Show that x ∗ = ( 1 , 2 ) is a fixed point of...

Sovardipk

Sovardipk

Answered

2022-07-01

Show that x = ( 1 , 2 ) is a fixed point of the system
x 1 = 2 + 3 x 1 2 x 2 x 1 2 + 2 x 1 x 2 x 2 2
x 2 = 3 + 4 x 1 3 x 2 x 1 2 + 2 x 1 x 2 x 2 2
Determine W s ( x ) and W u ( x ), the global stable and unstable manifolds of the fixed point x = ( 1 , 2 ) and give a parametric representation of those manifolds.

Answer & Explanation

Ronald Hickman

Ronald Hickman

Expert

2022-07-02Added 18 answers

Plug in ( 1 , 2 ) to see that you get 0, so the point is an equilibrium.
Calculate the jacobian of the right hand side:
D f ( x 1 , x 2 ) = ( 3 2 x 1 + 2 x 2 2 + 2 x 1 2 x 2 4 2 x 1 + 2 x 2 3 + 2 x 1 2 x 2 )
plug in our equilibrium point:
D f ( 1 , 2 ) = ( 3 2 + 4 2 + 2 4 4 2 + 4 3 + 2 4 ) = ( 5 4 6 5 )
Eigenvalues:
( 5 λ ) ( 5 λ ) + 24 = ( 5 λ ) ( 5 + λ ) + 24 = λ 2 1 = 0
Thus λ = ± 1 since our first eigenvalue is negative, constitutes to the stable manifold, and the other is positive, and constitutes to the unstable manifolds .
Now our corresponding eigenvectors are λ = 1 [ 1 , 1 ] and λ = 1 [ 1 , 2 3 ]
So our stable subspace:
E s = = span {eigenvector, for λ = 1 }= span { [ 1 , 2 3 ] } = { ( x , y ) R 2   | y = 2 3 x }
And unstable subspace:
E u = {eigenvector, for λ = 1 } = span { [ 1 , 1 ] } = { ( x , y ) R 2   | y = x }
Then the stable and unstable manifolds W s , W u are then tangential to E s , E u , at ( 1 , 2 ), But I am unsure on how to parameterise them, but they must depend and t. And l i m t W s ( t ) = ( 1 , 2 ), and l i m t W u ( t ) = ( 1 , 2 ).

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