BIPOLAR CONTROL (OR PARADOXICALLY UNILATERAL) OF IMBALANCED LIMIT-CYCLES AND STRANGE ATTRACTORS.: CHAOS AND HOMEOSTASIS

Abstract

The functioning of the mathematical model for the regulation of agonistic antagonistic couples (MRCAA) is first recalled: it intends to simulate normal and pathological states concerning some biological (im)balances, just as a control method allowing to reestablish the balances, if necessary. Using the MRCAA in the frame of AA networks, it was permitted to obtain strange attractors (SA). By approaching in that manner the problem of chaotic dynamics, we may understand that SA have important characteristics other than their aperiodicity. Their topology in the phase-space has equally to be considered. The position of a SA in the phase-space allows us to simulate the functioning of a biological system; if this does not correspond to the physiological position, we consider it as an imbalanced SA. Therefore, SA could take the place in biological modelling of limit-cycles, which were perhaps an approximation of these SA. Then, the problem of correction of these imbalances should be considered. Using the mathematical model of AA networks, it was possible to propose an outline as yet theoretical of this problem: how can we model imbalanced SA, then how can the balance be restored from a method already used in the correction of imbalanced limit-cycles? By this technique, SA have also became quasi-periodic attractors, but this was not the result which was particularly wished. This paper mainly concerns chaotic dynamics (CD) and the problems elicited by the control of corresponding systems. However, given that we will use a general model (or a model of function in the sense proposed by Rosen), i.e., the “model for the regulation of agonistic antagonistic couples” (MRAAC), it has seemed firstly necessary to recall the structure of this model and also to establish a comparison between the results of computer simulations with this model and the results of the control in concrete systems (bipolar or paradoxically unilateral therapies in bio-medicine).