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Mixed dynamics of 2-dimensional reversible maps with a symmetric couple of quadratic homoclinic tangencies

  • * Corresponding author: A. Delshams

    * Corresponding author: A. Delshams 

This work has been supported by the Russian Scientific Foundation grant: sections 1-4, 6 and 7 were carried out under the project 14-41-00044, and section 5 under the project 14-12-00811. AD, MG and JTL have been also partially supported by the MICIIN/FEDER grant MTM2015-65715-P and by the Catalan grant 2017SGR1049 (AD, JTL). MG has been partially supported by Juan de la Cierva-Formación Fellowship FJCI-2014-21229, the grant MTM2016-80117-P (MINECO/FEDER, UE) and the Knut and Alice Wallenberg Foundation grant 2013-0315. SG also thanks RFBR (grant 16-01-00364) and the Russian Ministry of Science and Education, project 1.3287.2017

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  • We study dynamics and bifurcations of 2-dimensional reversible maps having a symmetric saddle fixed point with an asymmetric pair of nontransversal homoclinic orbits (a symmetric nontransversal homoclinic figure-8). We consider one-parameter families of reversible maps unfolding the initial homoclinic tangency and prove the existence of infinitely many sequences (cascades) of bifurcations related to the birth of asymptotically stable, unstable and elliptic periodic orbits.

    Mathematics Subject Classification: 37-XX, 37G20, 37G40, 34C37.


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  • Figure 1.  Two different examples of planar reversible maps with symmetric nontransversal (quadratic tangency) heteroclinic cycles: (a) with a nontransversal symmetric heteroclinic orbit to a symmetric couple of saddle points, and (b) with a symmetric couple of nontransversal heteroclinic orbits to symmetric saddle points

    Figure 2.  Three examples of planar reversible maps with symmetric nontransversal homoclinic tangencies: (a) a symmetric quadratic homoclinic tangency; (b) a symmetric cubic homoclinic tangency; (c) a symmetric couple of nontransversal homoclinic (figure-8) orbits to the same symmetric saddle point

    Figure 3.  (a) An example of reversible map with a couple of symmetric homoclinic tangencies (homoclinic figure-8). (b) A neighbourhood of the contour $O\cup\Gamma_1\cup\Gamma_2$

    Figure 4.  (a) A reversible diffeomorphism with a symmetric transversal homoclinic orbit; (b) creation of a symmetric couple of nontransversal homoclinic orbits $\Gamma_1$ and $\Gamma_2$ (a "fish" configuration)

    Figure 8.  Domains of definitions and associated coordinates for the first return map ${T_{2m1k}} = T_2T_0^mT_1T_0^k$

    Figure 5.  A geometric structure of the homoclinic points $M_1^+$, $M_{1}^{-}$, $M_2^+$ and $M_2^-$ and their neighbourhoods in the figure-8 homoclinic configuration. Schematic actions of the first return maps: (a) $T_{1k} = T_1 T_0^k$, (b) $T_{2k} = T_2 T_0^k$ and (c) ${T_{2m1k}} = T_2 T_0^m T_1 T_0^k$

    Figure 6.  A geometric structure of the homoclinic points $M_1^+$, $M_{1}^{-}$, $M_2^+$ and $M_2^-$ and their neighbourhoods in the "fish" homoclinic configuration. Several schematic actions of the first return maps are represented: (a) $T_{1k} = T_1 T_0^k$, (b) $T_{2k} = T_2 T_0^k$ and (c) ${T_{2m1k}} = T_2 T_0^m T_1 T_0^k$

    Figure 7.  Domains of definition and range of the successor map from $\Pi_i^{+}$ into $\Pi_j^-$, $i, j = 1, 2$, under iterations of $T_{0}$ in the cases of (a) homoclinic figure-8; (b) homoclinic "fish"

    Figure 9.  Elements of the bifurcation diagram for the map $H$: painted regions correspond to the existence of symmetric elliptic and saddle fixed points of $H$

    Figure 10.  Two examples of creation of secondary homoclinic tangencies to the point $O$ together with their Smale horseshoes

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