For linear systems, using eigenvalues to judge the stability is a better way. For nonlinear systems, we usually judge the local stability of the systems near equalibriums by linearizing them. But this method is not always valid. For example, if the eigenvalues of the linearizing system are zeros or imaginary, we can not tell if this equalibrium is stable or unstable because the nonlinear system cannot be approximated by the linearizing system. In this case, we have to use Lyapunav functions.