Chaos as Irregular Hopping between Unstable Periodic Orbits and Its Network Representation

Syuji Miyazaki¹* and Yusuke Higuchi²

¹Graduate School of Informatics, Kyoto University, Kyoto 606-8501, Japan
²Faculty of Engineering, Kyoto University, Kyoto 606-8501, Japan
*E-mail address: syuji@acs.i.kyoto-u.ac.jp

(Received April 14, 2013; Accepted May 27, 2013)

Abstract. Based on a matrix representation of the generalized Frobenius-Perron operator describing large-deviation statistics of local expansion rates of one-dimensional chaotic map, directed graphs are constructed. Its network statistics reflect characteristic fluctuation in the vicinity of a specific bifurcation. In this treatment, chaos can be described as an irregular switching between finite specific unstable periodic orbits. Type-I intermittency is analyzed for the solvable Shobu-Ose-Mori map and the logistic map. Solvable irreducible and approximate redundant partitions are constructed to obtain directed graphs and degree distributions. The output degree distribution obtained from a redundant partition slightly fluctuates around that obtained from a irreducible partition in the case of a tent map. It is shown that the output degree distribution is a good candidate to capture characteristics of Type-I intermittency.

Keywords: Deterministic Chaos, Complex Network, Type-I Intermittency, Degree Distribution, Unstable Periodic Orbit