Circular causality.

Abstract

The problem of disentangling complex dynamic systems is addressed, especially with a view to identifying those variables that take part in the essential qualitative behaviour of systems. The author presents a series of reflections about the methods of formalisation together with the principles that govern the global operation of systems. In particular, a section on circuits, nuclei, and circular causality and a rather detailed description of the analytic use of the generalised asynchronous logical description, together with a brief description of its synthetic use (OreverseO logic). Some basic rules are recalled, such as the fact that a positive circuit is a necessary condition of multistationarity. Also, the interest of considering as a model, rather than a well-defined set of differential equations, a variety of systems that differ from each other only by the values of constant terms is emphasised. All these systems have a common Jacobian matrix and for all of them phase space has exactly the same structure. It means that all can be partitioned in the same way as regards the signs of the eigenvalues and thus as regards the precise nature of any steady states that might be present. Which steady states are actually present, depends on the values of terms of order zero in the ordinary differential equations (ODEs), and it is easy to find for which values of these terms a given point in phase space is steady. Models can be synthesised first at the level of the circuits involved in the Jacobian matrix (that determines which types and numbers of steady states are consistent with the model), then only at the level of terms of order zero in the ODE's (that determines which of the steady states actually exist), hence the title 'Circular casuality'.