System of equations: (y_1 y_2)=(2 0, 4 −1)(y_1 y_2)

Aldo Ashley 2022-10-29 Answered
System of equations:
( y ˙ 1 y ˙ 2 ) = ( 2 0 4 1 ) ( y 1 y 2 )
Looking at matrix A I can see a value not on the diagonal equal to zero. So the eigenvalues here are 2 and 1, meaning I have a saddle node. Next I look at the eigenvectors:
λ = 2 , [ 0 0 0 4 3 0 ]
λ = 1 , [ 3 0 0 4 0 0 ]
y = A e 2 x ( 3 4 ) + B e x ( 0 1 )
So I have an attractive trajectory on the y-axis, and a repulsive trajectory on the line y = 3 4 x
With the slope at origin being
y ˙ 2 y ˙ 1 = 4 y 1 y 2 2 y 1 = 0 0 = ±
Does all of this seem correct?
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Answers (1)

domwaheights0m
Answered 2022-10-30 Author has 11 answers
It is indeed correct, but the slope you computed is the slope of the orbit on the phase space, because each coordinate y 1 and y 2 have an exponencial growth/decay rate. also the origin is not a node, but a saddle equilibria
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