The universal metric properties of nonlinear transformations

@article{Feigenbaum1979TheUM,
  title={The universal metric properties of nonlinear transformations},
  author={Mitchell J. Feigenbaum},
  journal={Journal of Statistical Physics},
  year={1979},
  volume={21},
  pages={669-706}
}
  • M. Feigenbaum
  • Published 1 December 1979
  • Mathematics
  • Journal of Statistical Physics
AbstractThe role of functional equations to describe the exact local structure of highly bifurcated attractors ofxn+1 =λf(xn) independent of a specificf is formally developed. A hierarchy of universal functionsgr(x) exists, each descriptive of the same local structure but at levels of a cluster of 2r points. The hierarchy obeysgr−1(x)=−αgr(gr(x/α), withg=limr → ∞ gr existing and obeyingg(x) = −αg(g(x/α), an equation whose solution determines bothg andα. Forr asymptoticgr ∼ g − δ−rh* where δ > 1… 
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