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  • 1
    Electronic Resource
    Electronic Resource
    Some more universal scaling laws for critical mappings (1981)
    Springer
    Journal of statistical physics 26 (1981), S. 697-717 
    Details
    ISSN: 1572-9613
    Keywords: One-dimensional maps ; onset of turbulence ; Feigenbaum scaling laws ; critical phenomena ; universality ; sensitivity to initial conditions ; approach towards fractal attractors
    Source: Springer Online Journal Archives 1860-2000
    Topics: Physics
    Notes: Abstract We study such nonlinear mappingsx n +1=F(x n ;b cr) of an intervalI into itself for which the Feigenbaum scaling laws hold (i.e., for which bcr is an accumulation point of bifurcation points). Letx 0 be a random variable with some absolutely continuous distribution inI. We show in particular that (i) the geometric average distance ofx n from the nearest point of the attractor decreases liken −1.93387; (ii) the geometric average of ¦∂x n /∂x 0¦ increases liken 0.60; (iii) the geometric mean distance ¦x n −y n ¦ between the iterates of two close-by pointsx 0,y 0 asymptotically tends towards a value ∼¦x 0−y 0¦0.77. These-and other-properties are also borne out from a simple probabilistic model which depicts the evolution as a random walklike process.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01010934
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    Paper (German National Licenses)
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  • 2
    Electronic Resource
    Electronic Resource
    Scaling laws for invariant measures on hyperbolic and nonhyperbolic atractors (1988)
    Springer
    Journal of statistical physics 51 (1988), S. 135-178 
    Details
    ISSN: 1572-9613
    Keywords: Dynamical systems ; generalized dimensions and entropies ; Liapunov exponents ; scaling functions ; hyperbolicity ; phase transitions
    Source: Springer Online Journal Archives 1860-2000
    Topics: Physics
    Notes: Abstract The analysis of dynamical systems in terms of spectra of singularities is extended to higher dimensions and to nonhyperbolic systems. Prominent roles in our approach are played by the generalized partial dimensions of the invariant measure and by the distribution of effective Liapunov exponents. For hyperbolic attractors, the latter determines the metric entropies and provides one constraint on the partial dimensions. For nonhyperbolic attractors, there are important modifications. We discuss them for the examples of the logistic and Hénon map. We show, in particular, that the generalized dimensions have singularities with noncontinuous derivative, similar to first-order phase transitions in statistical mechanics.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01015324
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    Paper (German National Licenses)
    Fulltext
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