Klim Safonov
Higher School of Economics
Publications:
Gonchenko S. V., Safonov K. A., Zelentsov N. G.
Antisymmetric Diffeomorphisms and Bifurcations of a Double Conservative Hénon Map
2022, vol. 27, no. 6, pp. 647667
Abstract
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 twodimensional 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 period2 points, among which there are both standard bifurcations (parabolic, perioddoubling and pitchfork) and singular ones (during transition through $c=0$).
