  We prove that on a K\"ahler manifold admitting an extremal metric $\omega$
and for any K\"ahler potential $\varphi_0$ close to $\omega$, the Calabi flow
starting at $\varphi_0$ exists for all time and the modified Calabi flow
starting at $\varphi_0$ will always be close to $\omega$. Furthermore, when the
initial data is invariant under the maximal compact subgroup of the identity
component of the reduced automorphism group, the modified Calabi flow converges
to an extremal metric near $\omega$ exponentially fast.
