Coding Rule for Periodic Orbits in the One-dimensional Map

Yoshihiro Yamaguchi

Teikyo Heisei University, Ichihara, Chiba 290-0193, Japan
E-mail address: chaosfractal@iCloud.com

(Received March 20, 2018; Accepted October 17, 2018)

Abstract. A new coding rule for periodic orbits in unimodal one-dimensional maps is derived. The best-known example of a family of unimodal maps is the logistic map. The band merging is observed in the bifurcation diagram of the logistic map. Let be the critical value at which 2^k-band merges into 2^k−1-band. At , the diverging orbit appears and thus 1-band disappears. The relations for k ≥ 0 hold. Let s_q be the code for periodic orbit of period q in the parameter interval . Assume that the code s_q represented by symbols 0 and 1 is known. In the interval , there exists the periodic orbit of period 2^k × q (k ≥ 1). Let its code be s_{2^k × q}. Let D be the doubling operator defined by the substitution rules as 0 ⇒ 11 and 1 ⇒ 01. The following coding rule is derived. Operating k times of D to s_q, the code s_{2^k × q} is determined.

Keywords: One-dimensional Map, Bifurcation Diagram, Coding Rule, Periodic Orbits, Doubling Operator