213 - A19 - Linear Dynamical System

4 days ago by Professor213

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("<font size=+2 color=blue>Enter N below. Click Update when you are finished.</font>")) @interact def _(Size_of_System=[2..10],update=false): global N N = Size_of_System pretty_print(html("<font size=+2 color=blue>The size of your square system will be "+str(N)+"$\\times$"+str(N)+"</font>")) pretty_print(html("<font size=+2 color=blue>Evaluate the cell below and enter the system data.</font>")) 
       
Enter N below. Click Update when you are finished. None
Size_of_System 
update 

Click to the left again to hide and once more to show the dynamic interactive window
%auto print pretty_print(html("<font size=+2 color=blue>Enter the transition matrix for your dynamical system and the starting state below.<p> You can tab to navigate between cells. Click Update when you are finished.</font><p>")) @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 

Click to the left again to hide and once more to show the dynamic interactive window
%auto print 'Using the data' show(A) b0 = copy(b) show(b0) 
        
Using the data


                                
                            

                            
                                

                                    
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.
    • 


    





  
    

    
    






    


        
        
        
            
                
            
            
                
A 

            
            

            

        

        
        
        

            

                

                    
                               
                    
                    
                        

                            

                                
                                    
[24/25  3/50]
[ 1/25 47/50]

                                
                            

                            
                                

                                    
[24/25  3/50]
[ 1/25 47/50]

                                

                            
                                

                                    
                                

                        

                    		
                

            

        

        


    




  
    

    
    






    


        
        
        
            
                
            
            
                
%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 
        

          

            
              

            		
            
            
              
            
          

        



              

        

          

            

              
                

              		
            

            

              
            

          

        

              

            

            

          

        

                                
                            

                            
                                

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

                                

                            
                                

                                    
                                

                        

                    		
                

            

        

        


    




  
    

    
    






    


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

            
            

            

        

        
        
        

            

                

                    
                               
                    
                    
                        

                            

                                
                                    

                                
                            

                            
                                

                                    

                                

                            
                                

                                    
                                

                        

                    		
                

            

        

        


    




  
    

    
    






    


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

            
            

            

        

        
        
        

            

                

                    
                               
                    
                    
                        

                            

                                
                                    
1837/1748

                                
                            

                            
                                

                                    
1837/1748

                                

                            
                                

                                    
                                

                        

                    		
                

            

        

        


    




  
    

    
    






    


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