03856414

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03856414 Gille, Jean-Charles W\c{e}grzyn, Stefan Vidal, Pierre Grammaires des mots cr\'eateurs et propri\'et\'es des syst\`emes \'evolutifs correspondants \'etudi\'ees par la transformation de Carson discr\`ete. Podstawy Sterowania 13, 271-278 (1983). 1983 Panstwowe Wydawnictwo Naukowe, Warszawa FR developmental systems creative word feedback evolution matrix discrete Carson transform Developmental systems (organisms) are considered, made up of cells subjected to elementary operations. The six elementary operations are: stagnation (a cell remains itself), transformation (a cell is differentiated into another), bifurcation (a cell is transformed into two cells with or without change of direction) and generation (a cell generates another cell with or without change of direction). The operations are grouped in a creative word which controls the development of the system. The creative word may belong to types of different structures. If the operations are considered as instructions, these structures are expressed by different formal-language grammars: word without feedback (tree-shaped graph, grammar $L\sb 1)$, word with feedback towards the first cell (grammar $L\sb 2)$, word with feedback towards any cell (grammar $L\sb 3)$. The growth of the system (i.e., the number of the cells of each type at the k-th stage of development) is obtained by rising the evolution matrix M of the system to the k-th power or by using the discrete Carson transform (or equivalently the z- transform). For systems without feedback (grammar $L\sb 1)$ the eigenvalues of M are zero and one, the growth occurs as a polynomial function of k. In the present paper it is shown that for systems with feedback towards the first cell (grammar $L\sb 2)$ the growth occurs as an exponential function of k: some eigenvalues of M exceed one, the transform of the number of cells at the k-th stage is obtained from a formula similar to the transfer function of a sampled-data servomechanism. In two subsequent papers (to appear in Int. J. Syst. Sci.) the authors investigate in what manner the rate of growth is controlled by the dominant eigenvalue of M, which depends essentially on the operations lying inside the feedback loop and which can be evaluated.