# Meaning of Smooth Dynamical System?

What does smooth dynamical system mean? It is the title of a paper that 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. Apr 1, 2014 at 12:21
• Smooth usually means that the function is continuously differentiable. Apr 1, 2014 at 12:45

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.
On a logarithmic scale, such iteration flows might well be already familiar to you from the nesting property of Chebyshev polynomials, $T_m (T_n(x))= T_{mn}(x)$, since $T_{\exp (t)}(x)= \cos (e^t \arccos (x))$, in suitable intervals. (Note that for continuous iteration trajectory index t these functions are not polynomials anymore, in general, as they are for integer values of expt.)
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.
A system is called as smooth dynamical system when, the function is continuous in its domain & the nth order derivative for the system exists.All dynamical systems, are modeled mathematically evolving on Smooth manifolds $$C^n$$ where n belongs to natural numbers& they should have a one to one correspondence with the members of the set.