By H Nagashima; Y Baba

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**Additional resources for Introduction to Chaos: Physics and Mathematics of Chaotic Phenomena**

**Example text**

Let a = { I,},? I denote the corresponding open covering. It may happen that not all of I , are required to cover the interval I and it may be covered more efficiently without using all of the members of (In)n21. ] covering the interval I . ) Given a map f and a set E , the inverse image of E by f is the set of points in I that are mapped to E under f and denoted by f - ' E (= { x l f ( x ) E E } ) . The set f-'E is an open set when f is continuous and E is an open set. 9). 9. The inverse image of the tent map.

R )with R < 1 . It asymptotically approaches x = 0. 8. 6. An orbit of the logistic map L R ( x )with 1 < R < 3. r = 1 - f . 3. = I - f . The point xo is unstable since If’(xo)l = R > 1 while X I is stable. 6 shows that the orbit approaches X I . 3 ( = R I )< R < 1 f i (=R2). The inequality ILX(x)l > 1 implies that the fixed point ,XI is no longer stable. 7. This is the pitchfork bifurcation and the bifurcation point is R = 3. + are obtained by solving the equation L i ( x ) = x giving the fixed point of the twice-iterated map L i ( x ) .

14(b) shows the topological entropy of LR (the smooth curve) and the Lyapunov numberI4 (the spiky curve), both computed numerically, for the same range of the parameter R. ) By comparing these two figures, one finds that ( 1 ) h ( f ) monotonically increases after R = R , = 3 . 5 7 . . (the bifurcation changes from 2"-type period to 2"(2m 1)-type period at this value of R ) , where the chaotic behaviour takes place for the first time, while it is flat at the windows and (2) the Lyapunov number is negative at the windows.