Global Attractors and Basic Turbulence

  • Björn Birnir
Part of the NATO ASI Series book series (NSSB, volume 329)


We consider a dynamical system defined by a nonlinear partial differential equation (PDE) on a Banach space X. If for reasonably smooth initial data there exist unique solutions exhibiting complex nonlocal irregular behaviour, we say that the system is turbulent. This turbulence will be called basic turbulence if it can be described by a low-dimensional dynamical system of ordinary differential equations (ODE’s). It was proposed by Ruelle and Takens1, in 1971, that chaotic dynamics on a strange attractor was the cause of basic turbulence, but two fundamental problems have hampered progress for 20 years. The first one is that the estimates on the dimension of the ODE system that have been obtained for turbulent systems, have typically indicated that this system is too large to be useful. A system of a hundred or more ODE’s is no more useful that the discretization of the original PDE and its numerical orbits. The second problem is even more fundamental; the center manifold used by Ruelle and Takens1 is local, in phase space, whereas turbulence is necessarily a global phenomenon.


Unstable Manifold Global Attractor Strange Attractor Center Manifold Separable Banach Space 
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Copyright information

© Springer Science+Business Media New York 1994

Authors and Affiliations

  • Björn Birnir
    • 1
    • 2
  1. 1.Department of MathematicsUniv. of CaliforniaSanta BarbaraUSA
  2. 2.Univ. of Iceland Science InstituteReykjavíkIceland

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