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Some Codimension-Two Bifurcations for Maps, Leading to Chaos

  • G. Iooss
Part of the Springer Series in Synergetics book series (SSSYN, volume 24)

Abstract

Consider a family of mappings F μ in a Banach space, depending on a real parameter μ, and assume that 0 is a fixed point of F°. The derivative at this point being noted T°, an elementary bifurcation occurs in each of these cases:
  1. i)

    1 is a simple eigenvalue of T°, the remaining part of its spectrum being of modulus less than 1. This is the “saddle-node” bifurcation where, while μ crosses 0, two fixed points (a saddle point and a node) meet together and disappear. By some extra nonlocal phenomenon this can produce “intermittency” of a simple kind [1].

     
  2. ii)

    −1 is a simple eigenvalue of T°, the remaining part of its spectrum being of modulus less than 1. This is the “flip” bifurcation where, while μ crosses 0, the fixed point is changed from a node to a saddle, and two periodic points appear of period 2 for μ on one side of 0, they are node (resp.saddle) if the fixed point is a saddle (resp.node). A succession of such bifurcations seems to be frequent in physics (for instance in hydrodynamics [2]).

     
  3. iii)

    λ° and \({\bar \lambda _ \circ }\) are simple eigenvalues of T° on the unit circle, \(\lambda _ \circ ^n\, \ne \,1\) for n=l,2,3,4, the remaining part of the spectrum being of modulus less than 1. This is the “Hopf bifurcation” for maps which lead, while μ. crosses 0, to the creation of an invariant circle under F μ , growing from 0, attracting (resp. repelling) if it appears on the side where the fixed point is repelling (resp. attracting). The dynamic on the invariant circle depends on μ and is related to the rotation number of the diffeomorphism of the circle (the restriction of F μ to the invariant circle) [3], [4].

     

Keywords

Hopf Bifurcation Phase Portrait Periodic Point Chaotic Behavior Homoclinic Orbit 
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.

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Copyright information

© Springer-Verlag Berlin Heidelberg 1984

Authors and Affiliations

  • G. Iooss
    • 1
  1. 1.Math. DépartementUniversité de NiceNiceFrance

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