Autonomous natural systems, such as biological, social or neural systems, are open in the sense that they have exchanges with their surroundings, and their configuration changes in time, with destruction or rejection of some components, and emergence of new elements, either taken from the outside or internally generated. Thus their dynamical study cannot be confined to that of transformations on a fixed space, whatever be its dimension.

The Evolutive Systems (or ES) introduced in (Ehresmann and Vanbremeersch, 1987) model these systems. They describe the successive configurations of the system at each date of its timescale, and the transformations between them.

The configuration of the system at a time *t* of its timescale is represented by a category, say K* _{t}*, which describes the state of the system at

The dynamics is characterized by the intrinsic change of the configuration, and not by the motions of its components as seen by an external observer; these motions are taken into account only through their internal consequences, e.g. information or energy transfers (chemical or metabolic reactions).

The change from *t* to *t' *> *t *is modelled by a partial functor *k*(*t,t'*) from the state-category K* _{t}* at

The transition specifies what the components and links existing at *t* have become at *t'*, as we could indicate on two successive photos of an organism how a particular cell has changed. Formally, if an object N* _{t} *of K

To ensure that the successive states of an element, say N* _{t}*, are uniquely defined, we assume that the transitions are

1. If N* _{t}* has a new state N

2. Conversely, if N* _{t}* has a state N

Formally, an E*volutive System* (or ES) is defined by the following data:

1. its *timescale* which is a (finite or infinite) subset of the real numbers,

2. for each date *t *of its timescale, its *state-category* K_{t},

3. for each *t'* > *t*, the *transition* *k(t,t') *which is a partial functor from K_{t}** **to** **K* _{t'}* , these transitions being transitive in the above sense.

In classical models, a component of a system, say a cell in an organism, is supposed to remain 'the same' at the various times. On the contrary, in an ES, the cell as such is not represented by a unique object, but by the sequence of its successive states. Formally, a* component* N of the ES is a maximal sequence (N* _{t}*)