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 .  

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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. 

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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;

 

(1.1)

 

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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]]);

 

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]]);

 

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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.  

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