Dynamical Systems:
Hamiltonian
Dynamics
Separatrix
splitting
Bifurcation
Theory
Normal
form theory
Mathematics
in Biology
Hamiltonian
Dynamics :
A rigid body is an object (represented by its gravity center)
in which the distances between all its component
particles remain fixed under the application of a force or torque and
therefore conserves its shape during its motion. This surely
is an idealisation as the mere definition already contradicts the
principles of special relativity. Typical
motion in integrable Hamiltonian systems is quasi-periodic (if motions
are bounded, which is always the case for rigid body dynamics).
Geometrically speaking, this implies that the motion takes place on
invariant tori in phase space. In the dynamics of the rigid body the
toral angles have very intuitive meanings of rotation, precession and
nutation. For very low energies there may be pendulum-like motions as
well. In [14], in my list of
publications, we study the case of the Lagrange top.
In the selected preprint section [6], the techiniques in [18] applied to the bifurcation theory of limit cycles in planar vector fields. The setting consists of families that unfold a given Hamiltonian in a dissipative way. This leading part is of Morse type, which leads to the following three cases.The first concerns a Hamiltonian function that is regular on an annulus, and the second a Hamiltonian function with a nondegenerate minimum defined on a disc. In the third case the Hamiltonian function has a nondegenerate saddle-point with a saddle-connection. The first case, by an appropriate scalings, recovers the generic theory of the saddle-node of limit cycles and its cuspoid degeneracies, while the second case similarly recovers the generic theory of limit cycles subordinate to the codimension k-Hopf bifurcations k = 1,2,... The third case case enables a novel study of generic bifurations of limit cycles subordinate to homoclinic bifurcations. We now describe how the above analytic result is applied to bifurcations of limit cycles. For appropriate 1-dimensional Poincare maps, the fixed points correspond to the limit cycles. The fixed point sets (or zero-sets of the associated displacement functions) are studied by contact equivalence singularity theory. The cases where the Hamiltonian is defined on an annulus or a disc, directly can be reduced to catastrophe theory. In the third case, the displacement functions are known to have compensator expansions, whose first approximations are Dulac expansions. Application of our analytic result implies that in an exponentially narrow horn near a homoclinic loop, the bifucation theory of limit cycles again reduces to catastrophe theory.
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.