On the relationship between complex dynamics and complex geometrical structure
University of New South Wales, School of Mathematics
Luis da Costa
Ecole de Technologie Superieure (ETS) in Montreal (Quebec), Canada.
Last modified: April 25, 2006
Should dynamical properties of a system be directly mirrored in its geometric representation?
Is there a way to qualify a systems' geometrical structure as complex, once we know that the dynamics used to generate it is complex?
To have a way to approach these questions, the following case study is considered. First, we consider a probabilistic dynamics (a set of interval maps associated with probability distributions) that exhibits complex behaviour; i.e. under different sets of parameters the dynamics exhibits behaviour in all main dynamical regimes: periodic, chaotic with the so called "edge of chaos" in between. Then, we generate networks (graphs) out of the dynamics' parameters responsible for producing dynamical different regimes. We use a fairly simple method that resembles the Markov partition method but it also allows dealing with non-Markovian dynamical processes.
Once the graphs are generated, we would like to know if it is possible to structurally distinguish between them. In this sense, we present several measures of structural complexity (in networks and of the dynamics itself) and discuss their relevance to our problem. Here we stress the importance of considering a non extensive statistical framework such as the one introduced in Tsallis,1998.Then we test to see if these measures can be used as classifiers of the generated structures.
In the end we come back to the initial question and try to establish correspondences between complex dynamics and complex geometrical structure by means of the complexity measures introduced and discussed in the previous part and as such of a mainly ergodic theoretic analysis of the dynamics.