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Parameter renormalization of maps based on potential function
1Faculty of Engineering, Chiba University, 1-33 Yayoi-cho, Inage-ku, Chiba-shi 263, Japan.
Chaos (Woodbury, N.Y.)
|June 1, 1997
Summary
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.
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.
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