  The study of dynamical tunnelling in a periodically driven anharmonic
potential probes the quantum-classical transition via the experimental control
of the effective Planck's constant for the system. In this paper we consider
the prospects for observing dynamical tunnelling with ultracold atoms in
magnetic microtraps on atom chips. We outline the driven anharmonic potentials
that are possible using standard magnetic traps, and find the Floquet spectrum
for one of these as a function of the potential strength, modulation, and
effective Planck's constant. We develop an integrable approximation to the
non-integrable Hamiltonian and find that it can explain the behaviour of the
tunnelling rate as a function of the effective Planck's constant in the regular
region of parameter space. In the chaotic region we compare our results with
the predictions of models that describe chaos-assisted tunnelling. Finally we
examine the practicality of performing these experiments in the laboratory with
Bose-Einstein condensates.
