Stability and instability for dynamic equations on time scales

In this paper we examine the stability and instability of the equilibrium solution x = 0 to the first-order system of dynamic equations xΔ=f(t,x),t≥t0,x∈D⊂Rn,where t is from a so-called time scale T with t0 ∈ Tand D is a compact set. Our methods involve the existence of a positive definite Liapunov function V, such that its delta-derivative VΔ satisfies certain integral, definite or semidefinite sign properties. Finally, we use Liapunov functions to develop an invariance principle regarding solutions to the above dynamic equation.