Hilbert's sixteenth problem for polynomial Liénard equations

This article reports on the survey talk 'Hilbert's Sixteenth Problem for Liénard equations,' given by the author at the Oberwolfach Mini-Workshop 'Algebraic and Analytic Techniques for Polynomial Vector Fields.' It is written in a way that it is accessible to a public with heterogeneous mathematical background. The article reviews recent developments and techniques used in the study of Hilbert's 16th problem where the main focus is put on the subclass of polynomial vector fields derived from the Liérd equations

Canard cycles with three breaking mechanisms

Títol del volum: Mathematical Sciences with Multidisciplinary ApplicationsThis article deals with relaxation oscillations from a generic balanced canard cycle Γ subject to three breaking parameters of Hopf or jump type. We prove that in a rescaled layer of Γ there bifurcate at most five relaxation oscillations

Bifurcation of the separatrix skeleton in some 1-parameter families of planar vector fields

Agraïments: The author is supported by the Ramón y Cajal grant RYC-2011-07730This article deals with the bifurcation of polycycles and limit cycles within the 1-parameter families of planar vector fields X_m^k, defined by =y^3-x^2k 1,=-x my^4k 1, where m is a real parameter and k1 integer. The bifurcation diagram for the separatrix skeleton of X_m^k in function of m is determined and the one for the global phase portraits of (X^1_m)_mR is completed. Furthermore for arbitrary k1 some bifurcation and finiteness problems of periodic orbits are solved. Among others, the number of periodic orbits of X_m^k is found to be uniformly bounded independent of mR and the Hilbert number for (X_m^k)_mR, that thus is finite, is found to be at least one

Global phase portraits of some reversible cubic centers with noncollinear singularities

Agraïments: The first author also is supported by the Ramón y Cajal grant number RYC-2011-07730.The results in this paper show that the cubic vector fields ˙x = −y + M(x, y) − y(x2 + y2), y˙ = x + N(x, y) + x(x2 + y2), where M, N are quadratic homogeneous polynomials, having simultaneously a center at the origin and at infinity, have at least 61 and at most 68 topologically different phase portraits. To this end the reversible subfamily defined by M(x, y) = −γxy, N(x, y) = (γ − λ)x2 + α2λy2 with α, γ ∈ R and λ 6= 0, is studied in detail and it is shown to have at least 48 and at most 55 topologically different phase portraits. In particular, there are exactly 5 for γλ 0. Furthermore, the global bifurcation diagram is analyzed

Global phase portraits of some reversible cubic centers with collinear or infinitely many singularities

Agraïments: Furthermore the first author also is supported by the Juan de la Cierva grant number JCI-2007-49-764 and the second author also is partially supported by ICREA Academia.We study the reversible cubic vector fields of the form ˙x = −y + ax2 + bxy + cy2 − y(x2 + y2), y˙ = x + dx2 + exy + fy2 + x(x2 + y2), having simultaneously a center at infinity and at the origin. In this paper the subclass of these reversible systems having collinear or infinitely many singularities are classified with respect to topological and diffeomorphic equivalence

Cyclicity of unbounded semi-hyperbolic 2-saddle cycles in polynomial Lienard systems

The paper deals with the cyclicity of unbounded semi-hyperbolic 2-saddle cycles in polynomial Liénard systems of type (m, n) with m < 2n +1, m and n odd. We generalize the results in [1] (case m = 1), providing a substantially simpler and more transparant proof than the one used in [1]

Cyclicity of unbounded semi-hyperbolic 2-saddle cycles in polynomial Liénard systems

The paper deals with the cyclicity of unbounded semi-hyperbolic 2-saddle cycles in polynomial Lienard systems of type (m, n) with m < 2n+1, m and n odd. We generalize the results in [1] (case m = 1), providing a substantially simpler and more transparant proof than the one used in [1].status: publishe

Detecting alien limit cycles near a Hamiltonian 2-saddle cycle

This paper aims at providing an example of a cubic Hamiltonian 2-saddle cycle that after bifurcation can give rise to an alien limit cycle; this is a limit cycle that is not controlled by a zero of the related Abelian integral. To guarantee the existence of an alien limit cycle one can verify generic conditions on the Abelian integral and on the transition map associated to the connections of the 2-saddle cycle. In this paper, a general method is developed to compute the first and second derivative of the transition map along a connection between two saddles. Next, a concrete generic Hamiltonian 2-saddle cycle is analyzed using these formula's to verify the generic relation between the second order derivative of both transition maps, and a calculation of the Abelian integral.status: publishe