Discrete Dynamical Systems -- Adapted from VLA Module
A discrete dynamical system is a sequence of vectors related to one another by a square matrix A as follows:
, for k = 0, 1, 2, ...
If we think of k as representing time in some units, we can think of a dynamical system as describing the evolving behavior over time of the set of variables represented by the vectors . We refer to this behavior as the "dynamics" of the "system" of vectors .
Example 1: To determine the long-term behavior of the vectors we'll examine the eigenvector decomposition:
|>||Ex1 := Matrix([[525/1000,15/100],[-1875/10000,975/1000]]);|
First we find the eigenvalues and associated eigenvectors of A:
|>||h,P := Eigenvectors(Ex1);|
Therefore we have the eigenvector decomposition
Since both eigenvalues are less than one in absolute value, then both and have limit 0. Now the vector tends toward the zero vector in the limit every , so both the x-coordinate population and the y-coordinate population tends to 0. We sometimes say that the points are "attracted to the origin." Furthermore, since , then as long as is not 0, then the points will be close to for large k, which tells us that approaches the origin along a line with direction vector ie y=(5/2) x.
We plot the "trajectory" of the vectors , which is a plot of the points
Note: while we don't typically need the coefficients and , we could compute them as follows if we need them in applications:
|>||x0 := Vector([1.5,1]);|
|>||a := P^(-1).x0;|
Example 2: To determine the long-term behavior of the vectors we'll examine the eigenvector decomposition:
|>||Ex2 := Matrix([[9/10,-12/100],[-5/10,8/10]]);|
|>||h,P := Eigenvectors(Ex2);|
The eigenvector decomposition is , where the values of and depend on . Since the larger eigenvalue of B is 11/10, the points eventually move away from the origin and get arbitrarily far away. An associated eigenvector is , which tells us that as long as is nonzero, then eventually the trajectory follows the line through the origin y=-x.
However, note what happens below when the initial vector is an eigenvector associated with the smaller eigenvalue (ie =0). In this case we die off along the y= x line.
|>||x0 := Vector([.4,1]):|
Example 3: To determine the long-term behavior of the vectors we'll examine the eigenvector decomposition:
|>||Ex3 := Matrix([[9/10,2/10],[1/10,8/10]]);|
|>||h,P := Eigenvectors(Ex3);|
The larger eigenvalue is 1 so, for most starting positions, the system tends towards the line y=1/2 x associated to the eigenvector . We can say a bit more than this as =1 and so
which is the line parallel to (since changes over time) and through the tip of ie the place on y=1/2 x given by the starting position and a parallel transport there. So eventually we hit that line and then stay fixed there forevermore (unless
=0 in which case we die off along the y=-x line). For most starting populations, in the limit the populations will achieve the ratio of 2 of population a for each 1 of population b.
Example 4: Here is a system whose dynamics are more complicated:
|>||Ex4 := Matrix([[8/10,5/10],[-1/10,1]]);|
We sometimes say that the points are "attracted to the origin along a spiral." The spiral behavior is caused by the presence of complex eigenvalues.