Boundedness and Other Theories for Complex Systems

Introduction

We have developed the concept of boundedness for explaining in a self-consistent way the properties of complex systems. The goal of the present chapter is to compare the concept of boundedness with two established approaches to the subject-matter: self-organized criticality and deterministic chaos. The purpose is not to make a critical review but rather to demonstrate the fundamental differences between each of them and the concept of boundedness. By means of substantiating this problematic in such a way we hope to demonstrate not only the merits of our approach, but to focus on the methodological importance of the general frame through which we view an enormous class of empirical observations. Such a general frame offers unique choice that combines general view and logic and turns out crucial for substantiating an approach with far going consequences. The high non-triviality of the matter is induced by the range of its application: it encompasses an enormous diversity of phenomena and systems each of which belongs to a different branch of science. It is believed that in the different scientific domains different rules and laws operate so that any systematic approach to them starts with the question: is there a general interdisciplinary rule which generates that peculiar behavior? If the answer is positive then it implies a coexistence of specific and universal properties.

Our answer to this question is affirmative: we assert that there exists a general approach based on the concept of boundedness. The theoretical background of this concept does not allow it to be considered as a law in its traditional understanding. It assumes an operational protocol. The fundamental difference between the idea of operational protocol and the traditional understanding of a law is that the former neither specifies the environment nor requires its exact reoccurrence along with the reoccurrence of a given phenomenon. To remind, the traditional understanding of a law is set on establishing certain, usually quantitative, relations among a number of variables so that the relation to be invariant on reoccurrence of the event. These considerations entail the strong pre-supposition that the local environment of the event reoccurs also with the reoccurrence of the event. Instead, we assume that there exists certain quantitative relations which stay invariant even in an ever-changing environment. The difference with the traditional approach is that our assertion drives us closer to the notion of homeostasis than to the idea of thermodynamical equilibrium. It should be highlighted that we understand the notion of homeostasis in its most general meaning: it is the property of a system, either open or closed, to regulate its internal environment and tends to maintain a stable, constant condition. Thereby, we have made two fundamental steps: (i) the first one consists of imposing the notion of homeostasis, borrowed from biology, in explanation of the behavior of physical systems which are supposed to be sufficiently described by the idea of thermodynamical equilibrium. To remind that thermodynamical equilibrium is the state associated with the maximum entropy and which is a global attractor for every initial condition; on the contrary, the homeostasis is self-organized pattern which is robust to environmental fluctuations in a certain domain of control parameters; thus it is neither the state of maximum entropy nor its properties are explicitly related to the notion of entropy; moreover it is released from the requirement to be a global attractor and thus opens the door to “adaptation” of homeostasis on changing domains of control parameters. (ii) the second step consists of adopting the idea that the matter organization has hierarchical structure which goes in both directions: bottom up and top down so that there exists general operational protocol that governs it.