Antisymmetric Diffeomorphisms and Bifurcations of a Double Conservative Henon Map

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 o T-2, where T-2 = h o T (-1)(1) o h. We also discuss how this method can be used to construct normal forms of Poincar ' e 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 H ' enon map H of the form <overline>x = M + cx - y(2); y = M + c <overline>y - <overline>x(2). We construct this map by the proposed method for the case when T1 is the standard H ' enon map and the involution h is h : (x, y) -> (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).