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By Stein M.R., Dennis R.K. (eds.)

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At this point, the logistic map is surjective on the interval I = [-2,2]: Every point y E I is the image of two different points, x1,xz E I . Moreover, I is then an invariant set since f (1)= I . This is a one-to-one transformation between I and itself, FULLY DEVELOPED CHAOS IN THE LOGISTIC MAP 33 which is a diffeomorphism everywhere except at the endpoints x = f 2 , where the ) not differentiable. 25) a piecewise linear map known as the tent map. 5 1 -. 5 2 xn Fig. 25). 9 shows that the graph of the tent map is extremely similar to that of the logistic map (Fig.

The changes that are allowed are limited by topological arguments. Each different sequence of basis sets describing the transition from the laminar to the hyperbolic limit describes a differentroute to chaos. Each different route to chaos is a different path in a forcing diagram, shown in Fig. 8. 12 lNTRODUCTlON During this transition the underlying branched manifold is robust: It generally does not change. Large changes in control parameter values can cause changes in the underlying branched manifold.

At a = a2 = there is another period-doubling bifurcation where the period-2 orbit gives place to a period-4 orbit. The period-doublingbifurcations occurring at a = al and a = a2 are the first two members of an infinite series, known as the period-doubling cascade, in which an orbit of period 2n is created for every integer n. The bifurcation at a = a3 leading to a period-8 orbit is easily seen in the bifurcation diagram of Fig. 3, the one at a = a4 is hardly visible, and the following ones are completely indiscernible to the naked eye.

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