On the number of zeros of Abelian integrals for a polynomial Hamiltonian irregular at infinity

Up to now, most of the results on the tangential Hilbert 16th problem have been concerned with the Hamiltonian regular at infinity, i.e., its principal homogeneous part is a product of the pairwise different linear forms. In this paper, we study a polynomial Hamiltonian which is not regular at infinity. It is shown that the space of Abelian integral for this Hamiltonian is finitely generated as a R[h] module by several basic integrals which satisfy the Picard-Fuchs system of linear differential equations. Applying the bound meandering principle, an upper bound for the number of complex isolated zeros of Abelian integrals is obtained on a positive distance from critical locus. This result is a partial solution of tangential Hilbert 16th problem for this Hamiltonian. As a consequence, we get an upper bound of the number of limit cycles produced by the period annulus of the non-Hamiltonian integrable quadratic systems whose almost all orbits are algebraic curves of degree k+n, under polynomial perturbation of arbitrary degree.     

On the 16th Hilbert problem for algebraic limit cycles

For a polynomial planar vector field of degree n?2 with generic invariant algebraic curves we show that the maximum number of algebraic limit cycles is 1+(n−1)(n−2)/2 when n is even, and (n−1)(n−2)/2 when n is odd. Furthermore, these upper bounds are reached.     

Non-nested configuration of algebraic limit cycles in quadratic systems

This work deals with algebraic limit cycles of planar polynomial differential systems of degree two. More concretely, we show among other facts that a quadratic vector field cannot possess two non-nested algebraic limit cycles contained in different irreducible invariant algebraic curves.     

Small limit cycles bifurcating from fine focus points in quartic order Z3-equivariant vector fields

This paper is concerned with the number of limit cycles for a quartic polynomial Z₃-equivariant vector fields. The system under consideration has a fine focus point at the origin, and three fine focus points which are symmetric about the origin. By the computation of the singular point values, sixteen limit cycles are found and their distributions are studied by using the new methods of bifurcation theory and qualitative analysis. This is a new result in the study of the second part of the 16th Hilbert problem. It gives rise to the conclusion: H(4)?16, where H(n) is the Hilbert number for the second part of Hilbert's 16th problem. The process of the proof is algebraic and symbolic. As far as know, the technique employed in this work is different from more usual ones, the calculation can be readily done with using computer symbol operation system such as Mathematica.     

Abelian integrals and period functions for quasihomogeneous Hamiltonian vector fields

In this paper, we study the number of zeros of Abelian integrals and the monotonicity of period functions for planar quasihomogeneous Hamiltonian vector fields. The result for Abelian integrals extends the recent work of Li et al. [C. Li, W. Li, J. Llibre, Z. Zhang, Polynomial systems: A lower bound for the weakened 16th Hilbert problem, Extracta Math. 16 (3) (2001) 441–447] and Llibre and Zhang [J. Llibre, X. Zhang, On the number of limit cycles for some perturbed Hamiltonian polynomial systems, Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Anal. 8 (2) (2001) 161–181].     

On the 16th Hilbert problem for limit cycles on nonsingular algebraic curves

We give an upper bound for the maximum number N of algebraic limit cycles that a planar polynomial vector field of degree n can exhibit if the vector field has exactly k nonsingular irreducible invariant algebraic curves. Additionally we provide sufficient conditions in order that all the algebraic limit cycles are hyperbolic. We also provide lower bounds for N.     

Configurations of limit cycles in Liénard equations

We show that every finite configuration of disjoint simple closed curves in the plane is topologically realizable as the set of limit cycles of a polynomial Liénard equation. The related vector field X is Morse–Smale. Moreover it has the minimum number of singularities required for realizing the configuration in a Liénard equation. We provide an explicit upper bound on the degree of X, which is lower than the results obtained before, obtained in the context of general polynomial vector fields.     

