 1204]. The second author is supported by Portuguese National Funds through FCT - Fundação para a Ciência e a Tecnologia within the project PTDC/MAT/117106/2010 and by CAMGSD.