Reduction to finite dimensions of continuous systems having only a few amplified modes

  • G. Zimmermann
III. Instability and Turbulence
Part of the Lecture Notes in Physics book series (LNP, volume 235)


In the approach of Guckenheimer and Knobloch the amplitudes of trajectories on the unstable manifold 0 are the pivotal quantities. This places a certain restriction on the applicability of this approach, as only neighbourhoods of 0 of the unstable manifold of 0 are accessible, which have a one-to-one projection into their tangent at 0, the linear space spanned by the amplified modes. This restriction may be lifted, using the arc lengths of trajectories instead.


Trivial Solution Unstable Manifold Continuous System Finite Dimension Degenerate Form 


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  1. [1]
    Guckenheimer, J., and Knobloch, E.: Nonlinear Convection in a Rotating Layer: Amplitude Expansions and Normal Forms, Geophys. Astrophys. Fluid Dynamics 23 (1983), 247–272Google Scholar
  2. [2]
    Marsden, J., and McCracken, M.: The Hopf Bifurcation and its Application, Springer, 1976Google Scholar
  3. [3]
    Knobloch, E., and Proctor, M.R.E.: Non-linear Periodic Convection in Double-Diffusive Systems, J. Fluid Mech. 108 (1981), 291–316.Google Scholar

Copyright information

© Springer-Verlag 1985

Authors and Affiliations

  • G. Zimmermann
    • 1
  1. 1.Institut für MeteorologieUniversität MainzWest Germany

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