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Dynamical System

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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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