Universal vector scaling in one-dimensional maps

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Abstract:

A description of scaling structures in two-extremum, two-parameter, one-dimensional maps is presented. Examples of systems for which such map models apply are given. In parameter space, subharmonic bifurcation displays an infinite binary tree of features connected to a Cantor set of normal (Feigenbaum) cascades. The cascade limit lines are attached to the tree limits, tricritical points. The scaling behavior in the vicinity of the tricritical points is conveniently described with reference to sets of superstable lines and superstable-orbit-segment lines. The parameter-plane scaling with respect to these line sets is related to the renormalization-group treatment of scaling in function space. Power-law conjugacy is used to provide some physical insight into the nature of the second relevant vector that appears in the theory. © 1984 The American Physical Society.