-
Sahar Qandeel,
-
"On Devaney's Definition of Chaos"
-
The term chaos was first introduced in the paper of the mathematician James A. Yorke titled "Period Three Implies Chaos". Yorke defined the chaotic function and proved that for a continuous map f on a closed interval I, if there exists periodic point in I of period 3 then f is a chaotic function. During the last few decades, research in chaotic dynamical systems flourished and has gotten a lot of interest. At the close of the eighties, Robert L. Devaney published his popular book "An Introduction to Chaotic Dynamical Systems", making chaos theory popular that it entered universities as a course in dynamical systems. In 1989, Devaney published his definition of chaotic functions on a metric space laying the basis for later investigations. Let X be a metric space. A map f : X ! X is said to be chaotic on X if : 1. f is transitive. 2. The set of periodic points is dense. 3. f has sensitive dependence on initial conditions. In this talk, we propose the following modications on Devaney's denition of chaos. Denition 0.1. Let X be a metric space with metric d, and let f : X ! X be a continuous map. We say that f has a weakly sensitive dependence on initial conditions if there is a positive real number , such that for every point x 2 X and every neighborhood N of x there exist a point y 2 N and nonnegative integers n;m such that d(fn(x); fm(y)) > . Denition 0.2. Let X be a Topological space, and let f : X ! X be a continuous map. f is called weakly transitive if for every nonempty open subsets U and V of X, there exists a natural number k such that fk(U) \ V 6= or U \ fk(V ) 6= .