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Regular and Chaotic Dynamics

, Volume 19, Issue 4, pp 435–460 | Cite as

Scientific heritage of L.P. Shilnikov

  • Valentin S. Afraimovich
  • Sergey V. Gonchenko
  • Lev M. Lerman
  • Andrey L. Shilnikov
  • Dmitry V. Turaev
Article

Abstract

This is the first part of a review of the scientific works of L.P. Shilnikov. We group his papers according to 7 major research topics: bifurcations of homoclinic loops; the loop of a saddle-focus and spiral chaos; Poincare homoclinics to periodic orbits and invariant tori, homoclinic in noautonous and infinite-dimensional systems; Homoclinic tangency; Saddlenode bifurcation — quasiperiodicity-to-chaos transition, blue-sky catastrophe; Lorenz attractor; Hamiltonian dynamics. The first two topics are covered in this part. The review will be continued in the further issues of the journal.

Keywords

Homoclinic chaos global bifurcations spiral chaos strange attractor saddle-focus homoclinic loop saddle-node saddle-saddle Lorenz attractor hyperbolic set 

MSC2010 numbers

37-01 37-02 01A65 37C29 37D45 

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Copyright information

© Pleiades Publishing, Ltd. 2014

Authors and Affiliations

  • Valentin S. Afraimovich
    • 1
  • Sergey V. Gonchenko
    • 2
  • Lev M. Lerman
    • 2
  • Andrey L. Shilnikov
    • 2
    • 3
  • Dmitry V. Turaev
    • 4
  1. 1.Universidad Autónoma de San Luis PotosíSan Luis PotosíMéxico
  2. 2.Lobachevsky State University of Nizhny NovgorodNizhny NovgorodRussia
  3. 3.Neuroscience Institute and Department of Mathematics and StatisticsGeorgia State UniversityAtlantaUSA
  4. 4.Imperial CollegeLondonUK

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