# 2.6] What is a map? (nonlinear science)

## Description

This article is from the Nonlinear Science FAQ, by James D. Meiss jdm@boulder.colorado.edu with numerous contributions by
others.

# 2.6] What is a map? (nonlinear science)

A map is simply a function, f, on the phase space that gives the next state,

f(z) (the image), of the system given its current state, z. (Often you will

find the notation z' = f(z), where the prime means the next point, not the

derivative.)

Now a function must have a single value for each state, but there could be

several different states that give rise to the same image. Maps that allow

every state in the phase space to be accessed (onto) and which have precisely

one pre-image for each state (one-to-one) are invertible. If in addition the

map and its inverse are continuous (with respect to the phase space coordinate

z), then it is called a homeomorphism. A homeomorphism that has at least one

continuous derivative (w.r.t. z) and a continuously differentiable inverse is

a diffeomorphism.

Iteration of a map means repeatedly applying the map to the consequents of the

previous application. Thus we get a sequence

n
z = f(z ) = f(f(z )...) = f (z )
n n-1 n-2 0

This sequence is the orbit or trajectory of the dynamical system with initial

condition z_0.

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