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Parameter renormalization of maps based on potential function.

Ikuo Matsuba1

  • 1Faculty of Engineering, Chiba University, 1-33 Yayoi-cho, Inage-ku, Chiba-shi 263, Japan.

Chaos (Woodbury, N.Y.)
|June 1, 1997
PubMed
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This study presents a systematic method for deriving renormalization group equations for one-dimensional maps, revealing critical behavior in periodic doubling. A potential function approach simplifies determining universal constants and accumulation points, showing Feigenbaum-like scaling.

Area of Science:

  • Nonlinear Dynamics
  • Statistical Mechanics
  • Chaos Theory

Background:

  • Renormalization group (RG) methods are crucial for understanding critical phenomena.
  • One-dimensional maps exhibit complex behaviors, including period-doubling bifurcations.

Purpose of the Study:

  • To develop a systematic method for deriving parameter RG equations for 1D maps.
  • To investigate the critical behavior of periodic doubling using a potential function approach.
  • To analyze parameter reduction transformations for multi-parameter systems.

Main Methods:

  • Derivation of parameter renormalization group equations.
  • Introduction of a formal potential function for one-parameter cases.
  • Application of parameter reduction transformations for two-parameter cases.

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Main Results:

  • Accumulation points correspond to local potential maxima.
  • Universal constants and accumulation points are accurately determined.
  • The potential function exhibits scaling properties similar to the Feigenbaum function.
  • Parameter reduction transformations aid in identifying fixed points.

Conclusions:

  • The potential function method provides an effective framework for analyzing critical phenomena in 1D maps.
  • The study confirms universal scaling behaviors in parameter spaces.
  • The findings offer insights into the dynamics of chaotic systems.