After having the Jacobian matrix, I want to show whether the eigenvalues of this matrix are less than one or not.
$ J=\left(\begin{array}{cccc} a & 0 & 0 & b \\ c & d & 0 & 0 \\ 0 & 0 & e & f \\ g & g & g & h \end{array}\right) $
(Here $a,b,...,h $ are real constants which all are non-zero)
After finding $\left|J-\lambda I\right|=0$ and simplifying one can get the characteristics polynomial as $ \lambda^4+(-h-e-d-a)\lambda^3+(ad+ae+ah-bg+de+dh+eh-fg)\lambda^2+(-ade-adh-aeh+afg-bcg+bdg+beg-deh+dfg)\lambda+head-adfg+cgbe-gdbe=0 $
I have tried to find the solutions of the polynomial, and have difficulty to show whether the values are less than one or not.
Is there any relevant theorem for showing that the eigenvalues (in absolute value) less than one or not (for stability of dynamical system)?
Thank you!
Without further assumptions you cannot conclude that the eigenvalues are less than one in absolute value. For example, if $b = c = f = g = 0$, then the eigenvalues are $a, d, e, h$.