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On Topological and Hyperbolic Properties of Systems with Homoclinic Tangencies

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Part of the book series: Nonlinear Systems and Complexity ((NSCH,volume 12))

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Abstract

We study dynamical properties of a set Λ of trajectories from a small neighbourhood of a non-transversal Poincaré homoclinic orbit. We show that this problem has no univalent solution, as it takes place in the case of a transversal homoclinic orbit. Here different situations are possible, depending on the character of the homoclinic tangency, when Λ is trivial or contains topological (hyperbolic) horseshoes. In this chapter we find certain conditions for existence of both types of dynamics and give a description (in term of the symbolic dynamics) of the corresponding non-trivial hyperbolic subsets from Λ.

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Notes

  1. 1.

    Note that the case \(\sigma=1\) (i.e. O is a neutral saddle) is very specific and we do not consider it here. We only refer the reader to papers [9, 12, 21] in which various cases of neutral saddles (\(\sigma=1\)) with homoclinic tangencies were analysed; see also papers [10, 14, 15] in which area-preserving maps were considered.

  2. 2.

    Note that if \(\gamma<0\), then condition (2) can hold only when \(W^s_{loc}\) contains l u . Thus, in this case, any isolated one-sided tangency is, in fact, a tangency “from above”. On the other hand, if \(\gamma>0\), condition (2) allows a big variety of non-isolated tangencies.

  3. 3.

    For example in the case of tangency “from above”, the topological (geometrical) horseshoe of map T k (for every value of k from an infinite sequence of integers) contains an T k -invariant subset Δ k such that the system \({T_k}|\Delta_k\) is uniformly hyperbolic and topologically conjugate to a subshift of finite type with positive topological entropy.

  4. 4.

    We thank D. Turaev who attracted our attention to the interesting fact that the Katok theorem can be directly applied to the sectionally dissipative case. See also [35].

  5. 5.

    Therefore, in problems of such type, it is not reasonable to use a C 1-linearization (which, by the way, does not always exist in the multidimensional case). This can lead to non-repairable mistakes in the proofs or to absurd results, and, in the best case, only very rough topological properties can be established [32, 4].

  6. 6.

    We include also sequences with \(j_i=\infty\) or \(j_{i+1}=\infty\). Then such sequences contain infinite strings from zeros either on the left or, respectively, right end and correspond either α- or ω-asymptotic orbit to the fixed point \((\dots,0,\dots,0,\dots)\).

  7. 7.

    Thus, the diffeomorphisms with partial description in the main case \(\sigma\neq 1\) are such that conditions A–D are valid and the following combinations of signs of the parameters \(\lambda_1,\gamma, c\) and d are excluded: (1) those ones which correspond to the trivial class, i.e. n is even and (i) \(\gamma>0, d<0\) if \(\sigma<1\), (ii) \(\lambda_1>0, dc>0\) if \(\sigma>1\); and (2) those ones which correspond to the complete class, i.e (iii) \(\gamma>0,\lambda_1>0, c<0,d>0\) with even n and (iv) \(\gamma>0,\lambda_1>0, c<0\) with odd n.

  8. 8.

    This result was proved also in [6] for the sectionally dissipative case \(\sigma<1\).

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Acknowledgements

The authors thank M. Malkin and D. Turaev for very fruitful discussions. AG and SG have been partially supported by the Russian Scientific Foundation Grant 14-41-00044 and the grants of RFBR No.13-01-00589, 13-01-97028-povolzhye and 14-01-00344. ML was partially supported by a NSC research grant.

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Authors and Affiliations

  1. Research Institute of Applied Mathematics and Cybernetics, Nizhny Novgorod State University, Nizhny Novgorod, Russia

    Sergey Gonchenko

  2. Department of Calculated Mathematics and Cybernetics, Nizhny Novgorod State University, Nizhny Novgorod, Russia

    Alexander Gonchenko

  3. Department of Applied Mathematics & Center of Mathematical Modeling and Scientific Computing, National Chiao Tung University, 300, Hsinchu, Taiwan

    Ming-Chia Li

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  1. Sergey Gonchenko
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Correspondence to Sergey Gonchenko .

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  1. Autonomous University of San Luis Potosí, San Luis Potosi, Mexico

    Hernán González-Aguilar

  2. Autonomous University of San Luis Potosí, San Luis Potosi, Mexico

    Edgardo Ugalde

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Gonchenko, S., Gonchenko, A., Li, MC. (2015). On Topological and Hyperbolic Properties of Systems with Homoclinic Tangencies. In: González-Aguilar, H., Ugalde, E. (eds) Nonlinear Dynamics New Directions. Nonlinear Systems and Complexity, vol 12. Springer, Cham. https://doi.org/10.1007/978-3-319-09864-7_2

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