213 - A6 - Linear Dynamical System -- Sage

This worksheet allows the user to input a transition matrix and initial vector and then iterate for any desired number of time steps.

John Travis

Mississippi College

%hide %auto N=2 print pretty_print(html("Enter N below. Click Update when you are finished.")) @interact def _(Size_of_System=[2..10],update=false): global N N = Size_of_System pretty_print(html("The size of your square system will be "+str(N)+"$\\times$"+str(N)+"")) pretty_print(html("Evaluate the cell below and enter the system data."))

Enter N below. Click Update when you are finished. None

Size_of_System
update

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%auto print pretty_print(html("Enter the transition matrix for your dynamical system and the starting state below. You can tab to navigate between cells. Click Update when you are finished.")) @interact def _(Transition_Matrix=zero_matrix(QQ,N,N),Initial_State=zero_matrix(QQ,N,1),auto_update=False): global A global b A = Transition_Matrix b = Initial_State

Enter the transition matrix for your dynamical system and the starting state below.

You can tab to navigate between cells. Click Update when you are finished.

None

Transition_Matrix

Initial_State

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%auto print 'Using the data' show(A) b0 = copy(b) show(b0)

Using the data

$\newcommand{\Bold}[1]{\mathbf{#1}}\left(\begin{array}{rr} \frac{47}{50} & \frac{1}{25} \\ \frac{3}{50} & \frac{24}{25} \end{array}\right)$

$\newcommand{\Bold}[1]{\mathbf{#1}}\left(\begin{array}{r} 10 \\ \frac{4}{5} \end{array}\right)$

Using the data

Assignment: For a given dynamical systems problem (should be stated here):

Perform one step below to make certain your dynamical system is working properly
Perform a small number of steps (say 5-10) and notice any drastic changes in any of the various populations
Perform a large number of steps and notice whether the populations tend to stabilize about a given value.
Write up your results in a text cell below. Be certain to cut-and-paste the transition matrix and initial values into another cell for me to see what you used. A copy is output below the interactive cell below for you to grab and save.

[3/4 1/5]
[1/4 4/5]

[3/4 1/5]
[1/4 4/5]

%auto @interact def _(steps = slider(1,100,1,1)): global A,b b = copy(b0) print b terms = range(b.nrows()) colors = rainbow(b.nrows()) P = Graphics() # S = Graphics() for term in terms: P += point2d((0,b[term][0]),color=colors[term]) for k in range(1,steps): b = A*b # S += point2d((b[0][k],b[1][k])) for term in terms: P += point2d((k,b[term][0]),color=colors[term],size=30) show(P) # show(S) print pretty_print(html('Final values:\n'),b.n())

steps

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%auto show(A) show(b0) show(b)

Traceback (click to the left of this block for traceback)
...
NameError: name 'A' is not defined

Traceback (most recent call last):
  File "<stdin>", line 1, in <module>
  File "_sage_input_7.py", line 10, in <module>
    exec compile(u'open("___code___.py","w").write("# -*- coding: utf-8 -*-\\n" + _support_.preparse_worksheet_cell(base64.b64decode("c2hvdyhBKQpzaG93KGIwKQpzaG93KGIp"),globals())+"\\n"); execfile(os.path.abspath("___code___.py"))
  File "", line 1, in <module>
    
  File "/tmp/tmpOPWNHm/___code___.py", line 2, in <module>
    show(A)
NameError: name 'A' is not defined

(-5/92*3/95+100/95)

1837/1748

1837/1748

A = matrix([[95/100,3/100],[5/100,97/100]]) C = A.inverse() x = matrix([[582],[418]]) x0 = C*x x1 = A*x0 x2 = A*x1 x3 = A*x2 print(x3) C^3*x3

[343878/625]
[281122/625]
[600]
[400]

[343878/625]
[281122/625]
[600]
[400]

213 - A6 - Linear Dynamical System

432 days ago by Professor213

Click to the left again to hide and once more to show the dynamic interactive window

Click to the left again to hide and once more to show the dynamic interactive window

Click to the left again to hide and once more to show the dynamic interactive window