Spring 2005
Converting a map that depends on older states to a simple map
In some maps, the state at step n+1 depends on not
just the state at step n, but also on the states
at steps n-1, n-2, etc.
These notes give examples of converting such a map
into a higher dimensional map in which the state at
step n+1 depends only on the state at step n.
Example 1
Suppose we have the map
|
xn+1 = f(xn,xn-1,xn-2,xn-3)
|
We define
|
an+1
|
=
|
xn
|
|
bn+1
|
=
|
an
|
|
cn+1
|
=
|
bn
|
Then we have
xn-1 = an,
xn-2 = bn, and
xn-3 = cn.
The map can now be written
|
xn+1
|
=
|
f(xn,an,bn,cn)
|
|
an+1
|
=
|
xn
|
|
bn+1
|
=
|
an
|
|
cn+1
|
=
|
bn
|
The state of the system is now described by a four dimensional vector,
and the state at step n+1 is a function of the state at
step n only.
Example 2
Now consider the system
|
xn+1
|
=
|
f(xn,xn-1,xn-2,yn,yn-1)
|
|
yn+1
|
=
|
g(xn,xn-1,xn-2,yn,yn-1)
|
We define
|
an+1
|
=
|
xn
|
|
bn+1
|
=
|
an
|
|
cn+1
|
=
|
yn
|
Then we have
xn-1 = an,
xn-2 = bn, and
yn-1 = cn.
With these definitions,
the new state vector is
(xn,yn,an,bn,cn),
and the map is
|
xn+1
|
=
|
f(xn,an,bn,yn,cn)
|
|
yn+1
|
=
|
g(xn,an,bn,yn,cn)
|
|
an+1
|
=
|
xn
|
|
bn+1
|
=
|
an
|
|
cn+1
|
=
|
yn
|
