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Network analysis of chaotic systems through unstable periodic orbits
Miki U Kobayashi1, Yoshitaka Saiki2
1Faculty of Economics, Rissho University, 4-2-16 Osaki, Shinagawa-ku, Tokyo 141-8602, Japan.
Chaos (Woodbury, N.Y.)
|September 3, 2017
Summary
Unstable periodic orbits with many connections in a network model capture chaotic motion properties. This network analysis reveals scale-free characteristics in chaotic systems.
Area of Science:
- Complex Systems Science
- Nonlinear Dynamics
- Network Theory
Background:
- Chaotic motion is characterized by irregular transitions near unstable periodic orbits (UPOs).
- UPOs are crucial for determining statistical properties of chaos, like natural measures and fractal dimensions.
- Identifying which UPOs best approximate averaged chaos or turbulence properties remains unclear.
Purpose of the Study:
- To model the irregular transition process of chaotic motion using a network framework.
- To identify the characteristics of UPOs that effectively represent time-averaged properties of chaos.
- To investigate the network properties of UPOs within this model.
Main Methods:
- Constructed a network model where unstable periodic orbits act as nodes.
- Analyzed the connections (links) between UPOs within the network.
- Examined the degree distribution of the network to identify scale-free properties.
Main Results:
- Unstable periodic orbits with a higher number of links in the network model tend to capture time-averaged properties of chaos.
- The degree distribution of the UPO network exhibits a scale-free property.
- This suggests a hierarchical organization within the UPO structure of chaotic systems.
Conclusions:
- Network analysis provides a novel approach to understanding chaotic dynamics.
- UPOs with extensive connectivity are key to characterizing averaged chaotic behavior.
- The observed scale-free property offers insights into the fundamental structure of chaos.
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