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

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    $\begingroup$ 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. $\endgroup$ Apr 1, 2014 at 12:21
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    $\begingroup$ Smooth usually means that the function is continuously differentiable. $\endgroup$
    – Kyle Kanos
    Apr 1, 2014 at 12:45

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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.

The theory that makes this possible is the celebrated Schröder's equation (1870) which allows easy calculation of functional square (and higher) roots, and renormalization trajectories, since it turns out that the original Gell-Mann-Low Renormalization Group equation (1954) is, in fact, this very equation of Schröder, RG.

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.)

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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.

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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.

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