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).

• (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).

• (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.

Consider the two dimensional discrete dynamical system defined by the function F(x,y) = r²*A(x,y), where r² = (x² + y²) and A is the matrix that is rotation by pi/4 counterclockwise.

• (a) Sketch the phase portrait of F.
• (b) Are there any periodic points? What are the invariant sets of F and their basins of attraction? What is the basin of attraction of infinity? (i.e., which points have orbits that tend to infinity?)
• (c) In what regions of the plane does F contract areas? Expand areas? Is this what you would have expected by just looking at the phase portrait of F? (To answer this question you need to calculate the determinant of the Jacobian matrix of F.)
• (d) Find two sets D1 and D2 that are invariant under F, i.e., such that F(D1) in D1, and F(D2) in D2. What is the set Ai = int_1^infinity Fⁿ(Di) in each case? (Here, int_a^b Fⁿ(D) means the intersection of the sets F^a(D),...,F^b(D).) What are the basins of attraction of the sets A1 and A2?

Sketch the orbits of 0, 3, i, -i, (2+i), (2-i) under iteration by the function g(z) = (1/2)z + 1. What do you conclude about the dynamics of this function?

Sketch the orbits of the points 1/2, 1/2*exp(i*pi), 2*exp(i*pi), and 2*exp(i*pi/4) under the quadratic function q₀(z) = z². (Here, exp(x) is the exponential function e^x.)

Show that for each n = 1,2,..., the points z_(n,m) = exp(it_m) , m = 0,1,... 2ⁿ-1, where t_m = m*(2*pi)/(2ⁿ-1), are periodic points of q₀(z) of period n. From this deduce that periodic points of q₀(z) "fill out" J₀. Note that all these periodic points are repelling (i.e., unstable) periodic points of q₀(z).

• What is the Julia set of f_c(z) = z+c for various (complex) c? What is the Julia set of g(z) = zⁿ, where n>2?
• Find some points in the escape set and some points in the prisoner set for the function h(z) = z² + z.

(z is a complex number)

Verify that there are two fixed points of q_c(z) = z² + c for c = -(1/2) + i(1/2) and that one fixed point is attracting (stable) and the other is repelling (unstable).

Let c=0 and let W = W₁ U W₂ be the IFS for q₀(z) (as described in Section 13.7).

• (a) Plot the points of the "chaos game" using the sequence {1,1,2,1,2} starting with the points z₀ = 9, z₀ = 0.5i, and z₀ = 1.
• (b) Plot the game points using the same initial points z₀ given in (a) but use the sequences {1,1,1,1,1} and {2,2,2,2,2}.
Where do the game points go? What are the fixed points of W₁ and W₂?
• (c) Locate the addresses on J₀ (via the IFS given above).
• (d) What image of J₀ may result if a random sequence is not used in this chaos game?