Some Connections Between Individual And Collective Dynamics

Abstract

The concepts of chaos and complex dynamics have been studied exhaustively during several decades, not only due to the intrinsic richness of this theory, but also due to its many applications for modeling and solving real life problems. When studying the evolution in time of a dynamical system, if we assume that this dynamic is given by a defined function f which contains values in a space X, then for each x within X, its orbit under iteration x, f(x), f(2)(x), f(3)(x), ... describes how the 'point' x moves within X via the dynamics induced by f However, sometimes we need to know not only this individual dynamic, but also how a set of points moves in the space X (e.g., when studying migratory, demographic or epidemiological phenomena), which leads us to the analysis of collective dynamics problems. In this article we study some of the connections between individual and collective dynamics and, in this context, the fundamental questions we intend to answer are: What is the relationship between the dynamics of individual and collective movements? Does individual chaos or complex individual dynamics imply collective chaos or complex collective dynamics? And what about the reverse implication? Does it hold? Keywords KeyWords Plus:VALUED DISCRETE-SYSTEMS; CHAOS; TRANSITIVITY