(untitled) (Communications in Mathematical Research, 2021)
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.