Dynamical systems of interacting units: Information transport and higher order structures

Abstract

How can one model, characterize and analyse systems consisting of a large number of interacting units? This thesis addresses various related aspects of this issue using dynamical systems theory. The thesis is constituted by four papers presenting the main findings as well as an introductory part giving the context of the work.

In Paper I, we use evolutionary game theory to study a social dilemma where a group can obtain a collective benefit, but only if a threshold on the number of members willing to cooperate at a personal cost is met. The replicator dynamics of our model has interesting features, including catastrophic events. We also generalize the analysis to other n-player games.

In Paper II, we introduce several new concepts and principles for describing and studying hierarchicalmultilevel systems with interactions both between groups and among individual units. We illustrate these concepts through defining a class of systems called higher order cellular automata, and discuss new phenomena and patterns of behaviour found in these systems.

In Paper III, we introduce a local information measure for one-dimensional lattice systems in order to characterize the coherent structures emerging from the dynamics of higher order cellular automata as well as ordinary cellular automata. A further objective is to investigate to what extent information can be viewed as a local quantity in such systems. We demonstrate the applicability of our informationmeasure to these problems.

Paper IV is a continuation of Paper III. We show that local information is a locally conserved quantity in the important class of surjective cellular automata and provide bounds on the information transport.

The novel concepts and findings of the thesis may lead to an improved understanding of various phenomena found in systems of interacting units, and in particular systems with amultilevel dynamics.