By Leipholz H.
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Extra info for Stabilitaetstheorie: eine Einfuehrung in die Stabilitaet dynamischer Systeme und fester Koerper
B) Generation of two new stable fixed points in f 2 via a pitchfork bifurcation. (The bifurcation diagram looks like a pitchfork, see p. 159) Figure 26 shows fr (x) together with its second iterate fr2 (x) for r > r1 . We note four properties of f 2 (the index r is dropped for convenience): 1. It has three extrema with f 2 = f [ f (x)] f (x) = 0 at x0 = 1/2, because f (1/2) = 0, and at x1, 2 = f −1 (1/2), because f [ f [ f −1 (1/2)]] = f (1/2) = 0. 2. A fixed point x∗ of f (x) is also a fixed point of f 2 (x) (and all higher iterates).
L fR∞ δ f + O[(δ f )2 ] . 35) We observe that, according to eqs. 35) becomes approximately: T n fR (x) ∼ = g(x) + (R − R∞ )Lng δ f (x) for n 1. 37) 4 44 Universal Behavior of Quadratic Maps Figure 30: Parametrization of r by n and µ (schematically), i. , rn = Rn,1 =(n, ˆ 1) and Rn = Rn,0 =(n, ˆ 0). This equation can be further simplified if we expand δ f (x) with respect to the eigenfunctions ϕν of Lg , Lg ϕv = λv ϕv ; δ f = ∑ c v ϕv ; v = 1, 2 . . 39) v and assume that only one of the eigenvalues λν is larger than unity, i.
If a fixed point x∗ becomes unstable with respect to f (x), it becomes also unstable with respect to f 2 (and all higher iterates) because | f (x∗ )| > 1 implies | f 2 (x∗ )| = | f [ f (x∗ )] f (x∗ )| = | f (x∗ ) <2 > 1. 4. For r > 3, the old fixed point x∗ in f 2 becomes unstable, and two new stable fixed points x¯1 , x¯2 are created by a pitchfork bifurcation (see Fig. 26b). The pair x¯1 , x¯2 of stable fixed points of f 2 is called an attractor of f (x) of period two because any sequence of iterates which starts in [0, 1] becomes attracted by x¯1 , x¯2 in an oscillating fashion as shown in Fig.
Stabilitaetstheorie: eine Einfuehrung in die Stabilitaet dynamischer Systeme und fester Koerper by Leipholz H.