A survey of incursive, hyperincursive and anticipative systems

Daniel M.Dubois

Centre for Hyperincursion and Anticipation in Ordered Systems

CHAOS asbl, Institute of Mathematics B37, University of Liège

Grande Traverse 12, B-4000 LIEGE 1, Belgium

Daniel.Dubois@ulg.ac.be

This paper is a survey of anticipative systems related to the incursive and hyperincursive computation of nonlinear systems and control of chaos. The words "incursivity" and "hyperincursivity" have been proposed by D. Dubois in the year 1992 for defining a new type of computation in which future states of systems are taken into account to compute the current present time states. The first example, developed since 1990, was the "Fractal Machine", a hyperincursive cellular automaton defined by a frame, a composition rule and a path in the frame. The simpler fractal machine is a serial fractal automaton where the future state of an automaton depends of itself at the preceding time and of the preceding automaton at the future time. This generates the Sierpinski fractal.

The second example was the incursive discrete Lotka-Volterra non-linear model of an autocatalytic reaction or of a predator-prey ecosystem. This gives rise to an orbital stability.

A third example was the suppression of chaos in the Verhulst map with an incursive control given by a model of itself.љ There is a paradox if an anticipatory system contains a model of itself, because the model of itself must include also the model of itself and so on until infinity.љ There is an infinity of embedded models in each other. Mathematically, there is a means to resolve this paradox. The goal or objective of such an anticipatory system is not explicitly imposed from outside the system like in control theory but is determined by the system itself. A fourth example was the flip-flop memory logical gate which can be represented by a hyperincursive system because, for some inputs, two different outputs exist.

A fifth example is the Hyperincursive Verhulst map, with multiple solutions, which theoretically defines an infinite chaotic memory.

All these systems deal with what is called "strong anticipation", because the future states are anticipated from these systems themselves and not from models of these systems. Systems which compute their current states from potential future states, anticipated from models of these systems, deal with what is called "weak anticipation". In anticipatory systems, the knowledge of future states cannot be always approximated by using predictions of future states: any natural or artificial systems can build their future. The weak and strong anticipations are related to what is called exo-anticipation and endo-anticipation. An exo-anticipation is an anticipation made by a system about external systems. This anticipation can be based on a model of the environment, for the prediction of the future events. An endo-anticipation is an anticipation built by a system by itself, embedded in the system. This anticipation is based on the system self-dynamics.

After the survey of these systems, it is shown how to control chaos with three anticipative controllers: the incursive controller, the model predictive controller, and the incursive model predictive controller. The incursive controller, as well as the incursive model predictive controller, finds by itself the setpoint which is the unstable equilibrium of the chaos map. Indeed, the incursive control does not use an explicit setpoint, but an implicit setpoint given by the unstable equilibrium state which is stabilized.

All the concepts developed in this paper could give new avenues of research and development in the field of nanosystems that deals with systems at nanoscale level self-organization exhibiting nonlinearities and chaos.