Systems of linear nonautonomous differential equations - Instability and eigenvalues with negative real part

Sammanfattning: For an autonomous system of linear differential equations we are able to determine stability and instability with classical criteria, by looking at the eigenvalues. If the system is stable, all the eigenvalues have negative real part and if the system is unstable, there exist at least one eigenvalue with positive real part. However, if it were to be nonautonomous, the criterion fails. There exist examples where the systems are stable, yet the eigenvalues have real part with different or positive signs. Also for the unstable systems there exist examples where the matrices can have eigenvalues with strictly negative real part. In this thesis we examine the instability of linear nonautonomous systems of differential equations, following the article of Josić and Rosenbaum $\cite{1}$. They discuss a unified method for constructing two dimensional examples which we'll review and attempt to generalize to higher dimensions.