Prev: Energy of whole atom radiates at fusion/fission
Next: The Schwartz space
From: Arno Narque on 25 Jul 2010 18:39
Hello,

I have got a quite urgent questions regarding stability analysis of a
three dimensional of differential equations. I don't quite understand
how this works in that case or more which method the author uses.

I have got the Jacobi matrix of the dynamic system which is linearized
around a fixed point (steady state):

r+wh_k & wh_q−c_q & 0 \\
−qr_k − qr_hh_k & −qr_hh_q & 0 \\
.. & . & p\\

The author of the paper says that the first two variables in the third
line are positive and real (.) but can be omitted since they are not
involved in the calculation of the eigenvalues. Ok now I calculate the
eigenvalues:

det(M−λI)== [r + wh_k − λ_1][−qr_hh_q − λ_2][p − λ_3] − [wh_q −
c_q][−qr_k − qr_hh_k][p − λ_3] = 0

One of the eigenvalues is obvious: p = λ_3 where by definition p>0.
The author now says that for the system to be stable, he says that is a
sufficient condition, it must be shown that the eigen values λ_1 and
λ_2 have the opposite sign, therefore they are negative. To show this,
the aouthor says, it is enough to show that the first determinant of
the second order of the jacobi matrix is positive. Therefore that

det | r+wh_k & wh_q−c_q \\ −qr_k − qr_hh_k & −qr_hh_q \\| >0

I don't get on which method this stability analysis is based. Is it
some sort of the Routh-Hurwitz criterion?
The paper where I found this problem is: Optimal Taxation of Capital
Income in General Equilibrium with Infinite Lives:
http://ideas.repec.org/a/ecm/emetrp/v54y1986i3p607-22.html . This
question refers to the appendix.

It would be so great if somebody of you could help me out!!! I'd be so
grateful! It would be also cool if somebody could suggest me something
like "stability analysis for dummies ;-)" Thank you in advance,

Yours,

Arno




From: Arno Narque on 25 Jul 2010 18:41
correction: the first determinant of the second order has to be
negative to show that the other eigenvalues are negative.

From: Arno Narque on 26 Jul 2010 05:13
push! please, i would so appreciate your help!


From: Frederick Williams on 26 Jul 2010 10:46
Arno Narque wrote:
>
> push! please, i would so appreciate your help!

This sort of thing:

det(M−λI)== [r + wh_k − λ_1][−qr_hh_q − λ_2][p − λ_3] −
[wh_q −
c_q][−qr_k − qr_hh_k][p − λ_3] = 0

is unreadable.

--
I can't go on, I'll go on.
From: Arno Narque on 26 Jul 2010 11:30
On 2010-07-26 16:46:20 +0200, Frederick Williams said:

> Arno Narque wrote:
>>
>> push! please, i would so appreciate your help!
>
> This sort of thing:
>
> det(M−λI)== [r + wh_k − λ_1][−qr_hh_q − λ_2][p − λ_3] −
> [wh_q −
> c_q][−qr_k − qr_hh_k][p − λ_3] = 0
>
> is unreadable.

really? in my post this is displayed perfectly!

 |  Next  |  Last
Pages: 1 2
Prev: Energy of whole atom radiates at fusion/fission
Next: The Schwartz space