Dynamical System

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 is any continuous function, then the evolution of a variable can be given by the formula

(1)

This equation can also be viewed as a difference equation

(2)

so defining

(3)

gives

(4)

which can be read "as changes by 1 unit, changes by ." This is the discrete analog of the differential equation

(5)

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