“Homeostasic” control of imbalanced strangeattractors with or without asking to a change in chaotic dynamics

Abstract

Previously, we showed how the problem of the strange attractors (SA) did not only concern the nature of their dynamics and how the aim of the control did not depend only on the fact that such a dynamics was considered as favorable or wrong for the functioning of the system modeled by these SA. The topology in the phase space of a SA is as much important, because the ergodic theory allows us to take inot account the mean values of the variable trajectories in this SA. As far as biomathematical models are concerned, these mean values may correspond either to physiological values or to pathological values. So, control should in certain cases not try to make disappear (or reestablish) a chaotic dynamics (CD), but to displace the SA in order that the mean values would return to a physiological state. This paper reports a theoretical study of such type of SA control with a model so-called “model for the regulation of agonistic antagonistic (AA) couples” or “AA networks”, which include equations giving the values of control variables. Then it would be possible to reestablish the balance between the variables with or without making disappear the CD of the attractor. In the last case, the control variables have themselves a CD.