Magdalena Caubergh

Generalized Lienard Equations, Cyclicity and Hopf-Takens Bifurcations

We investigate the bifurcation of small-amplitude limit cycles in generalized Liénard equations. We use the simplicity of the Liénard family, to illustrate the advantages of the approach based on Bautin ideals. Essentially, this Bautin ideal is generated by the so-called Lyapunov quantities, that are computed for generalized Liénard equations and used to detect the presence of a Hopf-Takens bifurcation. Furthermore, the cyclicity is computed exactly.

Global Classification of a class of Cubic Vector Fields whose canonical regions are period annuli

We study cubic vector fields with inverse radial symmetry, i.e. of the form ẋ = δx - y + ax2 + bxy + cy2 + σ(dx - y)(x2 + y2), ẏ = x + δy + ex2 + fxy + gy2 + σ(x + dy) (x2 + y2), having a center at the origin and at infinity; we shortly call them cubic irs-systems. These systems are known to be Hamiltonian or reversible. Here we provide an improvement of the algorithm that characterizes these systems and we give a new normal form. Our main result is the systematic classification of the global phase portraits of the cubic Hamiltonian irs-systems respecting time (i.e. σ = 1) up to topological and diffeomorphic equivalence. In particular, there are 22 (resp. 14) topologically different global phase portraits for the Hamiltonian (resp. reversible Hamiltonian) irs-systems on the Poincare disc. Finally we illustrate how to generalize our results to polynomial irs-systems of arbitrary degree. In particular, we study the bifurcation diagram of a 1-parameter subfamily of quintic Hamiltonian irs-systems. Moreover, we indicate how to construct a concrete reversible irs-system with a given configuration of singularities respecting their topological type and separatrix connections.

Algebraic curves of maximal cyclicity

The paper deals with analytic families of planar vector fields, studying methods to detect the cyclicity of a non-isolated closed orbit, i.e. the maximum number of limit cycles that can locally bifurcate from it. It is known that this multi-parameter problem can be reduced to a single-parameter one, in the sense that there exist analytic curves in parameter space along which the maximal cyclicity can be attained. In that case one speaks about a maximal cyclicity curve (mcc) in case only the number is considered and of a maximal multiplicity curve (mmc) in case the multiplicity is also taken into account. In view of obtaining efficient algorithms for detecting the cyclicity, we investigate whether such mcc or mmc can be algebraic or even linear depending on certain general properties of the families or of their associated Bautin ideal. In any case by well chosen examples we show that prudence is appropriate.

GLOBAL PHASE PORTRAITS OF SOME REVERSIBLE CUBIC CENTERS WITH COLLINEAR OR INFINITELY MANY SINGULARITIES

We study the reversible cubic vector fields of the form ẋ = -y + ax2 + bxy + cy2 - y(x2 + y2), ẏ = 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 equivalence.

Canard Cycles with Three Breaking Mechanisms

This article deals with relaxation oscillations from a generic balanced canard cycle (Gamma) subject to three breaking parameters of Hopf or jump type. We prove that in a rescaled layer of (Gamma) there bifurcate at most five relaxation oscillations.

Global phase portraits of some reversible cubic centers with noncollinear singularities

The results in this paper show that the cubic vector fields ẋ = -y + M(x, y) - y(x2 + y2), ẏ = 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 ≠ 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 five for γλ 0. Furthermore, the global bifurcation diagram is analyzed.