Stiff ODE

Master J

A stiff ODE is defined as one for which the magnitude of the maximum eigenvalue of its Jacobian is much greater than that of the mininmum.

It is the real part of the eigenvalue which controls the error in an approximation when a numerical scheme is used to solve the ODE. If it is negative, the error decays away and the approximation approaches the true value for higher iterations.

My question is, is a positive real part also indicitive of a stiff ODE (in the definiton above)? If the error doesn't decay but grows instead, and some components dominate after a certain time, is this still a stiff ODE?