CoCalc Public FilesTA Sandbox / Daniel's Sandbox / HW5-P2.sagewsOpen with one click!
Authors: Kristin McCully, Daniel Sun
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Description: HW5-p2, spring 2016
Compute Environment: Ubuntu 18.04 (Deprecated)
var("R") var("J"
R J
#preliminary plot of the vector field # Exercise 2: Plot of eigenvector lines (eigenlines) eigenvector1 = plot(-1.5*R,(R,-10,10),legend_label="Dominant Eigenvector") eigenvector2 = plot(1/3*R,(R,-10,10),color="red",ymin=-10,ymax=10,legend_label="Non-dominant Eigenvector") show(eigenvector1 + eigenvector2)
# We can always reconstruct the matrix if we know the eigenvalue-eigenvector pairs E = matrix([[-2,3],[3,1]]) E.inverse()*vector([1,0])
(-1/11, 3/11)
c1 = -1*-1/11*vector([-2,3])-4*3/11*vector([3,1]) c1
(-38/11, -9/11)
E.inverse()*vector([0,1])
(3/11, 2/11)
c2 = -1*3/11*vector([-2,3])-4*2/11*vector([3,1])
# Reconstructing the matrix A = column_matrix(QQ,[c1,c2]) A
[-38/11 -18/11] [ -9/11 -17/11]
# The eigenvalues and eigenvectors of the this matrix match the original ones that we were given. A.eigenvectors_right()
[(-1, [ (1, -3/2) ], 1), (-4, [ (1, 1/3) ], 1)]
# Plotting the linear differential system, the trajectories "rush" in towards the dominant eigenline and then decay slowly towards the origin. Rprime(R,J) = A[0,0]*R + A[0,1]*J Jprime(R,J) = A[1,0]*R + A[1,1]*J system = [Rprime,Jprime] show(plot_vector_field([Rprime,Jprime],(R,-10,10),(J,-10,10)) + plot(-1.5*R,(R,-10,10),legend_label="dominant eigenvector")+plot(1/3*R,(R,-10,10),color="red",ymin=-10,ymax=10,legend_label="non-dominant eigenvector"), svg=false) show(Rprime) show(Jprime)
(R,J)  1811J3811R\displaystyle \left( R, J \right) \ {\mapsto} \ -\frac{18}{11} \, J - \frac{38}{11} \, R
(R,J)  1711J911R\displaystyle \left( R, J \right) \ {\mapsto} \ -\frac{17}{11} \, J - \frac{9}{11} \, R