Note on a paper of J. Llibre and G. Rodríguez concerning algebraic limit cycles

In a recent paper of Llibre and Rodríguez (J. Differential Equations 198 (2004) 374-380) it is proved that every configuration of cycles in the plane is realizable (up to homeomorphism) by a polynomial vector field of degree at most 2(n+r)-1, where n is the number of cycles and r the number of primary cycles (a cycle C is primary if there are no other cycles contained in the bounded region limited by C). In this letter we prove the same theorem by using an easier construction but with a greater polynomial bound (the vector field we construct has degree at most 4n-1). By using the same technique we also construct R³ polynomial vector fields realizing (up to homeomorphism) any configuration of limit cycles which can be linked and knotted in R³. This answers a question of R. Sverdlove.     

Limit cycles near hyperbolas in quadratic systems

In this paper we introduce the notion of infinity strip and strip of hyperbolas as organizing centers of limit cycles in polynomial differential systems on the plane. We study a strip of hyperbolas occurring in some quadratic systems. We deal with the cyclicity of the degenerate graphics DI2a from the programme, set up in [F. Dumortier, R. Roussarie, C. Rousseau, Hilbert's 16th problem for quadratic vector fields, J. Differential Equations 110 (1994) 86-133], to solve the finiteness part of Hilbert's 16th problem for quadratic systems. Techniques from geometric singular perturbation theory are combined with the use of the Bautin ideal. We also rely on the theory of Darboux integrability.     

A Transfer Principle in the Real Plane from Nonsingular Algebraic Curves to Polynomial Vector Fields

For every nonsingular algebraic curve C of degree m in the real plane a polynomial vector field of degree 2m–1 is constructed, which has exactly the ovals of C as attracting limit cycles. Therefore, every progress on the algebraic part of Hilbert's 16th problem automatically yields progress on its dynamical part.     

Limit cycles and Lie symmetries

Given a planar vector field U which generates the Lie symmetry of some other vector field X, we prove a new criterion to control the stability of the periodic orbits of U. The problem is linked to a classical problem proposed by A.T. Winfree in the seventies about the existence of isochrons of limit cycles (the question suggested by the study of biological clocks), already answered by Guckenheimer using a different terminology. We apply our criterion to give upper bounds of the number of limit cycles for some families of vector fields as well as to provide a class of vector fields with a prescribed number of hyperbolic limit cycles. Finally we show how this procedure solves the problem of the hyperbolicity of periodic orbits in problems where other criteria, like the classical one of the divergence, fail.     

Configurations of limit cycles and planar polynomial vector fields

We show that every finite configuration of disjoint simple closed curves of the plane is topologically realizable as the set of limit cycles of a polynomial vector field. Moreover, the realization can be made by algebraic limit cycles, and we provide an explicit polynomial vector field exhibiting any given finite configuration of limit cycles.     

Multiplicity of limit cycles and analytic m-solutions for planar differential systems

This work deals with limit cycles of real planar analytic vector fields. It is well known that given any limit cycle Γ of an analytic vector field it always exists a real analytic function f₀(x,y), defined in a neighborhood of Γ, and such that Γ is contained in its zero level set. In this work we introduce the notion of f₀(x,y) being an m-solution, which is a merely analytic concept. Our main result is that a limit cycle Γ is of multiplicity m if and only if f₀(x,y) is an m-solution of the vector field. We apply it to study in some examples the stability and the bifurcation of periodic orbits from some non-hyperbolic limit cycles.     

The hyperelliptic limit cycles of the Liénard systems

For Liénard systems , with fm and gn real polynomials of degree m and n respectively, in [H. Zoladek, Algebraic invariant curves for the Liénard equation, Trans. Amer. Math. Soc. 350 (1998) 1681-1701] the author showed that if m?3 and m+1<n<2m there always exist Liénard systems which have a hyperelliptic limit cycle. Llibre and Zhang [J. Llibre, Xiang Zhang, On the algebraic limit cycles of Liénard systems, Nonlinearity 21 (2008) 2011-2022] proved that the Liénard systems with m=3 and n=5 have no hyperelliptic limit cycles and that there exist Liénard systems with m=4 and 5<n<8 which do have hyperelliptic limit cycles. So, it is still an open problem to characterize the Liénard systems which have an algebraic limit cycle in cases m>4 and m+1<n<2m. In this paper we will prove that there exist Liénard systems with m=5 and m+1<n<2m which have hyperelliptic limit cycles.     

