I'm doing a research project on Chaotic Encryption using the logistic map(?). I'm still in my early stages, and my professor made the following question:
What's the difference between the logistic map and the logistic equation?
The thing is, I'd say: the map is the 'function' described by the 'equation' $x_{n+1} = \lambda x_n(1-x_n)$.
Would this be an accurate answer? I've never asked myself this question.
The logistic equation usually refers to the differential equation $$ \frac{dx}{dt} = r x \left( 1 - \frac{x}{K} \right) , $$ i.e., a continuous-time dynamical system which gives you a function $x(t)$, $t \in \mathbf{R}$, given an initial value $x(0)$.
The logistic map is the function on the right-hand side, $$ f(x) = r x \left( 1 - \frac{x}{K} \right) , $$ and usually when talking about the logistic map one is interested in the discrete-time dynamical system obtained by iteration of this map, $$ x_{n+1} = f(x_n) , $$ which gives you a sequence $(x_n)_{n \in \mathbf{N}}$ given an initial value $x_0$.