In a Hierarchical Evolutive System, a complex object emerging as the colimit of a pattern may later on take its own identity, independent from the initial pattern, as long as the change of its internal organization is progressive enough for the unity of the component to be temporally recognized on this organization. For instance, the components of a cell change in time though the cell remains 'the same'. The stability span measures the rate of change.

Let N be a component of level n+1 in a Hierarchical System. At time t, its state Nt is the colimit of at least one pattern P of linked objects of level n of the state-category Kt . At a later time t', the component and the pattern acquire new states Nt' and Pt'. In some cases Nt' is still a colimit of Pt', but not always: from t to t', a number of components of P can be suppressed, or at least can break their links to N (a member of an association can resign); conversely new components can be associated to N (new members) and replace other components. However during some period there will exist a representative sub-pattern Qt of P whose successive states still represent a decomposition of level n of the corresponding states of N (shortly, Qt 'remains' a decomposition of N). For instance, for a population of proteins this period is related to its half-life.

The stability span of N at t is the largest real dt such that there exists a decomposition Qt of level n of Nt in the category Kt whose successive states Qt', for t' between t and t+dt, remain decompositions of the states Nt' of N at t'.

If the stability span is long enough, the change of N is gradual and ensures a temporal continuity of its internal organization allowing to recognize that the object is the same. To measure the change in a decomposition P of N from t to a much later time s, we have to compare the new state Ps of P to a decomposition R of Ns of level n: both are temporally connected by a sequence of patterns, each representing a decomposition of states of N between t and s, with two successive patterns being deduced one from the other while remaining decompositions of N during a whole stability span. 

 

Complex Identity

In particular, let us suppose that N has emerged at t as the colimit of P (say, in a complexification process). Up to t+dt, the evolutions of N and of P are well correlated, but later on they may totally differ (e.g., P can be entirely suppressed). We say that N has taken its own n-complex identity, independent from P; its successive states have gradually changing decompositions of level n which are only preserved during a stability span.

The variation of the stability span of a component gives information on its rate of change. The spans are long if the system remains stable, while they shorten during development or recession phases. For instance, the decrease of the stability spans is one of the important signs in the theory of aging of an organism by a 'cascade of de-resynchronizations' proposed in (Ehresmann & Vanbremeersch 1993).