0000000000149615
AUTHOR
Michèle Pelletier
A generalization of Françoise's algorithm for calculating higher order Melnikov functions
Abstract In [J. Differential Equations 146 (2) (1998) 320–335], Francoise gives an algorithm for calculating the first nonvanishing Melnikov function Ml of a small polynomial perturbation of a Hamiltonian vector field and shows that Ml is given by an Abelian integral. This is done under the condition that vanishing of an Abelian integral of any polynomial form ω on the family of cycles implies that the form is algebraically relatively exact. We study here a simple example where Francoise's condition is not verified. We generalize Francoise's algorithm to this case and we show that Ml belongs to the C [ log t,t,1/t] module above the Abelian integrals. We also establish the linear differentia…
Synthèse temps-minimale au voisinage d'une cible de codimension un: cas exceptionnel plat en dimension trois
Resume Les trajectoires satisfaisant au systeme un x = X + uY et atteignant une cible N de codimension un en temps minimal peuvent arriver tangentiellement a N : c'est le cas exceptionnel. Nous prouvons qu'alors la synthese locale ne peut pas etre consideree comme une famille de syntheses planes et que de plus le systeme peut ne pas etre localement controlable. Nous illustrons ces proprietes sur un exemple de reacteurs chimiques.
Hamiltonian monodromy from a Gauss-Manin monodromy
International audience
Singular systems in dimension 3: Cuspidal case and tangent elliptic flat case
We study two singular systems in R3. The first one is affine in control and we achieve weighted blowings-up to prove that singular trajectories exist and that they are not locally time optimal. The second one is linear in control. The characteristic vector field in sub-Riemannian geometry is generically singular at isolated points in dimension 3. We define a case with symmetries, which we call flat, and we parametrize the sub-Riemannian sphere. This sphere is subanalytic.
A note on a generalization of Françoise's algorithm for calculating higher order Melnikov functions
In [J. Differential Equations 146 (2) (1998) 320–335], Françoise gives an algorithm for calculating the first nonvanishing Melnikov function M of a small polynomial perturbation of a Hamiltonian vector field and shows that M is given by an Abelian integral. This is done under the condition that vanishing of an Abelian integral of any polynomial form ω on the family of cycles implies that the form is algebraically relatively exact. We study here a simple example where Françoise’s condition is not verified. We generalize Françoise’s algorithm to this case and we show that M belongs to the C[log t, t, 1/t] module above the Abelian integrals. We also establish the linear differential system ver…