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

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn

Exercise 2.3.10. (5 points) Which of the discrete dynamical systems below tend to a limiting steady state vector, and if so, what is it?

NOTE: The initial state is (0,1,0). We could take three state vectors, (1,0,0), (0,1,0), and (0,0,1) and see what happens after many stages -- say n=10, 100. Stacking them together simply gives the identity matrix, and since 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, all we have to do is examine the powers of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUYjNiMtRiw2JVEiQUYnRi9GMkYr -- the columns should tend to a constant vector. Observe that each vector is a probability distribution vector. Furthermore, this answers something slightly different from the question.

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

QyQtSSV3aXRoRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiNJLkxpbmVhckFsZ2VicmFHRiUhIiI=

print();

(a) This matrix appears to stabilize to steady state given by columns of A^100, that is, approximately (0.17241,0.51724,0.31034). This is also the case if we calculate A^100*x0

x0 :=

LSZJJ1ZlY3Rvckc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjSSdjb2x1bW5HRig2Iy9JJCVpZEdGKCIqWywlcDg=

print();

print();

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

print();

NiYqJClJIkFHNiIiIiYiIiIqJClGJSIjNUYoKiQpRiUiI11GKCokKUYlIiQrIkYo

print();

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn

(b) This matrix appears to flip back and forth, and does not stabilize to a steady state. However, the middle column is constant, so if the initial state of the discrete dynamical system is (0,1,0), then the dds is instantly stable.

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

print();

NicqJClJIkFHNiIiIiYiIiIqJClGJSIjXUYoKiQpRiUiJCsiRigqJClGJSIkLCJGKCokKUYlIiQtIkYo

print();

(c) This matrix appears to stabilize to steady state given by columns of A^100, which appear to be approximately (0,0,1).

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

print();

NiUqJClJIkFHNiIiIzUiIiIqJClGJSIjXUYoKiQpRiUiJCsiRig=

print();

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn

(d) This matrix appears to stabilize to steady state given by columns of A^100, which appear to be approximately (0,0,0).

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

print();

NiUqJClJIkFHNiIiIzUiIiIqJClGJSIjXUYoKiQpRiUiJCsiRig=

print();

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn

Exercise 12. (3 points for calculation, 2 for graph). Compute the total number of walks in the graph with adjacency matrix below of length at most 3. We will eyeball the resulting matrix below and add up all the entries in the result, since the (i,j)th entry gives all walks of length at most 3. To get the final answer, we'll be sneaky and multiply on left and right by a row and column of 1's. This will give us the total sum of all entries, which turns out to be 132.

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

print();

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

print();

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

print();

LUkld2l0aEc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjSSZwbG90c0dGJw==

For the picture of this digraph we used an external routine to create an eps file, then clicked Insert, then Image.

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