I was trying to understand the rigorous definition of sensitive dependence on initial conditions which is as follows -
$f : X \mapsto X$ where $X$ is a metric space.
If there exists $\epsilon > 0$ such that $\forall x \in X \forall \delta>0 $ there exists $y \in X$ with $d(x,y)<\delta$ and $d(f^n(x),f^n(y))$ for some $n > 0$.
Source - Definition in the Book
Linear Chaos, Definition 1.21
A First Course in Discrete Dynamical Systems, Definition 11.18
An Intuitive explanation, visualization of the above rigorous definition would be very much helpful.