Stability for time varying linear dynamic systems on time scales

We study conditions under which the solutions of a time varying linear dynamic system of the form xΔ(t)=A(t)x(t) are stable on certain time scales. We give sufficient conditions for various types of stability, including Lyapunov-type stability criteria and eigenvalue conditions on “slowly varying'' systems that ensure exponential stability. Finally, perturbations of the unforced system are investigated, and an instability criterion is also developed.