On the period of the limit cycles appearing in one-parameter bifurcations

The generic isolated bifurcations for one-parameter families of smooth planar vector fields {X_μ} which give rise to periodic orbits are: the Andronov-Hopf bifurcation, the bifurcation from a semi-stable periodic orbit, the saddle-node loop bifurcation and the saddle loop bifurcation. In this paper we obtain the dominant term of the asymptotic behaviour of the period of the limit cycles appearing in each of these bifurcations in terms of μ when we are near the bifurcation. The method used to study the first two bifurcations is also used to solve the same problem in another two situations: a generalization of the Andronov-Hopf bifurcation to vector fields starting with a special monodromic jet; and the Hopf bifurcation at infinity for families of polynomial vector fields.     

Hilbert's 16th problem for classical Liénard equations of even degree

Classical Liénard equations are two-dimensional vector fields, on the phase plane or on the Liénard plane, related to scalar differential equations . In this paper, we consider f to be a polynomial of degree 2l−1, with l a fixed but arbitrary natural number. The related Liénard equation is of degree 2l. We prove that the number of limit cycles of such an equation is uniformly bounded, if we restrict f to some compact set of polynomials of degree exactly 2l−1. The main problem consists in studying the large amplitude limit cycles, of which we show that there are at most l.     

Upper bounds for the number of orbital topological types of planar polynomial vector fields modulo limit cycles

The purpose of this paper is to find an upper bound for the number of orbital topological types of nth-degree polynomial fields in the plane. An obstacle to obtaining such a bound is related to the unsolved second part of the Hilbert 16th problem. This obstacle is avoided by introducing the notion of equivalence modulo limit cycles. Earlier, the author obtained a lower bound of the form $2^{cn^2 } $ . In the present paper, an upper bound of the same form but with a different constant is found. Moreover, for each planar polynomial vector field with finitely many singular points, a marked planar graph is constructed that represents a complete orbital topological invariant of this field.     

Weak infinitesimal Hilbert’s 16th problem

The following weak infinitestimal Hilbert’s 16th problem is solved. Given a real polynomial H in two variables, denote by M(H, m) the maximal number possessing the following property: for any generic set {γ _i } of at most M(H,m) compact connected components of the level lines H = c _i of the polynomial H, there exists a form θ = P dx + Q dy with polynomials P and Q of degrees no greater than m such that the integral ∫ _H=c θ has nonmultiple zeros on the connected components {γ _i }. An upper bound for the number M(H,m) in terms of the degree n of the polynomial H is found; this estimate is sharp for almost every polynomial H of degree n. A multidimensional version of this result is proved. The relation between the weak infinitesimal Hilbert’s 16th problem and the following question is discussed: How many limit cycles can a polynomial vector field of degree n have if it is close to a Hamiltonian vector field?     

An explicit expression for the first return map in the center problem

The classical center-focus problem posed by H. Poincaré in 1880's asks about the characterization of planar polynomial vector fields such that all their integral trajectories are closed curves whose interiors contain a fixed point, a center. In this paper, we present a method allowing for the first time to obtain an explicit expression for the first return map in the center problem.     

Polynomial differential systems having a given Darbouxian first integral

The Darbouxian theory of integrability allows to determine when a polynomial differential system in has a first integral of the kind f₁^λ₁?f_p^λ_pexp(g/h) where f_i, g and h are polynomials in , and for i=1,…,p. The functions of this form are called Darbouxian functions. Here, we solve the inverse problem, i.e. we characterize the polynomial vector fields in having a given Darbouxian function as a first integral.On the other hand, using information about the degree of the invariant algebraic curves of a polynomial vector field, we improve the conditions for the existence of an integrating factor in the Darbouxian theory of integrability.