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In the following, we shall consider only the case r = 1, which is representative for r > 1/2. What can we say about a sequence of iterates if there are no stable fixed points? First of all we notice that points which are close together, become more and more separated during the first iterations, as shown in Fig. 16. If we plot the nth iterate ∆n (x), we see from Fig. 16 d n that again it is piecewise linear and has the slope dx ∆ (x) = 2n , except for the countable −n n set of points j · 2 where j = 0, 1 .

From Fig. 6) 22 3 Piecewise Linear Maps and Deterministic Chaos Figure 12: Definition of the Liapunov exponent. which, in the limits ε → 0 and N → ∞, leads to the correct formal expression for λ(x0 ): 1 f N (x0 + ε) − f N (x0 ) log N→∞ ε→0 N ε 1 d f N (x0 ) . 7) This means that eλ(x0 ) is the average factor by which the distance between closely adjacent points becomes stretched after one iteration. The Liapunov exponent also measures the average loss of information (about the position of a point in [0, 1]) after one iteration.

Uniformly covers the interval [0, 1], and the system is ergodic. As in the case of the Liapunov exponent, we will later study invariant density for more complicated maps and show that it is not always a constant. 21) is defined by 1 N−1 ∑ xˆi+m xˆi N→∞ N i=0 C(m) = lim where ¯ xˆi = f i (x0 ) − x; 1 N−1 i ∑ f (x0 ) . 36) From this definition follows that C(m) yields another measure for the irregularity of the sequence of iterates x0 , f (x0 ), f 2 (x0 ). . It tells us how much the deviations of the iterates from their average value, xˆi = xi − x¯ that are m steps apart (i.

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Acoustic Topography Varying with the Position of the Organ Pipe by Barus C.


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