stability in the periodic orbit and in the singular point <mover> x &#x02D9;<!-- ˙ --

taghdh9 2022-06-24 Answered
stability in the periodic orbit and in the singular point
x ˙ = y + λ x ( 36 9 x 2 y 2 ) y ˙ = 9 x + λ y ( 36 9 x 2 y 2 ) z ˙ = 6 z λ 2 x 2 y 2 z 3
I want to analyze the stability in the periodic orbit and in the singular point, so for the singular point I take the derived matrix of the linear part, and I got the eigenvalues, wich are λ 1 = 6, λ 2 = 3 i , λ = 3 i . I wanted to use Andronov-Vitt but I have two eigenvalues with no real part so I can´t, does anybody can help me with this?
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Answers (1)

scoseBexgofvc
Answered 2022-06-25 Author has 20 answers
It appears that you have an issue with the Jacobian and hence the eigenvalues.
There is one critical point at ( x , y , z ) = ( 0 , 0 , 0 )
Evaluating J ( x , y , z ) at this critical points yields the eigenvalues:
λ 1 = 6 , λ 2 = 36 λ 3 i , λ 3 = 36 λ + 3 i
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