Robert Roussarie

On the number of limit cycles which appear by perturbation of separatrix loop of planar vector fields

Consider a fami ly of vector fCelds x~ on the plane. This fami ly depends on a parameter ~ ~ /R A, for some A ~ /~, and is supposed to be 0 ~ in (m,~) 6 /i~ 2X /~A. Suppose that for ~ = O, the vector f i e l d X o has a separatrix loop. This means that X o has an hyperbol ic saddle point s o and that one of the stable separatr ix of 8 o coincides with one of the unstable one. The union of th is curve and s o is the loop ?. A return map is defined on one side of r .

Bifurcations of planar vector fields and Hilbert's sixteenth problem

Preface.- 1 Families of Two-dimensional Vector Fields.- 2 Limit Periodic Sets.- 3 The 0-Parameter Case.- 4 Bifurcations of Regular Limit Periodic Sets.- 5 Bifurcations of Elementary Graphics.- 6 Desingularization Theory and Bifurcation of Non-elementary Limit Periodic Sets.- Bibliography.- Index.

Generic 3-parameter families of vector fields on the plane, unfolding a singularity with nilpotent linear part. The cusp case of codimension 3

A cusp type germ of vector fields is a C ∞ germ at 0∈ℝ 2 , whose 2-jet is C ∞ conjugate to We define a submanifold of codimension 5 in the space of germs consisting of germs of cusp type whose 4-jet is C 0 equivalent to Our main result can be stated as follows: any local 3-parameter family in (0, 0) ∈ ℝ 2 × ℝ 3 cutting transversally in (0, 0) is fibre- C 0 equivalent to

Complex ecological models with simple dynamics: From individuals to populations

The aim of this work is to study complex ecological models exhibiting simple dynamics. We consider large scale systems which can be decomposed into weakly coupled subsystems. Perturbation Theory is used in order to get a reduced set of differential equations governing slow time varying global variables. As examples, we study the influence of the individual behaviour of animals in competition and predator-prey models. The animals are assumed to do many activities all day long such as searching for food of different types. The degree of competition as well as the predation pressure are dependent upon these activities. Preys are more vulnerable when doing some activities during which they are very exposed to predators attacks rather than for others during which they are hidden. We study the effect of a change in the average individual behaviour of the animals on interspecific relationships. Computer simulations of the whole sets of equations are compared to simulations of the reduced sets of equations.

Bifurcations of cuspidal loops

A cuspidal loop for a planar vector field X consists of a homoclinic orbit through a singular point p, at which X has a nilpotent cusp. This is the simplest non-elementary singular cycle (or graphic) in the sense that its singularities are not elementary (i.e. hyperbolic or semihyperbolic). Cuspidal loops appear persistently in three-parameter families of planar vector fields. The bifurcation diagrams of unfoldings of cuspidal loops are studied here under mild genericity hypotheses: the singular point p is of Bogdanov - Takens type and the derivative of the first return map along the orbit is different from 1. An analytic and geometric method based on the blowing up for unfoldings is proposed here to justify the two essentially different models for generic bifurcation diagrams presented in this work. This method can be applied for the study of a large class of complex multiparametric bifurcation problems involving non-elementary singularities, of which the cuspidal loop is the simplest representative. The proofs are complete in a large part of parameter space and can be extended to the complete parameter space modulo a conjecture on the time function of certain quadratic planar vector fields. In one of the cases we can prove that the generic cuspidal loop bifurcates into four limit cycles that are close to it in the Hausdorff sense.

Elementary graphics of cyclicity 1 and 2

In this paper we elaborate the techniques to prove for several elementary graphics that their cyclicity is one or two. We first prove two main results for Cinfinity vector fields in general. The first one states that a graphic through an arbitrary number of attracting hyperbolic saddles (hyperbolicity ratio r>1) and attracting semi-hyperbolic points (one negative eigenvalue) has cyclicity 1. A second result says that for a graphic with one hyperbolic and one semi-hyperbolic singularity of opposite character the cyclicity is two. We then specialize to graphics with fixed connections and show that 33 graphics appearing among quadratic systems and listed in a previous paper have a cyclicity at most two (five cases are done only under generic conditions).

Geometric Singular Perturbation Theory Beyond Normal Hyperbolicity

Geometric Singular Perturbation theory has traditionally dealt only with perturbation problems near normally hyperbolic manifolds of singularities. In this paper we want to show how blow up techniques can permit enlarging the applicability to non-normally hyperbolic points. We will present the method on well chosen examples in the plane and in 3-space.

Germs of diffeomorphisms in the plane

Summary, some motivation and acknowledgments.- Introduction, definitions, formal study and statement of the results.- Stability of type I-and type II-singularities.- Stability of type III-singularities.- Proof of the C? results.- Proof of the topological results.

A method of desingularization for analytic two-dimensional vector field families

It is well known that isolated singularities of two dimensional analytic vector fields can be desingularized: after a finite number of blowing up operations we obtain a vector field that exhibits only elementary singularities. In the present paper we introduce a similar method to simplify the periodic limit sets of analytic families of vector fields. Although the method is applied here only to reduce to families in which the zero set has codimension at least two, we conjecture that it can be used in general. This is related to the famouss Hiberts problem about planar vector fields.

Melnikov functions and Bautin ideal

The computation of the number of limit cycles which appear in an analytic unfolding of planar vector fields is related to the decomposition of the displacement function of this unfolding in an ideal of functions in the parameter space, called the Ideal of Bautin. On the other hand, the asymptotic of the displacement function, for 1-parameter unfoldings of hamiltonian vector fields is given by Melnikov functions which are defined as the coefficients of Taylor expansion in the parameter. It is interesting to compare these two notions and to study if the general estimations of the number of limit cycles in terms of the Bautin ideal could be reduced to the computations of Melnikov functions for some 1-parameter subfamilies.