By P. Auger, George J. Klir
Essentially the most primary and effective methods of conceptualizing complicated structures is to arrange them hierarchically. A hierarchically equipped procedure is represented by way of a community of interconnected subsystems, every one of which has its personal community of subsystems, and so forth, until eventually a few undemanding subsystems are reached that aren't extra decomposed. This unique and demanding ebook proposes a normal mathematical concept of a hierarchical approach and exhibits the way it might be utilized to very assorted subject matters corresponding to physics (Hamiltonian systems), biology (coupling the molecular and the mobile levels), ecology (coupling the person, inhabitants and ecosystem), and economics (coupling the sectoral, local and nationwide levels). the 1st try to advance mathematical frameworks for conceptualizing such platforms in basic terms seemed within the early Seventies, and it took one other decade ahead of those mathematical frameworks have been utilized to varied particular contexts of the sciences of the normal. a lot of this paintings has been pioneered by way of the writer, and he provides a extra thorough paintings so as to have a huge influence on moving present considering within the sciences of the typical to a extra holistic foundation during which a number of degrees of hierarchically geared up platforms are seen in an built-in type.
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Extra resources for Dynamics and Thermodynamics in Hierarchically Organized Systems. Applications in Physics, Biology and Economics
5). This E function must be regarded as a particular V<* global variable. Thus, in this case a a q corresponds to one local minimum of a global function Ε which is a a σ function E ( n p of the state variables n". The E function characterizes a a the intra-group dynamics. This function is supposed to own local minima for each of the stable steady state points with coordinates q? so that the trajectories are always going downhill. Figure 4 shows an example of constant Ε° contour map. The trajectories are going towards the local minimum points corresponding to local smaller values of the function.
Here, we describe such a model in a formalism quite similar to P. Delattre s but we consider particular systems which exhibit a hierarchical structure. In this way, we consider partitions of the system into groups and we define hierarchical partitions. Then, for a two level system, we show how it is possible to obtain weakly coupled systems of equations governing respectively the collective and the individual dynamics. We study particularly two aspects, how to jump from one level to the other and how a level can influence the others?
Only inter-group dynamics is able to make vary n . a a One must notice that the condition ( 2 . 3 3 ) is not necessary to obtain a time hierarchy accompanying the hierarchy in the transformations of elements. Indeed, we are going to assume that the state variables whose dynamics is governed by strong intra-group interactions, see ( 2 . 3 6 ) . e. for each set of group variables, an intra-group equilibrium. This intra-group equilibrium can correspond to a local minimum of an E function. Instead of the space (nj, n^ a n ^ ) , one can use the space ( n n ^ a N<*-1 n a n ( * ^ t0 r a w t * ^ ie a contour map.
Dynamics and Thermodynamics in Hierarchically Organized Systems. Applications in Physics, Biology and Economics by P. Auger, George J. Klir