Quadratic Lyapunov Function and Exponential Dichotomy on Time Scales

Abstract

In this paper, we study the relationship between exponential dichotomy and quadratic Lyapunov function for the linear equation $x^∆ = A(t)x$ on time scales. Moreover, for the nonlinear perturbed equation $x^∆ = A(t)x + f(t, x)$ we give the instability of the zero solution when $f$ is sufficiently small.