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This article is cited in 1 scientific paper (total in 1 paper)
Alexey Borisov Memorial Volume
Antisymmetric Diffeomorphisms and Bifurcations of a Double
Conservative Hénon Map
Sergey V. Gonchenkoab, Klim A. Safonovb, Nikita G. Zelentsova a Mathematical Center “Mathematics of Future Technologies”,
Lobachevsky State University of Nizhny Novgorod,
pr. Gagarin 23, 603022 Nizhny Novgorod, Russia
b Laboratory of Dynamical Systems and Applications,
National Research University Higher School of Economics,
ul. Bolshaya Pecherskaya 25/12, 603155 Nizhny Novgorod, Russia
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 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 $c=0$).
Keywords:
reversible diffeomorphism, parabolic bifurcation, period-doubling bifurcation.
Received: 21.09.2022 Accepted: 24.10.2022
Citation:
Sergey V. Gonchenko, Klim A. Safonov, Nikita G. Zelentsov, “Antisymmetric Diffeomorphisms and Bifurcations of a Double
Conservative Hénon Map”, Regul. Chaotic Dyn., 27:6 (2022), 647–667
Linking options:
https://www.mathnet.ru/eng/rcd1185 https://www.mathnet.ru/eng/rcd/v27/i6/p647
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