Extracting and Representing ualitative Behaviors of Cornplex Systerns ' in Phase Spaces

We develop a qualitative method for understanding and representing phase space structures of complex systems. To demonstrate this method, a program has been constructed that understands qualitatively different regions of the phase spaces and represents and extracts geometric shape information about these regions, using deep domain knowledge of dynamical system theory. Given a dynamical system specified as a system of governing equations, the program applies a successive sequence of operations to incrementally extract the qualitative information and generates a complete, high level symbolic descr' tion of the phase space structure, through a combination of numerical, combinatorial, and geometric computations and -spatial reasoning techniques. The high level description is sensible to human beings and manipulable by other programs. We are currently applying the method to a dfficult engineering design domain in which controllers for complex systems are to be automatically synthesized to achieve desired properties, based on the knowledge of the "shapes" of the systems.