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We provide the best lower bound for the number of critical periods of planar polynomial centers known up to now. The new lower bound is obtained in the Hamiltonian class and considering a single period annulus. This lower bound doubles the previous one from the literature, and we end up with at least n 2 − 2 (resp. n 2 − 2 n − 1) critical periods for planar polynomial systems of odd (resp. even) degree n. Key idea is the perturbation of a vector field with many cusp equilibria, whose construction is by itself a nontrivial construction that uses elements of catastrophe theory. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

We study the number of periodic orbits that bifurcate from a cubic polynomial vector field having two period annuli via piecewise perturbations. The cubic planar system ( x ′ , y ′ ) = ( − y ( ( x − 1 ) 2 + y 2 ) , x ( ( x − 1 ) 2 + y 2 ) ) has simultaneously a center at the origin and at infinity. We study, up to first order averaging analysis, the bifurcation of periodic orbits from the two period annuli, first separately and second simultaneously. This problem is a generalization of [24] to the piecewise systems class. When the polynomial perturbation has degree n , we prove that the inner and outer Abelian integrals are rational functions and we provide an upper bound for the number of zeros. When the perturbation is cubic, the same degree as the unperturbed vector field, the maximum number of limit cycles, up to first order perturbation, from the inner and outer annuli is 9 and 8, respectively. When the simultaneous bifurcation problem is considered, 12 limit cycles exist. These limit cycles appear in three types of configurations: ( 9 , 3 ), ( 6 , 6 ) and ( 4 , 8 ). In the non-piecewise scenario, only 5 limit cycles were found. [ABSTRACT FROM AUTHOR]

In this paper we study the centers of projective vector fields QT of three-dimensional quasi-homogeneous differential system dx/dt = Q(x) with the weight (m,m, n) and degree 2 on the unit sphere S2. We seek the sufficient and necessary conditions under which QT has at least one center on S2. Moreover, we provide the exact number and the positions of the centers of QT. First we give the complete classification of systems dx/dt = Q(x) and then, using the induced systems of QT on the local charts of S2, we determine the conditions for the existence of centers. The results of this paper provide a convenient criterion to find out all the centers of QT on S2 with Q being the quasi-homogeneous polynomial vector field of weight (m,m, n) and degree 2. [ABSTRACT FROM AUTHOR]

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 devoted to establish a new approach, namely parallelization, to compute the linear parts of the Lyapunov constants. More concretely, it is shown 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 z ˙ = i z + 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 gives 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 . [ABSTRACT FROM AUTHOR]

In this work, we are interested in isolated crossing periodic orbits in planar piecewise polynomial vector fields defined in two zones separated by a straight line. In particular, in the number of limit cycles of small amplitude. They are all nested and surrounding one equilibrium point or a sliding segment. We provide lower bounds for the local cyclicity for planar piecewise polynomial systems, M p c (n) , with degrees 2, 3, 4 , and 5. More concretely, M p c (2) ≥ 13 , M p c (3) ≥ 26 , M p c (4) ≥ 40 , and M p c (5) ≥ 58. The computations use parallelization algorithms. [ABSTRACT FROM AUTHOR]

In this work, we are interested in crossing limit cycles surrounding only one equilibrium point or a sliding segment. The studied systems are piecewise cubic polynomial defined in two zones separated by a straight line. In this class, we get at least 24 crossing limit cycles, all of them in only one nest, bifurcating from a cubic polynomial center. The computations use a parallelization algorithm. [ABSTRACT FROM AUTHOR]

Our interest is centered in the study of the number of limit cycles for nonsmooth piecewise linear vector fields on the plane when the switching curve is x y = 0. We consider the symmetric case. That is, one vector field defined in the odd quadrants and the other in the even ones. We deal with equilibrium points of center-focus type, with matrices in real Jordan form, in each vector field when the infinity is monodromic. In this case, we provide the center classification at infinity, we prove that the maximum order of a weak focus is five. Moreover, we show the existence of systems exhibiting five limit cycles bifurcating from infinity. [ABSTRACT FROM AUTHOR]

Abstract: Consider the vector field , where the set of critical points is formed by straight lines, not passing through the origin and parallel to one or two orthogonal directions. We perturb it with a general polynomial perturbation of degree and study the maximum number of limit cycles that can bifurcate from the period annulus of the origin in terms of and . Our approach is based on the explicit computation of the Abelian integral that controls the bifurcation and on a new result for bounding the number of zeroes of a certain family of real functions. When we apply our results for we recover or improve some results obtained in several previous works. [Copyright &y& Elsevier]

Abstract: We analyze whether a given set of analytic functions is an Extended Chebyshev system. This family of functions appears studying the number of limit cycles bifurcating from some nonlinear vector field in the plane. Our approach is mainly based on the so called Derivation–Division algorithm. We prove that under some natural hypotheses our family is an Extended Chebyshev system and when some of them are not fulfilled then the set of functions is not necessarily an Extended Chebyshev system. One of these examples constitutes an Extended Chebyshev system with high accuracy. [Copyright &y& Elsevier]

Abstract: Consider planar ordinary differential equations of the form ,, where is an algebraic curve. We are interested in knowing whether the existence of multiple factors for is important or not when we study the maximum number of zeros of the Abelian integral that controls the limit cycles that bifurcate from the period annulus of the origin when we perturb it with an arbitrary polynomial vector field. With this aim, we study in detail the case , where is a positive integer number and prove that has essentially no impact on the number of zeros of . This result improves the known studies on . One of the key points of our approach is that we obtain a simple expression of based on some successive reductions of the integrals appearing during the procedure. [Copyright &y& Elsevier]