On synchronous behavior in complex nonlinear dynamical systems

The purpose of this dissertation is to study synchronous behavior of certain nonlinear dynamical systems by the method of contraction theory. Contraction theory provides an elegant way to analyze the behavior of certain non- linear systems. Under sometimes easy to check hypotheses, systems can be shown to have the incremental stability property that trajectories converge to each other. This work provides a self contained introduction to some of the basic concepts and results in contraction theory. As we will discuss later, contractivity is not a topological, but is instead a metric property: it depends on the norm being used in contraction theory and in fact an appropriate choice of norms is critical. One of the main contributions of this dissertation is to generalize some of the existing results in the literature which are based on L2 norms to results based on non L2 norms using some modern techniques from nonlinear functional analysis. The focus of the first main part of this dissertation is on the application of con- traction theory and graph theory to synchronization in complex interacting systems that can be modeled as an interconnected network of identical systems. We base our approach on contraction theory, using norms that are not induced by inner products. Such norms are the most appropriate in many applications, but proofs cannot rely upon Lyapunov-like linear matrix inequalities, and different techniques, such as the use of the Perron-Frobenious Theorem in the cases of L1 or L? norms, must be introduced. On the second main part of this work, using the method of contraction theory based on non L2 norms, spatial uniformity for the asymptotic behavior of the solutions of a reaction diffusion PDE with Neumann boundary conditions will be studied.