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On Bifurcations of Homoclinic Tangencies in Area-Preserving Maps on Non-orientable Manifolds

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Difference Equations, Discrete Dynamical Systems and Applications (ICDEA 2012)

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Abstract

We study bifurcations of non-orientable area-preserving maps with quadratic homoclinic tangencies. We study the case when the maps are given on non-orientable two-dimensional manifolds. We consider one and two parameter general unfoldings and establish results related to the appearance of elliptic periodic orbits.

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Notes

  1. 1.

    The birth of 2-elliptic generic periodic orbits was proved in [17, 18] for the case of four-dimensional symplectic maps with homoclinic tangencies to saddle-focus fixed points.

  2. 2.

    In other words, any “homoclinic loop” (e.g. the curve \(\ell (O,M^+,O)\) composed from a connected pieces of manifolds \(W^u\) and \(W^s\) with border points O and \(M^+\)) is not contractible in \(\mathcal{M}\), see Fig. 1.

  3. 3.

    along with all derivatives by coordinates and parameters up to order either \((r-2)\) for \(n=1\) or \((r-2n-2)\) for \(n\ge 2\).

  4. 4.

    The rigorous proof requires quite delicate analytical considerations which are not presented here, see e.g. [20, 23, 24, 26].

  5. 5.

    see e.g. the papers [17, 19, 22–24, 26, 33, 34] in which the rescaling method was applied for the conservative case.

  6. 6.

    Of course, we lose, a little, in a smoothness, since the second order normal form is \(C^{r-2}\) only. However, we get more principally important information on form of the first return maps. On the other hand, our considerations cover also \(C^\infty \) and real analytical cases.

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Acknowledgments

The authors thank D.Turaev, L.Lerman and R.Ortega for fruitful discussions and remarks. This work has been partially supported by the Russian Scientific Foundation Grant 14-41-00044. S.G. has been supported partially by the grants of RFBR No.16-01-00324 and 14-01-00344. A.D. and M.G. have been partially supported by the Spanish MINECO-FEDER Grant MTM2012- 31714 and the Catalan Grant 2014SGR504.

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Authors and Affiliations

  1. Departament de Matemàtica Aplicada I, Universitat Politècnica de Catalunya, Barcelona, Spain

    Amadeu Delshams & Marina Gonchenko

  2. Institute of Applied Mathematics and Cybernetics, Nizhny Novgorod University, Nizhny Novgorod, Russia

    Sergey Gonchenko

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  1. Amadeu Delshams
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  2. Marina Gonchenko
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Correspondence to Marina Gonchenko .

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Editors and Affiliations

  1. Departament de Matemàtiques, Universitat Autònoma de Barcelona, Cerdanyola del Vallès, Barcelona, Spain

    Lluís Alsedà i Soler

  2. Department of Mathematics, University of Arizona, Tucson, Arizona, USA

    Jim M. Cushing

  3. Department of Mathematics, Trinity University, San Antonio, Texas, USA

    Saber Elaydi

  4. Faculty of Sciences, Department of Mathematics, University of Porto, Porto, Portugal

    Alberto A. Pinto

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Delshams, A., Gonchenko, M., Gonchenko, S. (2016). On Bifurcations of Homoclinic Tangencies in Area-Preserving Maps on Non-orientable Manifolds. In: Alsedà i Soler, L., Cushing, J., Elaydi, S., Pinto, A. (eds) Difference Equations, Discrete Dynamical Systems and Applications. ICDEA 2012. Springer Proceedings in Mathematics & Statistics, vol 180. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-52927-0_8

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