Prove that a random sequence with distribution function v(x) defined on the interval [0,1] such that v(x) > 0 for all x in [0,1] is ergodic. (See the handout, Random Sequences .)
Remark: The converse is not true. That is, an ergodic sequence need not be random (for example, an ergodic orbit of the logistic function f ₄(x); it has a 'distribution function' that is positive for all x in [0,1] but it is not a random sequence; cf. question #8 in Homework #4).

Can the Shadowing Lemma be used to justify the following two statements?

• The observed instability of periodic points of f(x)=4x(1-x) (i.e., the computer experiments) implies the instability of the exact (or true) periodic points of f(x).
• If f(x) has an observed ergodic orbit, then f(x) has an exact ergodic orbit.

Suppose { x₀, x₁, x₂, . . . } is a random sequence of numbers from [0,1] such that the distribution function v(x) > 0 for all x in [0,1] (see the Random Sequences handout). Is this orbit an ergodic orbit? Prove or disprove (i.e., prove the negation). Note that you will have to use the definition of an ergodic orbit. Is the converse true? That is, is an ergodic orbit a random sequence?

Complete the proof outlined in class: If x is a number in [0,1] such that [x]₂ contains every pattern of 0 ^'s and 1 ^'s, then the orbit of x under the tent transformation T is ergodic. Then prove that the orbit of x' = h(x) is ergodic for f₄ (refer to Figure 10.41).

Draw the bifurcation diagram and the final state diagram of g_a(x) = x³ + ax for a from -2.5 to +2 (you may want o refer to your answer to question #4 in Homework #4). Do not try to solve the equation g_a(x) = x to find the fixed points; instead, use the plots provided here of the graphs of g and g² for various a (a = 2, 1, 0, -1, -1.2, -1.5, -2, -2.5) to sketch the bifurcation diagram. You will, however, need to determine the stability of the periodic points (for the final state diagram). You may use the fact that g_a has no periodic points of prime period greater than 2 for these values of a.

Plots of the graphs of g_a(x).

Use the applet "Logistic Movie" to determine the period-doubling points b₁ (period 6), b₂ (period 12) and b₃ (period 24) in the period 3 window near a = 3.83 (see Figure 11.41). Use these values to estimate the Feigenbaum constant delta (see pages 611 and 636) for this period-doubling scenario. (You will have to estimate the b_i to 4 decimal places.)

A dynamical system depends on a parameter a. Initially, you observe a steady state (i.e., a period 1 orbit). As a increases you observe a period 2 oscillation appearing at a = a₁ = 7. Then at a = a₂ = 10 you observe that the period 2 orbits splits into a period 4 orbit. As a continues to increase a series of period-doublings occurs. Assuming Universality, at what a value would you expect to observe the onset of chaos? ('Assuming Universality' means assuming that the system will go through a series of period-doubling bifurcations as the parameter a changes, and that the distance (in a) between bifurcations is given by the Feigenbaum constant delta.)

Answer the following questions about the final state diagram of the logistic equation f = f _a (see for example, Figure 11.5 in the text). See the notes, Notes on the Logistic Functiuon - Self-Similarity and Universality.

• (a) Explain the upper and lower boundaries in the band region a greater than 3.57.
• (b) Explain the two bands at a = 3.65. First look at orbits of f ² with the applet "Graphical Iteration" and find two 'invariant boxes'. Show that all orbits (i.e., all initial points) eventually stay in one of the two boxes. However, when you look at one of these orbits (i.e., one of these initial points) under iteration by f (use the applet), the orbit fills out both bands (note that the 'gap' between the two bands contains the (unstable) fixed point of f).
• (c) Why is there only one band for a greater than 3.68? (Think of those 'invariant boxes' of f ².)
• (d*) Explain why it is that when the curve f ³(v_a) meets the curve v _a (i.e., for that a value; see Figure 11.39) a period 3 window appears in the final state diagram (see Figure 11.5). (Note that the third curve down in Figure 11.39 (top) is the curve f ³(v_a), not f ²(v_a) like I wrote on the handout.) Could you predict the locations of other period windows based on the curves f ^k(v_a)?

Write down the relative ordering of the following numbers using Charkovsky's ordering of the integers: 56, 31, 128, 160. That is, which one of these numbers is the 'smallest', the next 'smallest', ... , the 'largest' according to Charkovsky's ordering?

Prove that if f(x) (here f is an arbitrary continuous function) has an orbit of prime period 2 ^m, then f(x) has an orbit of period 2^kfor all k = 0,...,m. Prove this without using Charkovsky's Theorem (just sketch the graph of f ^k(x), i.e., use graphical analysis). Hint: Consider the graph of f ^p(x) where p = 2(m-1) (and note that f ^p has a period 2 orbit).

• (a) Use Charkovsky's Theorem to prove that if f(x) is a continuous function from [0,1] to [0,1] that has only finitely many distinct periodic points, then the period of any one of them must be 2 ^mfor some m.
• (b) State three periods other than odd ones that have the following property : If a function like the one in part (a) has a periodic orbit of that prime period, then that function has infinitely many distinct periodic orbits.

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