Changing parameters in a 3x3 Matrix trying to find the general solution.

matrices ordinary-differential-equations dynamical-systems

1,338

Solution 1

If you look at that matrix right, its determinant is obviously $(b-\lambda)(\lambda^2-a^2)$, which should make it a breeze to find eigenvalues and eigenvectors.

Solution 2

$X^{'}= \begin{pmatrix} 0&0&a \\ 0&b&0\\ a&0&0 \end{pmatrix}X$ , $X=\begin{pmatrix} x\\y\\z\end{pmatrix}$$$\begin{pmatrix} x'\\y'\\z'\end{pmatrix} =\begin{pmatrix} 0&0&a \\ 0&b&0\\ a&0&0 \end{pmatrix}\begin{pmatrix} x\\y\\z\end{pmatrix}$$then we have $$\begin{cases} x'=az & \text{} \\ y'=by& \text{} \\ z'=ax & \text{} \\ \end{cases}$$ w e have $y=e^{bt+d_1}$$$x'=az\to x''=az'\to x''=a^2x$$ easily find x then by substitution find z

1,338

Faust

Updated on March 05, 2020

Comments

Faust almost 3 years

Consider the system $$X^{'}= \begin{pmatrix} 0&0&a \\ 0&b&0\\ a&0&0 \end{pmatrix}X$$

depending on the two parameters a and b.

1) find the general solution of this system. \begin{pmatrix} -\lambda&0&a \\ 0&(b-\lambda)&0\\ a&0&-\lambda \end{pmatrix} $-\lambda(b-\lambda) (-\lambda)+ a(b-\lambda) (-\lambda) = -\lambda^{3} +(b+a) \lambda^{2}-ab\lambda $

$(\lambda^{2} -(b+a) \lambda+ab) (-\lambda)$

$(\lambda -[(b+a)/2])^{2} = -ab + [(a+b)^{2}/4]$

$\lambda_{1} = (-ab + [(a+b)/4])^{1/2}$ $\lambda_{2} = -(-ab + [(a+b)/4])^{1/2}$ $\lambda_{3} = 0$

im not sure how to force the eigenvalues out of this and subbing this mess into that equation for each eigenvalue is there an easier way to compute this?

EDIT as pointed out below this is a mess by expanding down the second column of the matrix we have $\lambda_{1} = b$ $\lambda_{2} = -a$ $\lambda_{3} = a$

Where $V_{1}=<0,1,0>$ $V_{2}=<1,0,(-1)>$ $V_{3}=<1,0,(1)>$

Why isnt this the general solution?

$C_{1} e^{bt} V_{1}$ + $C_{2} e^{-at} V_{2}$ + $C_{3} e^{at} V_{3}$

2) sketch the region in the ab plane where this system has different types of phase portraits.
Faust almost 10 years

sorry i don't really understand how u got all those things. or in general what that even says, perhaps its time for me to go to sleep...
Faust almost 10 years

im not sure how you got that but, that would mean $\lambda$ = b,a,-a Edit nvm im a moron
Gerry Myerson almost 10 years

Evaluate the determinant by expansion along the second row.
Gerry Myerson almost 10 years

@julien, $z'=ax$.
Julien almost 10 years

@GerryMyerson Yes, sure, I added a typo to a typo...Thanks.
Julien almost 10 years

Nice observation, Maisam. So doing this, we find necessary conditions on $x,y,z$. And we need to check that they work afterwards. +1.
Faust almost 10 years

i follow the train of logic to get to $x^{''}=a^{2}x$
M.H almost 10 years

@Faust7:use characteristics polynomial to solve it $(D^2-a^2)x=0$
Faust almost 10 years

Sorry i have never seen anything like this before i am not sure why your trying to find x?
Faust almost 10 years

are you talking about the vectors?
M.H almost 10 years

@Faust7:i offer look on this book differential equation and dynamical system an introduction to chaos of Morris sorry i dont have time to explain i must go university
Faust almost 10 years

haha, you have been more then helpful sir thank you very much for all your help!