# THE FIXED POINT AS A PERIOD-1 RECURRENT IN TOPOLOGICAL 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..