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We consider a dynamical system containing a large number of elements which can be in different states. In order to define a hierarchical partition of the system, strongly coupled states are put together creating groups of states. To each group is associated a single global variable. For such a two level system, we study dynamical aspects, and particularly the jumping from a microscopic level towards a more macroscopic level. In this way, we obtain dynamical equations governing the time behavior of the global variables which are shown to be slow-time varying with respect to microscopic variables. Each group reaches an intra-group equilibrium and the fast-varying microscopic variables are replaced by their time averages. This process leads to an important reduction in the number of variables. We also study interactions between the different levels of organization, which can be of two types, top-bottom and bottom-top couplings. In general, intra-group equilibrium depend upon the values of the global variables. This is top-bottom coupling. In many cases, the parameters of the global dynamical equations also depend upon the intragroup equilibrium. This is bottom-top coupling. Then, we present an ecological application by studying a model of coupled individual and population levels.
Kevwords. Hierarchically organized systems; Individual and group dynamics: Topbottom and bottom-top couplings; coupled individual and population levels.