Bifurcation of pre-images of differences of exponentially small functions [ Back ]

15:30 to 16:30
UAB - Dept. Matemàtiques (C1/-128)
Magdalena Caubergh


In recent articles of De Maesschalck, Dumortier, Panazzolo, Roussarie,..., bifurcation problems of limit cycles from slow-fast systems are reduced to the bifurcation problem of pre-images of differences of exponentially small functions. By the example in [DPR2007] it is evident that for such (generic) equations classical tools as the Implicit Function Theorem or Mather's Preparation Theorem from Catastrophe Theory do not apply.

From this point of view interest goes to the maximum number of pre-images of these types of difference functions and to the corresponding bifurcation diagram in function of the parameter.

In this talk, first I describe the general setting corresponding to the presence of one or more generic breaking mechanisms that allow the bifurcation of relaxation oscillations. Next I recall some known and new results corresponding to the case with only one breaking parameter. In particular the simplest generic case corresponding to only one breaking mechanism, in which a saddle-node bifurcation of pre-images occurs, will be presented in detail. Finally some results corresponding to the case with two or more breaking mechanisms are discussed.

This talk reports on a recent collaboration with R. Roussarie (Université de Bourgogne, France).

[DPR2007] F. Dumortier, D. Panazzolo, R. Roussarie, More limit cycles than expected in Liénard equations, Proceedings of the American Mathematical Society, 135, 1895--1904, 2007.