By Sternberg S.
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Extra info for Dynamical systems (Math 118)
They are initially attracting, and become superattracting at sn > bn and become unstable past the next bifurcation value bn+1 > sn when the period 2n points appear. 569946 The values of the bn are obtained by numerical experiment. We shall describe a method for computing the sn using Newton’s method. 2. LOCAL BIFURCATIONS. that this is still just the beginning of the story. 83. We shall come back to all of these points, but first discuss theoretical problems associated to bifurcations. 2 Local bifurcations.
Not restricted to the logistic family. 6692.... denote Feigenbaum’s constant, then the superstable values sr in the period doubling scenario satisfy sr = s∞ − Bδ −r + o(δ −r ) where s∞ and B depend on the specifics of the family, but δ applies to a large class of such families. There is another “universal” parameter in the story. Suppose that our family fµ consists of maps with a single maximum, Xm , so that Xm must be one of the points on any superstable periodic orbit. ) Let dr denote the difference between Xm an the next nearest point on the superstable 2r orbit; more precisely, define r−1 dr = fs2r (Xm ) − Xm .
Finally, since Lµ is decreasing on 32 CHAPTER 2. BIFURCATIONS [1 − µ1 , 1], any point in [1 − µ1 , 1) is mapped into (0, 1 − µ1 ] and so converges to the non-zero fixed point. In short, every point in (0, 1) is in the basin of attraction of the non-zero fixed point and (except for the points µ1 and the fixed point itself) eventually converge toward it in a “spiral” fashion. 4 µ = 3. Much of the analysis of the preceding case applies here. 2) now has a (double) root. But this root is 32 = 1 − µ1 .