This thesis reports theoretical and experimental investigation of some applications of the atom-optical quantum kicked rotor (AOQKR) using a Bose-Einstein condensate (BEC) of 87Rb atoms exposed to optical standing wave. The first project theoretically demonstrated that quantum accelerator modes (QAMs) induced by BEC subjected to an accelerated standing wave is reproduced by a phase-modulated optical lattice. We studied the stability of QAMs in phase diagram composed of area of stable islands to resemble the Arnold Tongue structure. This tool allows us to search stable transporting islands subjected to vary sequence of modulated phases, in which novel "accelerator mode" may be revealed. The second part of this thesis devoted to experimentally and theoretically testify the feasibility of using the overlap of two quantum states, subjected to two slightly-different Hamiltonians, to identify quantum chaos from the regular motion. We focused the last project on realization of quantum ratchets in momentum space by utilizing multiple-component initial state; we discussed some potential applications on phase detection and quantum walk due to the property of sensitive phase and relative high transport efficiency of the ratchets. We concluded the thesis by a chapter introducing an algorithm (split-step Fourier method) to solve time-dependent Schrodinger equation numerically. This generic algorithm overcomes the barrier of Raman-Nath limitation and thus allows us to simulate models for ratchet, fidelity, quantum walk etc.