Past exam of the mathematics course of the University of Cambridge / 2018 / iii / Paper 331 / 4 / b / i

A normal mode $x\propto e^{-i\omega t}$ has eigenvalue $\lambda =-i\omega$ of $A$. Setting $d_{\omega }=\omega -{\omega }_1$, the characteristic equation isThusOne branch then has positive imaginary frequency and hence positive exponential growth rate. If $4ac<b^2$, both frequencies are real and the distinct-eigenvalue system has bounded oscillatory motion. At equality there is no positive exponential growth rate, but a nontrivial Jordan block can cause algebraic growth, so exponential neutrality must not be confused with Lyapunov stability. The parameter $d$ listed in the PDF does not appear in the printed matrix and has no effect on this calculation.