Journal of Dynamics and Differential Equations

, Volume 17, Issue 1, pp 175–215 | Cite as

Homoclinic Shadowing

  • Brian A. Coomes
  • Hüseyin KoçakEmail author
  • Kenneth J. Palmer


A new method for rigorously establishing the existence of a transversal homoclinic orbit to a periodic orbit (or a fixed point) of diffeomorphisms in R n is presented. It is a computer-assisted technique with two main components. First, a global Newton’s method is devised to compute a suitable pseudo (approximate) homoclinic orbit to a pseudo periodic orbit. Then, a homoclinic shadowing theorem, which is proved herein, is invoked to establish the existence of a true transversal homoclinic orbit to a true periodic orbit near these pseudo orbits.


Transversal homoclinic orbits pseudo orbits chaos shadowing Newton’s method 


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Copyright information

© Springer Science+Business Media, Inc. 2005

Authors and Affiliations

  • Brian A. Coomes
    • 1
  • Hüseyin Koçak
    • 1
    Email author
  • Kenneth J. Palmer
    • 2
  1. 1.Departments of Mathematics and Computer ScienceUniversity of MiamiCoral GablesUSA
  2. 2.Department of MathematicsNational Taiwan UniversityTaipeiTaiwan

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