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Antisymmetric Diffeomorphisms and Bifurcations of a Double Conservative Hénon Map

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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 \(c=0\)).

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Notes

  1. Note that the map (1.3) was derived in [4] in the case of two-dimensional diffeomorphisms with a quadratic homoclinic tangency of invariant manifolds of a saddle fixed point of conservative type, i. e., when it has real multipliers \(\lambda\) and \(\lambda^{-1}\), where \(|\lambda|\neq 1\).

  2. Note that a neighborhood of a generic elliptic periodic point of a reversible map has a different structure than in the conservative case, since, in addition to a continuum of KAM-curves [25], it also contains infinitely many periodic sinks and sources [26], which is impossible in the conservative case.

  3. For a generic symmetric fixed point with multipliers \(\lambda_{1,2}=1\), the linearization matrix consists of one Jordan cell, i. e., it has a single eigenvector corresponding to \(\lambda=1\). Moreover, for a parabolic bifurcation, the eigenvector is tangent to the line of fixed points of involution, and for a pitchfork bifurcation, the vector is orthogonal to this line. In our case, the matrix \(DH\) is diagonal and, hence, all vectors are its eigenvectors.

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Funding

This work was carried in the framework of the grant 19-11-00280 of the Russian Science Foundation. The work was also partially supported by the grant 0729-2020-0036 of the Ministry of Science and Higher Education of the Russian Federation (Section 3.2) and by the grant 19-71-10048 of the Russian Science Foundation (Section 3.3). S. Gonchenko and K. Safonov also thank the Foundation for the Development of Theoretical Physics and Mathematics “BASIS” for supporting scientific research.

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Authors and Affiliations

  1. Mathematical Center “Mathematics of Future Technologies”, Lobachevsky State University of Nizhny Novgorod, pr. Gagarin 23, 603022, Nizhny Novgorod, Russia

    Sergey V. Gonchenko & Nikita G. Zelentsov

  2. Laboratory of Dynamical Systems and Applications, National Research University Higher School of Economics, ul. Bolshaya Pecherskaya 25/12, 603155, Nizhny Novgorod, Russia

    Sergey V. Gonchenko & Klim A. Safonov

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  1. Sergey V. Gonchenko
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  2. Klim A. Safonov
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  3. Nikita G. Zelentsov
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Correspondence to Sergey V. Gonchenko, Klim A. Safonov or Nikita G. Zelentsov.

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MSC2010

37G10,37G25

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Gonchenko, S.V., Safonov, K.A. & Zelentsov, N.G. Antisymmetric Diffeomorphisms and Bifurcations of a Double Conservative Hénon Map. Regul. Chaot. Dyn. 27, 647–667 (2022). https://doi.org/10.1134/S1560354722060041

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