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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]

Given p, q \in Z\geq 2 with p \not = q, we study generalized Abel differential equations dx d\theta = A(\theta)xp+B(\theta)xq, where A and B are trigonometric polynomials of degrees n,m \geq 1, respectively, and we are interested in the number of limit cycles (i.e., isolated periodic orbits) that they can have. More concretely, in this context, an open problem is to prove the existence of an integer, depending only on p, q,m, and n and that we denote by \scrH p,q(n,m), such that the above differential equation has at most \scrH p,q(n,m) limit cycles. In the present paper, by means of a second order analysis using Melnikov functions, we provide lower bounds of \scrH p,q(n,m) that, to the best of our knowledge, are larger than the previous ones appearing in the literature. In particular, for classical Abel differential equations (i.e., p = 3 and q = 2), we prove that \scrH 3,2(n,m) \geq 2(n + m) 1. [ABSTRACT FROM AUTHOR]

The Bendixson–Dulac theorem provides a criterion to find upper bounds for the number of limit cycles in analytic differential systems. We extend this classical result to some classes of piecewise differential systems. We apply it to three different Liénard piecewise differential systems The first is linear, the second is rational and the last corresponds to a particular extension of the cubic van der Pol oscillator. In all cases, the systems present regions in the parameter space with no limit cycles and others having at most one. [ABSTRACT FROM AUTHOR]

We study the number of limit cycles of planar piecewise linear differential systems separated by a branch of an algebraic curve. We show that for each n E N there exist piecewise linear differential systems separated by an algebraic curve of degree n having [n/2] hyperbolic limit cycles. Moreover, when n = 2, 3, we study in more detail the problem, considering a perturbation of a center and constructing examples with 4 and 5 limit cycles, respectively. These results follow by proving that the set of functions generating the first order averaged function associated to the problem is an extended complete Chebyshev system in a suitable interval. [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]

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. [ABSTRACT FROM AUTHOR]

In this paper we deal with planar piecewise linear differential systems defined in two zones. We consider the case when the two linear zones are angular sectors of angles α and 2 π − α , respectively, for α ∈ ( 0 , π ) . 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 α , we prove the existence of systems with four limit cycles up to fifth order and, for α = π / 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. [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]

Consider real or complex polynomial Riccati differential equations a ( x ) y ˙ = b 0 ( x ) + b 1 ( x ) y + b 2 ( x ) y 2 with all the involved functions being polynomials of degree at most η . We prove that the maximum number of polynomial solutions is η + 1 (resp. 2) when η ≥ 1 (resp. η = 0 ) and that these bounds are sharp. For real trigonometric polynomial Riccati differential equations with all the functions being trigonometric polynomials of degree at most η ≥ 1 we prove a similar result. In this case, the maximum number of trigonometric polynomial solutions is 2 η (resp. 3) when η ≥ 2 (resp. η = 1 ) and, again, these bounds are sharp. Although the proof of both results has the same starting point, the classical result that asserts that the cross ratio of four different solutions of a Riccati differential equation is constant, the trigonometric case is much more involved. The main reason is that the ring of trigonometric polynomials is not a unique factorization domain. [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]

This paper proves the uniqueness of limit cycles for sewing planar piecewise linear systems with two zones separated by a straight line, Σ, and only one Σ-singularity of monodromic type. The proofs are based in an extension of Rolle's Theorem for dynamical systems on the plane. [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]

We study the center-focus problem as well as the number of limit cycles which bifurcate from a weak focus for several families of planar discontinuous ordinary differential equations. Our computations of the return map near the critical point are performed with a new method based on a suitable decomposition of certain one-forms associated with the expression of the system in polar coordinates. This decomposition simplifies all the expressions involved in the procedure. Finally, we apply our results to study a mathematical model of a mechanical problem, the movement of a ball between two elastic walls. [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 the weak 16th Hilbert problem, the Poincaré-Pontryagin-Melnikov function, M 1 (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 1 (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. [ABSTRACT FROM AUTHOR]

