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On Topological and Hyperbolic Properties of Systems with Homoclinic Tangencies

  • Sergey GonchenkoEmail author
  • Alexander Gonchenko
  • Ming-Chia Li
Chapter
Part of the Nonlinear Systems and Complexity book series (NSCH, volume 12)

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.

Keywords

Invariant Manifold Homoclinic Orbit Finite Order Symbolic Dynamic Homoclinic Point 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Notes

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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Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  • Sergey Gonchenko
    • 1
    Email author
  • Alexander Gonchenko
    • 2
  • Ming-Chia Li
    • 3
  1. 1.Research Institute of Applied Mathematics and CyberneticsNizhny Novgorod State UniversityNizhny NovgorodRussia
  2. 2.Department of Calculated Mathematics and CyberneticsNizhny Novgorod State UniversityNizhny NovgorodRussia
  3. 3.Department of Applied Mathematics & Center of Mathematical Modeling and Scientific ComputingNational Chiao Tung UniversityHsinchuTaiwan

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