# Meaning of Smooth Dynamical System?

What does smooth dynamical system mean? It is the title of a paper I am supposed to read in non linear systems.

-
Can you provide a reference to the specific paper? The Wikipedia article on dynamical systems states that smooth can mean different things in different contexts. – Daniel Geisler Apr 1 '14 at 12:21
Smooth usually means that the function is continuously differentiable. – Kyle Kanos Apr 1 '14 at 12:45

A dynamical system is a system that evolves by a rule over time. As opposed to modeling a system with a PDE, the system is modeled by some type of iterated function, $f^t(x)$. Smooth implies that the function being iterated is differentiable, so we are not talking about an iterated system like cellular automata.
Many dynamical systems involve recursive sequences of simple functions, $f(f(f(...f(x))))$ which are iterated n times to lead to some discrete iteration sequence (a Picard sequence) with often tractable properties. It is possible however, to analytically continue the discrete iteration index n to a continuous non integer one, fractional, infinitesimal, or negative, through the miracle of functional conjugation and to produce continuous orbits, $f_t(x)$, whose velocity and continuous phase-space trajectories may be monitored.
Typically, you use it, e.g., to extend the Beverton-Holt map $f(x)=x/(2−x)$ to its continuous iterate $$f_t(x)=\frac{x}{2^t + x(1-2^t)} ~,$$ where f(x) is the above for t=1. The resulting evolution surface is thus smooth.