*Forma,* Vol. 28 (No. 1), pp. 1-5, 2013

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

Syuji Miyazaki^{1}* and Yusuke Higuchi^{2}

^{1}Graduate School of Informatics, Kyoto University, Kyoto 606-8501, Japan

^{2}Faculty 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

[Full text] (PDF 124 KB)