Weak-foci of high order and cyclicity

Agraïments: This work was done when H. Liang was visiting the Department of Mathematics of Universitat Autònoma de Barcelona. He is very grateful for the support and hospitality. The first author is supported by the NSF of China (No. 11201086 and No. 11401255) and the Excellent Young Teachers Training Program for colleges and universities of Guangdong Province, China (No. Yq2013107).Agraïments: The second author is partially supported by UNAB13-4E-1604.A particular version of the 16th Hilbert's problem is to estimate the number, M(n), of limit cycles bifurcating from a singularity of center-focus type. This paper is devoted to finding lower bounds for M(n) for some concrete n by studying the cyclicity of different weak-foci. Since a weak-focus with high order is the most current way to produce high cyclicity, we search for systems with the highest possible weak-focus order. For even n, the studied polynomial system of degree n was the one obtained by QiuYan2009 where the highest weak-focus order is n^2 n-2 for n=4,6, 18. Moreover, we provide a system which has a weak-focus with order (n-1)^2 for n 12. We show that Christopher's approach Chr2006, aiming to study the cyclicity of centers, can be applied also to the weak-focus case. We also show by concrete examples that, in some families, this approach is so powerful and the cyclicity can be obtained in a simple computational way. Finally, using this approach, we obtain that M(6) 39, M(7) 34 and M(8) 63

A proof of Perko's conjectures for the Bogdanov-Takens system

The Bogdanov-Takens system has at most one limit cycle and, in the parameter space, it exists between a Hopf and a saddle-loop bifurcation curves. The aim of this paper is to prove the Perko's conjectures about some analytic properties of the saddle-loop bifurcation curve. Moreover, we provide sharp piecewise algebraic upper and lower bounds for this curve

Limit cycles in planar piecewise linear differential systems with nonregular separation line

Agraïments: The first author is supported by FAPESP grant number 2013/24541-0 and CAPES grant number 88881.030454/2013-01 Program CSF-PVE and UNAB13-4E-1604.In this paper we deal with lanar piecewise linear differential systems defined in two zones. We consider the case when the two linear zones are angular sectors of angles and 2 - respectively, for (0,). We study the problem of determining lower bounds for the number of isolated periodic orbits in such systems using Melnikov functions. These limit cycles appear studying higher order piecewise linear perturbations of a linear center. It is proved that the maximum number of limit cycles that can appear up to a sixth order perturbation is five. Moreover, for these values of we prove the existence of systems with four limit cycles up to fifth order and, for =/2, we provide an explicit example with five up to sixth order. In general, the nonregular separation line increases the number of periodic orbits in comparison with the case where the two zones are separated by a straight line

On extended chebyshev systems with positive accuracy

Agraïments: The first author is supported by a FAPESP-BRAZIL grant 2013/16492-0. The second author is supported by UNAB13-4E-1604 grant.A classical necessary condition for an ordered set of n+1 functions F to be an ECT-system in a closed interval is that all the Wronskians do not vanish. With this condition all the elements of Span(F) have at most n zeros taking into account the multiplicity. Here the problem of bounding the number of zeros of Span(F) is considered as well as the effectiveness of the upper bound when some Wronskians vanish. For this case we also study the possible configurations of zeros that can be realized by elements of Span(F). An application to count the number of isolated periodic orbits for a family of nonsmooth systems is performed

Limit cycles coming from some uniform isochronous centers

Agraïments: The first author is supported by the NSF of China (No. 11201086), the Foundation for Distinguished Young Talents in Higher Education of Guangdong, China (No.2012LYM0087) and the Excellent Young Teachers Training Program for colleges and uni- versities of Guangdong Province, China (No. Yq2013107).This article is about the weak 16--th Hilbert problem, i.e. we analyze how many limit cycles can bifurcate from the periodic orbits of a given polynomial differential center when it is perturbed inside a class of polynomial differential systems. More precisely, we consider the uniform isochronous centers $$x= -y x^2 y (x^2 y^2)^n, y= x x y^2 (x^2 y^2)^n,$$ of degree 2n 3 and we perturb them inside the class of all polynomial differential systems of degree 2n 3. For n=0,1 we provide the maximum number of limit cycles, 3 and 8 respectively, that can bifurcate from the periodic orbits of these centers using averaging theory of first order, or equivalently Abelian integrals. For n=2 we show that at least 12 limit cycles can bifurcate from the periodic orbits of the center

Models de població del món i prediccions

En aquest treball pretenem predir com evolucionarà la població del món a partir de dades disponibles (estimades) de la població humana durant els darrers vint segles. Les fonts que hem utilitzat per a obtenir les dades en les que basarem el nostre estudi, són: el web del cens dels EEUU, el de la Divisió de Població del Departament d'Economia i Assumptes Socials de la ONU

First-order perturbation for multi-parameter center families

Altres ajuts: Acord transformatiu CRUE-CSICIn the weak 16th Hilbert problem, the Poincaré-Pontryagin-Melnikov function, M(h), is used for obtaining isolated periodic orbits bifurcating from centers up to a first-order analysis. This problem becomes more difficult when a family of centers is considered. In this work we provide a compact expression for the first-order Taylor series of the function M(h,a) with respect to a, being a the multi-parameter in the unperturbed center family. More concretely, when the center family has an explicit first integral or inverse integrating factor depending on a. We use this new bifurcation mechanism to increase the number of limit cycles appearing up to a first-order analysis without the difficulties that higher-order studies present. We show its effectiveness by applying it to some classical examples

The local cyclicity problem : Melnikov method using Lyapunov constants

In 1991, Chicone and Jacobs showed the equivalence between the computation of the first-order Taylor developments of the Lyapunov constants and the developments of the first Melnikov function near a non-degenerate monodromic equilibrium point, in the study of limit cycles of small-amplitude bifurcating from a quadratic centre. We show that their proof is also valid for polynomial vector fields of any degree. This equivalence is used to provide a new lower bound for the local cyclicity of degree six polynomial vector fields, so M(6) ≥ 44. Moreover, we extend this equivalence to the piecewise polynomial class. Finally, we prove that Mcp(4) ≥ 43 and Mcp(5) ≥ 65