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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]

In this paper we aim to find the highest number of critical periods in a class of planar systems of polynomial differential equations for fixed degree having a center. We fix our attention to lower bounds of local criticality for low degree planar polynomial centers. The main technique is the study of perturbations of reversible holomorphic (isochronous) centers, inside the reversible centers class. More concretely, we study the Taylor developments of the period constants with respect to the perturbation parameters. First, we see that there are systems of degree 3 ≤ n ≤ 16 for which up to first order at least (n 2 + n − 4) / 2 critical periods bifurcate from the center. Second, we improve this number for centers with degree from 3 to 9. In particular, we obtain 6 and 10 critical periods for cubic and quartic degree systems, respectively. [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]

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 study the criticality of some planar systems of polynomial differential equations having a center for various low degrees n. To this end, we present a method which is equivalent to the use of the first non-identically zero Melnikov function in the problem of limit cycles bifurcation, but adapted to the period function. We prove that the Taylor development of this first order function can be found from the linear terms of the corresponding period constants. Later, we consider families which are isochronous centers being perturbed inside the reversible centers class, and we prove our criticality results by finding the first order Taylor developments of the period constants with respect to the perturbation parameters. In particular, we obtain that at least 22 critical periods bifurcate for n = 6 , 37 for n = 8 , 57 for n = 10 , 80 for n = 12 , 106 for n = 14 , and 136 for n = 16. Up to our knowledge, these values improve the best current lower bounds. • Relation of Melnikov theory and Period constants. • The highest number of critical periods for low degree systems. • New isochronous polynomial systems. • Parallelization of the computations. [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]

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. [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: In this paper we study the number of limit cycles bifurcating from isochronous surfaces of revolution contained in , when we consider polynomial perturbations of arbitrary degree. The method for studying these limit cycles is based on the averaging theory and on the properties of Chebyshev systems. We present a new result on averaging theory and generalizations of some classical Chebyshev systems which allow us to obtain the main results. [Copyright &y& Elsevier]

This paper is mainly devoted to the study of the index of a map at a zero, and the index of a polynomial map over Rn. For semi-quasi-homogeneous maps we prove that the index at a zero coincides with the index at this zero of its quasi-homogeneous part. For a class of polynomial maps with finite zero set we provide a method which makes easier the computation of its index over Rn. Finally we relate the index and the multiplicity. [ABSTRACT FROM AUTHOR]