  We exhibit a fundamental relationship between measures of dynamical and
structural stability of equilibriums, arising from real dynamical systems. We
show that dynamical stability, quantified via systems local response to
external perturbations, coincides with the minimal internal perturbation able
to destabilize the equilibrium. First, by reformulating a result of control
theory, we explain that harmonic external perturbations reflect the spectral
sensitivity of the Jacobian matrix at the equilibrium, with respect to constant
changes of its coefficients. However, for this equivalence to hold, imaginary
changes of the Jacobian's coefficients have to be allowed. The connection with
dynamical stability is thus lost for real dynamical systems. We show that this
issue can be avoided, thus recovering the fundamental link between dynamical
and structural stability, by considering stochastic noise as external and
internal perturbations. More precisely, we demonstrate that a system's local
response to white-noise perturbations directly reflects the intensity of
internal white noise that it can accommodate before asymptotic mean-square
stability of the equilibrium is lost.