We consider piecewise quadratic perturbations of centers of piecewise quadratic systems in two zones determined by a straight line through the origin. By means of expansions of the displacement map, we are able to find isolated zeros of it, without dealing with the unsurprising difficult integrals inherent in the usual averaging approach. We apply this technique to non-smooth perturbations of the four families of isochronous centers of the Loud family, S 1 , S 2 , S 3 , and S 4 , as well as to non-smooth perturbations of non-smooth centers given by putting different S i 's in each zone. To show the coverage of our approach, we apply its first order, which is equivalent to averaging theory of the first order, in perturbations of the already mentioned centers considering all the straight lines through the origin. Then we apply the second order of our approach to perturbations of the above centers for a specific oblique straight line. Here in order to argue we introduce certain blow-ups in the perturbative parameters. As a consequence of our study, we obtain examples of piecewise quadratic systems with at least 12 limit cycles. By analyzing two previous works of the literature claiming much more limit cycles we found some mistakes in the calculations. Therefore, the best lower bound for the number of limit cycles of a piecewise quadratic system is up to now the 12 limit cycles found in the present paper. [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]

The aim of this paper is the study of the center-focus and cyclicity problems inside the class X of 3-dimensional vector fields that admit a first integral that leaves invariant any sphere centered at the origin. We classify the centers of linear, quadratic homogeneous and a family of quadratic vector fields F ⊂ X , restricted to one of these spheres. Moreover, we show the existence of at least 4 limit cycles in family F . [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]

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 We study the number of limit cycles bifurcating from a piecewise quadratic system. All the differential systems considered are piecewise in two zones separated by a straight line. We prove the existence of 16 crossing limit cycles in this class of systems. If we denote by H p (n) the extension of the Hilbert number to degree n piecewise polynomial differential systems, then H p (2) ≥ 16. As fas as we are concerned, this is the best lower bound for the quadratic class. Moreover, in the studied cases, all limit cycles appear nested bifurcating from a period annulus of a isochronous quadratic center. [ABSTRACT FROM AUTHOR]

A classical perturbation problem is the polynomial perturbation of the harmonic oscillator, ( x ′ , y ′ ) = ( − y + ε f ( x , y , ε ) , x + ε g ( x , y , ε ) ) . In this paper we study the limit cycles that bifurcate from the period annulus via piecewise polynomial perturbations in two zones separated by a straight line. We prove that, for polynomial perturbations of degree n , no more than N n − 1 limit cycles appear up to a study of order N . We also show that this upper bound is reached for orders one and two. Moreover, we study this problem in some classes of piecewise Liénard differential systems providing better upper bounds for higher order perturbation in ε , showing also when they are reached. The Poincaré–Pontryagin–Melnikov theory is the main technique used to prove all the results. [ABSTRACT FROM AUTHOR]

This paper is devoted to study the algebraic limit cycles of planar piecewise linear differential systems. In particular, we present examples exhibiting two explicit hyperbolic algebraic limit cycles, as well as some one-parameter families with a saddle-node bifurcation of algebraic limit cycles. We also show that all degrees for algebraic limit cycles are allowed. [ABSTRACT FROM AUTHOR]

We consider the family of planar differential systems depending on two real parameters This system corresponds to the normal form for the 1:2 resonance which exhibits a heteroclinic connection. The phase portrait of the system has a limit cycle which disappears in the heteroclinic connection for the parameter values on the curve We significantly improve the knowledge of this curve in a neighborhood of the origin. [ABSTRACT FROM AUTHOR]

We study planar polynomial differential equations that in complex coordinates write as z ˙ = A z + B z k z ¯ l + C z m z ¯ n . We prove that for each p ∈ N there are differential equations of this type having at least p limit cycles. Moreover, for the particular case z ˙ = A z + B z ¯ + C z m z ¯ n , which has homogeneous nonlinearities, we show examples with several limit cycles and give a condition that ensures uniqueness and hyperbolicity of the limit cycle. [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: We prove that a family of functions defined through some definite integrals forms an extended complete Chebyshev system. The key point of our proof consists of reducing the study of certain Wronskians to the Gram determinants of a suitable set of new functions. Our result is then applied to give upper bounds for the number of isolated periodic solutions of some perturbed Abel equations. [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]

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]

Planar piecewise linear systems with two linearity zones separated by a straight line and with a periodic orbit at infinity are considered. By using some changes of variables and parameters, a reduced canonical form with five parameters is obtained. Instead of the usual Bendixson transformation to work near infinity, a more direct approach is introduced by taking suitable coordinates for the crossing points of the possible periodic orbits with the separation straight line. The required computations to characterize the stability and bifurcations of the periodic orbit at infinity are much easier. It is shown that the Hopf bifurcation at infinity can have degeneracies of co-dimension three and, in particular, up to three limit cycles can bifurcate from the periodic orbit at infinity. This provides a new mechanism to explain the claimed maximum number of limit cycles in this family of systems. The centers at infinity classification together with the limit cycles bifurcating from them are also analyzed. [ABSTRACT FROM AUTHOR]

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]