Draw the phase portrait and classify the origin of the system:
xdot = [1 2; 2 1]x
det(A-lambda*I) = 0
The Attempt at a Solution
First find the eigenvalues and eigenvectors:
det(A-lambda*I) = (lambda+1)(lambda-3) = 0
we can see that the eigenvalues are:
lambda_1 = -1 and lambda_2 = 3
for lambda_1 = -1: 2*k1 + 2*k2 = 0
k1 = -k2
when k1 = 1, k2 = - 1 the related eigenvector is (1; -1)
for lambda_2 = 3: -2*k1 + 2*k2 = 0
k1 = k2
when k1 = 1, k2 = 1 the eigenvector is K2 = (1; 1)
since the matrix of coefficeints is a 2x2 matrix and since we found two linearly independent solutions,
the general solution of the system is:
X = c1*X1 + c2*X2 = c1*(1;-1)*exp(-t) + c2*(1; 1)*exp(t)
x = c1*exp(-t) + c2*exp(t)
y = -c1*exp(-t) + c2*exp(t)
We can classify the origin as neither a repeller nor an attractor.
Is this correct?
Also I feel like I need to provide more information when classifying the origin but I don't know what. For instance should I call the origin a saddle point because it has eigenvalues of different polarities?
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