Alain Jacquemard

PIECEWISE SMOOTH REVERSIBLE DYNAMICAL SYSTEMS AT A TWO-FOLD SINGULARITY

This paper focuses on the existence of closed orbits around a two-fold singularity of 3D discontinuous systems of the Filippov type in the presence of symmetries.

On singularities of discontinuous vector fields

Abstract The subject of this paper concerns the classification of typical singularities of a class of discontinuous vector fields in 4D. The focus is on certain discontinuous systems having some symmetric properties.

Invariant varieties of discontinuous vector fields

We study the geometric qualitative behaviour of a class of discontinuous vector fields in four dimensions around typical singularities. We are mainly interested in giving the conditions under which there exist one-parameter families of periodic orbits (a result that can be seen as one analogous to the Lyapunov centre theorem). The focus is on certain discontinuous systems having some symmetric properties. We also present an algorithm which detects and computes periodic orbits.

Computer analysis of periodic orbits of discontinuous vector fields

Computer algebra provides powerful tools to perform a precise and qualitative analysis of the phase portraits of a class of discontinuous vector fields of R 4 . We derive from the analysis of semialgebraic varieties various results about the existence of families of periodic orbits. The existence of both symmetric and asymmetric periodic orbits for reversible discontinuous vector fields is proved.

Bifurcation of Singularities Near Reversible Systems

In this paper we study generic unfoldings of certain singularities in the class of all C ∞ reversible systems on R 2.

Optimal control of an ensemble of Bloch equations with applications in MRI

The optimal control of an ensemble of Bloch equations describing the evolution of an ensemble of spins is the mathematical model used in Nuclear Resonance Imaging and the associated costs lead to consider Mayer optimal control problems. The Maximum Principle allows to parameterize the optimal control and the dynamics is analyzed in the framework of geometric optimal control. This leads to numerical implementations or suboptimal controls using averaging principle.