Bifurcation analysis of the Hénon map.
Bifurcation de cycles limites au voisinage de polycycles hyperboliques et génériques à trois sommets
Bifurcation of an invariant torus of a system of differential equations in the degenerate case.
Bifurcation of contracting singular cycles
Bifurcation of periodic orbits of time dependent Hamiltonian systems on symplectic manifolds.
Bifurcation set and limit cycles forming compound eyes in a perturbed Hamiltonian system.
In this paper we consider a class of perturbation of a Hamiltonian cubic system with 9 finite critical points. Using detection functions, we present explicit formulas for the global and local bifurcations of the flow. We exhibit various patterns of compound eyes of limit cycles. These results are concerned with the weakened Hilbert's 16th problem posed by V. I. Arnold in 1977.
Bifurcations de polycycles infinis pour les champs de vecteurs polynomiaux du plan
Bifurcations of limit cycles from cubic Hamiltonian systems with a center and a homoclinic saddle-loop.
It is proved in this paper that the maximum number of limit cycles of system⎧ dx/dt = y⎨⎩ dy/dt = kx - (k + 1)x2 + x3 + ε(α + βx + γx2)yis equal to two in the finite plane, where k > (11 + √33) / 4 , 0 < |ε| << 1, |α| + |β| + |γ| ≠ 0. This is partial answer to the seventh question in [2], posed by Arnold.
Branching of periodic orbits from Kukles isochrones.
Breaking Cycles on Surfaces.