The fixed point as a period-1 recurrent in opological dynamical systems

Abstract

The behavior of the dynamical orbit of a system by describing it relies on the method used. The paper uses the logistic function to illustrate and describe the fixed point of the periodic–like recurrence as a periodic -1 recurrent. The study is based on Theorem 1:(?,?)is a fixed (Stationary) recurrent point, iff for all ?∈? and an operator ?:?→? a continuous map and any neighborhood ?⊂? then,⋃≔{?∈?:?^?(?)≠∅}∴ ??(?)=?∈?,?∈?,?>?, Theorem 2: a point ? is periodic -1 or fixed point if ?(?)=?, ???(?)=???_?(?) and form a fixed (Stationary) recurrent point ?^?(?)=?∈?,?∈?,?>? and, Definition 7: a point ?∈ ? is said to be recurrent if for any neighborhood ? of?, there exists an integer ?≥?such that ?^?(?)∈? through the application of the logistic function. The application of the logistic function on the two theorems (Theorem 1 and Theorem 2) and Definition 7 explained that period-1 recurrent only exists when there is the existence of fixed point (periodic orbits) which depends solely on the initial point and the parameter ? of the logistic function..