  Based on the holographic principle and the Barrow entropy, Barrow holographic
dark energy had been proposed. In order to analyze the stability and the
evolution of Barrow holographic dark energy, we, in this paper, apply the
dynamical analysis and statefinder methods to Barrow holographic dark energy
with different IR cutoff and interacting terms. In the case of using Hubble
horizon as IR cutoff with the interacting term
$Q=\frac{\lambda}{H}\rho_{m}\rho_{D}$, we find this model is stable and can be
used to describe the whole evolution of the universe when the energy transfers
from the pressureless matter to the Barrow holographic dark energy. When the
dynamical analysis method is applied to this stable model, an attractor
corresponding to an accelerated expansion epoch exists and this attractor can
behave as the cosmological constant. Furthermore, the coincidence problem can
be solved in this case. Then, after using the statefinder analysis method to
this model, we find this model can be discriminated from the standard
$\Lambda$CDM model. Finally, we have discussed the turning point of Hubble
diagram in Barrow holographic dark energy and find the turning point does not
exist in this model.
