Hierarchical behavior in fit dynamical systems (Q1069883)

One considers a class of differentiable dynamical systems that fulfill a criterion of adaptation, or being fit. This criterion is found to be sufficient for the existence of hierarchical behavior in these systems, once they attain a certain degree of complexity (that means the number of independent state variables required to describe the phenotype). Commutative diagram methods are used to show that under reasonable conditions the hierarchical behavior takes the form of a closed dynamics of aggregate variables. This dynamics characterizes the system functions as a whole. Existence conditions for the aggregate dynamics are obtained, together with their differential form and their connections to the original system description. The results indicate that evolutionary processes are likely to produce systems that are hierarchically organized in terms of their function as well as their structure.