Global Dynamics

  • John H. Lowenstein


In the preceding chapters, we have achieved a detailed understanding, based on the organizing principles of renormalization and symbolic coding, of the local dynamics associated with renormalizable kicked-oscillator maps. We now wish to show how the whole recursive return map formalism can be extended (“lifted”) to the aperiodic orbits on the infinite plane of the global map W. The local scaling, characterized by the self-similar contraction of domains in the fundamental cell Ω, will be seen to have, if certain conditions are satisfied, a global counterpart in the form of asymptotic self-similar expansion. Just as the local contraction (within a single ergodic component) is associated with a scale factor ωK < 1, the global expansivity will have its own scale factor ωW > 1. Given that the temporal behavior scales with a factor ωT, it is not surprising that orbits will be found to tend to infinity for asymptotically long times with a power-law exponent log ωW/logωT.


Global Dynamics Local Orbit Invariant Sector Partial Path Sector Code 
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  1. Kouptsov K.L., Lowenstein J.H., and Vivaldi F. (2002) Quadratic rational rotations of the torus and dual lattice maps, Nonlinearity 15, 1795–1842.MathSciNetADSzbMATHCrossRefGoogle Scholar
  2. Lowenstein J.H., Poggiaspalla G., and Vivaldi F. (2005) Sticky orbits in a kicked-oscillator model, Dynamical Systems 20, 413–451.MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Higher Education Press, Beijing and Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • John H. Lowenstein
    • 1
  1. 1.Department of PhysicsNew York UniversityNew YorkUSA

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