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2.6] What is a map? (nonlinear science)




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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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