Dynamical system

http://mathworld.wolfram.com/DynamicalSystem.html

Dynamical System
EXPLORE THIS TOPIC IN the MathWorld Classroom

A means of describing how one state develops into another state over the course of time. Technically, a dynamical system is a smooth action of the reals or the integers on another object (usually a manifold). When the reals are acting, the system is called a continuous dynamical system, and when the integers are acting, the system is called a discrete dynamical system. If f is any continuous function, then the evolution of a variable x can be given by the formula
x_(n+1)=f(x_n).
(1)

This equation can also be viewed as a difference equation
x_(n+1)-x_n=f(x_n)-x_n,
(2)

so defining
g(x)=f(x)-x
(3)

gives
x_(n+1)-x_n=g(x_n)*1,
(4)

which can be read “as n changes by 1 unit, x changes by g(x).” This is the discrete analog of the differential equation
x^'(n)=g(x(n)).

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