38 citations · 123 across the 18 of their papers we have counts for
8 papers · 1 filter
Perturbation theory of the quadratic Lotka-Volterra double center
Jean-Pierre Françoise, Lubomir Gavrilov
We revisit the bifurcation theory of the Lotka-Volterra quadratic system \begin{eqnarray} X_0 :\left\{\begin{aligned} \dot{x}=& - y -x^2+y^2 ,\\ \dot{y}= &\;\;\;\;x - 2xy \end{alig…
Hilbert's 16th problem on a period annulus and Nash space of arcs
Jean-Pierre Françoise, Lubomir Gavrilov, Dongmei Xiao
This article introduces an algebro-geometric setting for the space of bifurcation functions involved in the local Hilbert's 16th problem on a period annulus. Each possible bifurcat…
Canard cycles in global dynamics
Alexandre Vidal, Jean-Pierre Françoise
Fast-slow systems are studied usually by "geometrical dissection". The fast dynamics exhibit attractors which may bifurcate under the influence of the slow dynamics which is seen a…
Analytical Study of a Triple Hopf Bifurcation in a Tritrophic Food Chain Model
J. -P. Francoise, J. Llibre
We provide an analytical proof of the existence of a stable periodic orbit contained in the region of coexistence of the three species of a tritrophic chain. The method used consis…
Enhanced delay to Bifurcation
J. -P. Francoise, C. Piquet, A. Vidal
This article provides an example of fast-slow system such that most orbits remain as close as possible to the unstable manifold of the fast dynamics for an arbitrarily long time.
Periodic motions galore: How to modify nonlinear evolution equations so that they feature a lot of periodic solutions
F. Calogero, J-P Francoise
A simple trick is illustrated, whereby nonlinear evolution equations can be modified so that they feature a lot - or, in some cases, only -- periodic solutions. Several examples (O…
Analytic properties of the return mapping of Lienard Equations
Jean-Pierre Francoise
An effective version of Bautin's theory with explicit estimates on the size of the neighborhood on which the number of limit cycles is controlled is given here for the case of Lien…
On the bifurcation of periodic orbits
Jean-Pierre Francoise
This article is a survey on recent contributions to an effective version of Bautin's theory about the bifurcation of periodic orbits (limit cycles). The analysis of Hopf bifurcatio…