By P. Auger, George J. Klir
The most basic and effective methods of conceptualizing complicated platforms is to prepare them hierarchically. A hierarchically geared up method is represented through a community of interconnected subsystems, each one of which has its personal community of subsystems, etc, till a few common subsystems are reached that aren't extra decomposed. This unique and critical e-book proposes a common mathematical concept of a hierarchical procedure and exhibits the way it will be utilized to very assorted themes resembling physics (Hamiltonian systems), biology (coupling the molecular and the mobile levels), ecology (coupling the person, inhabitants and ecosystem), and economics (coupling the sectoral, nearby and nationwide levels). the 1st try and enhance 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 average. a lot of this paintings has been pioneered through the writer, and he provides a extra thorough paintings so that it will have an immense impression on moving present pondering within the sciences of the ordinary to a extra holistic foundation during which a number of degrees of hierarchically equipped structures are considered in an built-in model.
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Additional resources for Dynamics and Thermodynamics in Hierarchically Organized Systems. Applications in Physics, Biology and Economics
Example text
21) is composed of A χ N equations. For instance, consider a g system Σ with 5 groups with 5 states inside each group. 21). g) Inter-level couplings, bottom-top and top-bottom interactions Jumping from one level to another leads to an important reduction of the number of equations. 42). Thus, a small arbitrary change in the frequencies rates becoming a ν ? + Δ ν ? 41). At the group level, each group is treated like a state and one obtains a collective graph governed by the group transition matrix A .
Now, we are going to discuss the first condition. a Figure 3 showed an example of converging trajectories (with several stable steady state points) and fig. 4 showed that each of these stable steady state points is surrounded by an area of convergence. Figure 5 showed hyperspheres in dotted lines centered on the steady state point in different situations. 8) is satisfied. 10) become more and more important. Thus, assuming the knowledge of the contour map of an E<* function, (resulting for instance of computer simulations), for each isolated group α and for each value of the group variables V as shown on fig.
N + n exp(A (t-τ)) e dt. no are the initial conditions. If all the eigenvalues of the matrix A are different, the behavior of the solution depends on the maximum eigenvalue x of this matrix. If the real part of x is negative, the m m solution evolves towards a steady state point - A * e. On the contrary, if g the real part of x is positive, o(t) grows exponentially with time. 21) is composed of A χ N equations. For instance, consider a g system Σ with 5 groups with 5 states inside each group. 21).