Antisymmetric Diffeomorphisms and Bifurcations of a Double Conservative Hénon Map

We propose a new method for constructing multidimensional reversible maps by only two input data: a diffeomorphism and an involution , i. e., a map (diffeomorphism) such that . We construct the desired reversible map in the form , where . We also discuss how this method can be used to construct norm...

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Bibliographic Details
Published in:Regular & chaotic dynamics 2022-11, Vol.27 (6), p.647-667
Main Authors: Gonchenko, Sergey V., Safonov, Klim A., Zelentsov, Nikita G.
Format: Article
Language:English
Subjects:
Bifurcations
Canonical forms
Dynamical Systems and Ergodic Theory
Isomorphism
Mathematics
Mathematics and Statistics
Poincare maps
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Summary:We propose a new method for constructing multidimensional reversible maps by only two input data: a diffeomorphism and an involution , i. e., a map (diffeomorphism) such that . We construct the desired reversible map in the form , where . 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 of the form . We construct this map by the proposed method for the case when is the standard Hénon map and the involution is . For the map , 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 ).
ISSN:1560-3547
1560-3547
1468-4845
DOI:10.1134/S1560354722060041