We develop a qualitative method for understanding and representing phase space structures of complex systems and demonstrate the method with a program, MAPS --- Modeler and Analyzer for Phase Spaces, using deep domain knowledge of dynamical system theory. Given a dynamical system, the program generates a complete, high level symbolic description of the phase space structure sensible to human beings and manipulable by other programs. Using the phase space descriptions, we are developing a novel control synthesis strategy to automatically synthesize a controller for a nonlinear system in the phase space to achieve desired properties.