On the general problem of stability for impulsive differential equations

Criteria for stability, asymptotical stability and instability of the nontrivial solutions of the impulsive system dx dt = f (t,x), t = θi (x), Δx|t=θi (x) = Ii (x), i ∈ N = {1, 2, . . .}, where Δx|t=θ := x(θ+)−x(θ), x(θ+) = limt→θ+ x(t) are obtained by Lyapunov’s second method. The construction of a reduced system in the neighbourhood of a nontrivial solution is a central auxiliary result of the paper.  2003 Elsevier Inc. All rights reserved.