Contraction and partial contraction : a study of synchronization in nonlinear networks

Abstract

This thesis focuses on the study of collective dynamic behaviors, especially the spontaneous synchronization behavior, of nonlinear networked systems. We derives a body of new results, based on contraction and partial contraction analysis. Contraction is a property regarding the convergence between two arbitrary system trajectories. A nonlinear dynamic system is called contracting if initial conditions or temporary disturbances are forgotten exponentially fast. Partial contraction, introduced in this thesis, is a straightforward but more general application of contraction. It extends contraction analysis to include convergence to behaviors or to specific properties (such as equality of state components, or convergence to a manifold). Contraction and partial contraction provide powerful analysis tools to investigate the stability of large-scale complex systems. For diffusively coupled nonlinear systems, for instance, a general synchronization condition can be derived which connects synchronization rate to net- work structure explicitly. The results are applied to construct flocking or schooling models by extending to coupled networks with switching topology. We further study the networked systems with different kinds of group leaders, one specifying global orientation (power leader), another holding target dynamics (knowledge leader). In a knowledge-based leader-followers network, the followers obtain dynamics information from the leader through adaptive learning. We also study distributed networks with non-negligible time-delays by using simplified wave variables and other contraction-oriented analysis. Conditions for contraction to be preserved regardless of the explicit values of the time-delays are derived.

(cont.) Synchronization behavior is shown to be robust if the protocol is linear. Finally, we study the construction of spike-based neural network models, and the development of simple mechanisms for fast inhibition and de-synchronization.