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With different Feigenbaum constants δ, n for p · 3 ). 2 Pitchfork Bifurcation and the Doubling Transformation In this section, we show that the “Feigenbaum route” in Fig. 23 is generated by pitchfork bifurcations that relate the emergence of new branches to a universal law of functional composition. By introducing the doubling transformation T (which describes this law), we show that the Feigenbaum constants α and δ are indeed universal. They appear as the (negative inverse) value of the eigenfunction of T at x = 1 and as the only relevant eigenvalue of the linearized doubling operator, respectively.

21) is still lacking, we can obtain a unique solution if we specify the nature of the maximum of g(x) at x = 0 (for example quadratic) and require that g(x) is a smooth function. 73 . 0267057x6 − . . 502807876 . . This establishes the universality of α. 4 Pitchfork Bifurcation and the Doubling Transformation 43 Linearized Doubling Transformation and δ What can we say about the scaling along the r-axis? The values r = Rn for which a 2n cycle becomes superstable, are determined by the condition that x = 1/2 is an element of the supercycle (see eq.

Even if two points x and x differ only after their nth digit an , this difference becomes amplified under the action of σ, and their nth iterates σn (x) and σn (x ) already differ in the first digit because σn (x) = (0, an . ) where σ2 (x) = σ[σ(x)], etc. 2. The sequence of iterates σn (x0 ) has the same random properties as successive tosses of a coin. To see this, we attach to σn (x0 ) the symbol R or L depending on whether the iterate is contained in the right or left part of the unit interval.

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