Sensitive dependence on parameters in nonlinear dynamics

Details Sensitive dependence on parameters in nonlinear dynamics Sensitive dependence on parameters in nonlinear dynamics

Jul 1985

adsabs.harvard.edu/cgi-bin/nph-data_query?bibcode=1985phrvl..55..351f&link_type=abstract

Physical Review Letters (ISSN 0031-9007), vol. 55, July 22, 1985, p. 351-354.

Mathematics

Dynamical Systems

78

Chaos, Dynamical Systems, Nonlinear Systems, Parameter Identification, Systems Stability, Branching (Mathematics), Fractals, Liapunov Functions, Numerical Stability, Robustness (Mathematics), Scaling Laws

Scientific paper

Two qualitatively different types of dynamical behavior can be so tightly interwoven that it becomes impossible to predict when a small change in parameters will cause a change in qualitative properties. For quadratic mappings of the interval, for example, the chaotic parameter values form a Cantor set of positive measure, broken up by periodic intervals. This set can be described by a global scaling law, which makes it possible to form a good estimate of the fraction of chaotic parameter values. Sensitive dependence on parameters occurs when the scaling exponent beta is less than 1. It is conjectured that beta displays universal behavior.

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