By Roberto Serra

This quantity describes our highbrow direction from the physics of complicated sys tems to the technology of man-made cognitive structures. It used to be intriguing to find that the various innovations and strategies which reach describing the self organizing phenomena of the actual global are appropriate additionally for comprehend ing cognitive techniques. numerous nonlinear physicists have felt the fascination of such discovery in recent times. during this quantity, we are going to restrict our dialogue to synthetic cognitive structures, with no trying to version both the cognitive behaviour or the fearful constitution of people or animals. at the one hand, such man made platforms are very important in step with se; nevertheless, it may be anticipated that their research will make clear a few common ideas that are appropriate additionally to organic cognitive platforms. the most objective of this quantity is to teach that nonlinear dynamical structures have a number of homes which lead them to quite appealing for achieving many of the targets of synthetic intelligence. the keenness which was once pointed out above needs to besides the fact that be certified by way of a serious attention of the constraints of the dynamical platforms technique. realizing cognitive tactics is a major medical problem, and the achievements reached to this point enable no unmarried way to declare that it's the in simple terms legitimate one. particularly, the procedure established upon nonlinear dynamical structures, that's our major subject, continues to be in an early level of development.

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23) UI,h} We have seen that, for one-dimensional systems, the contraction of the segments provides an indication on the stability of the attractors and, at the same time, on the type of dependence upon the initial conditions. We will now attempt to extend this result to the two-dimensional case in the simplest possible manner. Later on, however, we shall see that this extension is only a partial one which needs to be further studied. We will now determine the conditions for a contraction of the areas in the plane.

This may be immediately verified, as Fig. 2a shows, using a graphical representation of the results of successive iterations starting from an initial condition x(O) f:. O. In this interval of values for A, the origin is asymptotically stable for all possible initial conditions with 0 ~ x(O) ~ 1. When A > 1/2, however, there are two unstable fixed points, one of which being the origin. This can be verified graphically. ,. Y=X / / / lr-~\===71 / ---/--/ / O~-~-L~L--_ x(O) X (a) 1/2 x'(O) 1 xli) (b) fig.

1 o z Fig. 10. Graphical representation of a linear transformation From this transformation the unit square OMIMM2 (see Fig. e. by lao - /3-y1 Thus, if lao - /3-y1 < 1 the areas contract; they dilatate in the opposite case. Let us now consider a specific example of a two-dimensional map: the socalled "Henon map" (Schuster, 1984). This map may be considered as the result of a refinement of the logistic map, when the model takes into account the fact that the linear term (the generation term) must refer to the preceding time-lapse.