Bifurcation
Theory:
Starting from the time I was a Ph-D student, my interest is with the
dynamics that arise after perurbation of degenerate dynamical systems.
In particular, I have studied the case where a given vector field
possesses a degenerate homoclinic
orbit, See Ref [1,2,3,4,5,10].
We show the existence of chaos,
strange attractors [1, 3] and
the existence of infinitely many homoclinic
doubling cascades [2,4].

Techniques developed in [1,3] and those in [7,8] (see
section ('normal for theory' below) are used to investigate strange
attractor with a large entropy [10]. This
later results reveals the only knowledge of low order terms in
asymptotics are not always sufficient to describe/anticipate the
dynamics.

Mathematics
in Biology:
Classical normal form theory is used to investigated the dynamics of
Predator-Prey systems with their bifurcation pattern. A 2-dimensional
predator-prey model with five parameters is
investigated, adapted from the Volterra-Lotka system by a
non-monotonic response function. A description of the various
domains of structural stability and their bifurcations is given. The
bifurcation structure is reduced to four organising centres of
codimension 3. Research is initiated on time-periodic perturbations
by several examples of strange attractors. See [12,13].
In order to classify all possible Morse-Smale portrait, we proceed to a
surgery

applied to limit-cycle and elaborate a classification of the
Morse-Smale portraits 'Modulo' the limit cycles. This is of great help
to anticipate the dynamics and their bifurcation.

Normal Form
Theory
: Motivated by the research done in
homoclinic bifurcations, we are interested in the Dulac map of germ of a vector field.
A result is proposed in [6] int he
case of a saddle in the three-dimensional space. Other result concern a
direct (asymptotic) computation of a linearisation,
conjugating the germ

with its linear part. First results are obtained in the case of a
single vector field [7,8] and
later in terms of family [17].
These results ohold in any dimension. A non mooth normal form theory is
develop (which generalise that of Poincare-Dulac)
that eliminate resonant term by resonant term. This non smooth
normal form carries Dulac or compensator expansion. A Dulac expansion
is formed by monomial terms that may contain a
specific logarithmic factor. In compensator expansions this logarithmic
factor is deformed. In [18] such
expansions are studied in the frame of quasi periodic bifurcation. We
study analytic properties of compensator and Dulac expansions in a
single variable. We first consider Dulac expansions when the
power of the logarithm is either 0 or 1. Here we construct
an explicit exponential scaling in the space of coefficients,
which in an exponentially narrow horn, up to rescaling and division,
leads to a polynomial expansion.
A similar result holds for the compensator case.