Yuu Miino, Tetsushi Ueta, Calculation method for unstable periodic points in unimodal maps using symbolic dynamical system, Institute of Electronics, Information and Communication Engineers, Nonlinear Theory and Its Applications, IEICE, Vol.E13-N, No.2, pp.2-24, May 2022.
Abstract: In this study, we have focused on the unimodal maps and developed the numerical method to calculate the unstable periodic points (UPPs), based on the theory of the symbolic dynamical system. The core technique of the method is the definition of a non-deterministic map G. From the experimental result of two typical maps: logistic map and tent map, we have confirmed the proposed method works very well within the defined errors. Our method has the following advantages: the method converges rapidly as the period of the target UPP is larger; we can choose the target UPP regardless of its cause (any bifurcation is not a matter); we can find the UPPs that are always unstable in the given parameter range. The convergence of the method is guaranteed by two standpoints: the corresponding symbolic dynamical system, and the asymptotic stability of UPP of G. Hereby, the error of the convergence is scalable according to the numeric precision of the software.