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1Ph 106bHomework Assignment No. 3Due: Thursday, 31 January 20081. An ergodic map that is not chaotic, part II. Consider the invertiblemap on the unit intervalM(x)=x + α (mod 1) ≡x + α, if x + α<1,x + α − 1 , if x + α ≥ 1.where α ∈ (0, 1) is an irrational real number. This map is continuousif we regard it as a map from the circle to the circle, by identifyingthe points 0 and 1. (It just rigidly rotates the circle by the angle 2παradians.) In this exercise, you will show that M is ergodic, with auniform invariant density.Let A(x) be a smooth function on the circle, or equivalently a perio dicfunction of x with period 1. Then ergodicity means that a time averageover a long orbit can be replaced by an integral over phase space:hAitime= hAispace, wherehAitime= limN →∞1NN −1Xn=0A(Mn(x0)) ,hAispace=Z10dx A(x) ,Note that A(x), being periodic, can be expressed as a Fourier seriesA(x)=∞Xk=−∞˜Ake2πikx.When we say the map is “smooth” we mean that the Fourier coef-ficients˜Akdecrease rapidly for large k . For this problem, you mayassume that the sum is truncated to a sum over a finite number ofFourier modes,A(x)=kmaxXk=−kmax˜Ake2πikx.a) Evaluate the sum over n and take the limit N →∞to expresshAitimein terms of the˜Ak’s.2b) Evaluate the integral to express hAispacein terms of the˜Ak’s. Verifythat hAitime= hAispace.2. Conjugate maps. Recall that if M is an ergodic differentiable one-dimensional map defined on the unit interval I =[0, 1], the Lyapunovexponent h of M can be expressed ash =Z10dxρ(x)ln|M0(x)| ,where M0(x) denotes the first derivative of M and ρ(x) is a densityfunction invariant under M . A map˜M is said to be conjugate to Mif˜M can be expressed as˜M = g ◦ M ◦ g−1,where ◦ denotes composition of mappings, and g is a differentiable andinvertible map from I to I. Show that if M and˜M are conjugate theyhave the same Lyapunov exponent.3. Periodic orbits of the cat map. Recall that the “cat map” T is achaotic map from the 2-torus to the 2-torus that is continuous, invert-ible, and area preserving. If we represent the 2-torus as a unit squarewith opposite sides identified, the action of T on the p oint (x, y) ∈ I ×Ican be expressed asT :x → x0= x + y (mod 1) ,y → y0= x +2y (mod 1) .Find the periodic orbits of the cat map of length 1 and 2. How manyperiodic orbits are there of length 3?4. Stable orbits of a one-dimensional map. Consider the logistic mapon the unit intervalM(x)=rx(1 − x) ,where 0 ≤ r ≤ 4.a) Find (analytically) all fixed points of M in the unit interval, andfor each fixed point, whether it is stable or unstable.b)Forr =3.3, find the perio d-two orbit of M : x1→ x2→ x1(de-termine x1and x2numerically to four-digit accuracy). Find thestability coefficient M0(x1)M0(x2) of this orbit (also to four-digitaccuracy). Hint: It is useful to study the iterated map


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CALTECH PH 106B - Homework Assignment No. 3

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