Qualitative Analysis of Complex Dynamical Systems

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The qualitative properties of complex dynamical systems are investigated. Such systems could be mathematically represented by large interconnected systems of differential equations, integral equations, partial differential equations, or a combination of these types of equations. The work will investigate the existence and stability of oscillations, the stability of the equilibria, and the input-output properties of such systems. These topics are clearly of great importance in theoretical studies and in application to specific technological problems. The methods will involve functional analysis techniques, transformations, results from the theory of integral manifolds, comparison theory for differential and integral equations and the theory of resolvent operators for integral equations. It is expected that the results of this research will provide new methods to investigate oscillatory behavior of complex feedback systems. They will also provide new methods of analyzing stability properties of feedback control systems for large and complex dynamical systems.