Stability of a linear system with parameter

Question

The problem I am trying to deal with is a system of 3 linear equations with parameter, that are of the form

$x'=3y+2x$

$y'=3x+2z$

$z'=3h+2z$

From theorem we know that such a system only has an equilibrium at $(0,0,0)$ and we can determine its stability with eigen values of its Jacobian.

I am stumped when doing this as I have to solve a cubic that has the parameter $h$ involved. E.g [$\lambda^3-4\lambda^2+\lambda(h-9)+4(h-3)$]

How might I reasonably choose values of $h$ so that I can find the solutions to cubic.

Answer 1

Hint: Use the Hurwitz criterion (see page 3 of reference). For

$$\chi(\lambda) = a_3\lambda^3 + a_2\lambda^2 + a_1\lambda + a_0$$

For a cubic equation, you have to check if all the coefficients are positive and if

$$a_2a_1 > a_3a_0.$$

You can remember this as 'The product of the inner coefficients has to be larger than the product of the outer coefficients'. The resulting inequalities will constraint possible values of $h$.