  Controlling the translational motion of cold atoms using optical lattice
potentials is of both theoretical and experimental interest. By designing two
on-resonance time sequences of kicking optical lattice potentials, a novel
connection between two paradigms of nonlinear mapping systems, i.e., the kicked
rotor model and the kicked Harper model, is established. In particular, it is
shown that Hofstadter's butterfly quasi-energy spectrum in periodically driven
quantum systems may soon be realized experimentally, with the effective Planck
constant tunable by varying the time delay between two sequences of control
fields. Extensions of this study are also discussed. The results are intended
to open up a new generation of cold-atom experiments of quantum nonlinear
dynamics
