Klim Safonov

This paper deals with the problem of smoothness of the stable invariant foliation for a homoclinic bifurcation with a neutral saddle in symmetric systems of differential equations. We give an improved sufficient condition for the existence of an invariant smooth foliation on a cross-section transversal to the stable manifold of the saddle. It is shown that the smoothness of the invariant foliation depends on the gap between the leading stable eigenvalue of the saddle and other stable eigenvalues. We also obtain an equation to describe the one-dimensional factor map, and we study the renormalization properties of this map. The improved information on the smoothness of the foliation and the factor map allows one to extend Shilnikov’s results on the birth of Lorenz attractors under the bifurcation considered.

We propose a new method for constructing multidimensional reversible maps by only two input data: a diffeomorphism $T_1$ and an involution $h$, i.e., a map (diffeomorphism) such that $h^2 = Id$. We construct the desired reversible map $T$ in the form $T = T_1\circ T_2$, where $T_2 = h\circ T_1^{-1}\circ h$. We also discuss how this method can be used to construct normal forms of Poincaré maps near mutually symmetric pairs of orbits of homoclinic or heteroclinic tangencies in reversible maps. One of such normal forms, as we show, is a two-dimensional double conservative Hénon map $H$ of the form $\bar x = M + cx - y^2; \ y = M + c\bar y - \bar x^2$. We construct this map by the proposed method for the case when $T_1$ is the standard Hénon map and the involution $h$ is $h: (x,y) \to (y,x)$. For the map $H$, we study bifurcations of fixed and period-2 points, among which there are both standard bifurcations (parabolic, period-doubling and pitchfork) and singular ones (during transition through $c=0$).