Problems

{ } 2.10.
Problems for section 2.1.
2.1.
For the quadratic map (eq. (2.1)), show that the interval is a trapping region for all x in the unit interval and all .

2.2.
Use the transformation



to show that the quadratic map (eq. (2.1)) can be written as



or (using a different x-transformation) as



Specify the ranges to which x and are restricted under these transformations.

2.3.
Read the Tellus article by Lorenz mentioned in reference [1].

Section 2.2.
2.4.
Write a program to calculate the iterates of the quadratic map.

Section 2.3.
2.5.
For f(x) = 4 x (1 - x) and , calculate by the graphical method described in section 2.3.

2.6.
Show by graphical analysis that if and in the quadratic map, then as , . Further, show that if , then the fixed point at the origin is an attractor.

Section 2.4.
2.7.
Find all the attractors and basins of attraction for the map where m is a constant.

2.8.
The tent map  (see sections 2.1-2.2 of Rasband, reference [6]) is defined by



(a) Sketch the graph of the tent map for . Why is it called the tent map?

(b) Show that the fixed points for the tent map are



(c) Show that is always repelling and that is attracting when .

(d) For a one-dimensional map the Lyapunov exponent  is defined by



Show that for , the Lyapunov exponent for the tent map is . Hint: For the tent map use the chain rule of differentiation to show that



2.9.
Determine the local stability of orbits with and using graphical analysis as in Figure 2.9.

Section 2.5.
2.10.
Determine the parameter value(s) for which the quadratic map intersects the line y = x just once.

2.11.
Consider a period two orbit of a one-dimensional map,



(a) Use the chain rule for differentiation to show that .

(b) Show that the period two orbit is stable if , and unstable if .

2.12.
For , use Mathematica to find the locations of both period three orbits in the quadratic map.

2.13.
Consider a p-cycle of a one-dimensional map. If this p-cycle is periodic then it is a fixed point of . Let represent one orbit of period p. Show that this periodic orbit is stable if



Hint: Show by the chain rule for differentiation that for any orbit (not necessarily periodic),



Now assume the orbit is periodic. Then for any two points of a period p orbit, and , note that .

2.14.
Consider a seed near a period two orbit, , with slope near to . Show by graphical analysis that ``flips'' back and forth between the two points on the period two orbit. Show by graphical construction that this period two orbit is stable if , and unstable if .

2.15.
For , show that the quadratic map intersects the y = x line times, but only some of these intersection points belong to new periodic orbits of period n. For n = 1 to 10, build a table showing the number of different orbits of of period n.

2.16.
Prove (see Prob. 2.15) that for prime n, is an integer (this result is a special case of the Simple Theorem of Fermat) .

2.17.
(a) Derive the equation of the graph shown in Figure 2.12.

(b) Verify that the two intersection points shown in the figure occur at 3 and .

2.18.
Equation (2.21) follows directly from equation (2.12) and the chain rule discussion in Problem 2.13. Derive it using only equations (2.13), (2.19), and (2.20).

Section 2.6.
2.19.
Write a program to generate a plot of the bifurcation diagram for the quadratic map.

Section 2.7.
2.20.
Show that the period two orbit in the quadratic map loses stability at ; i.e., at this value of .

2.21.
Show that the ``equispaced'' orbits of the bouncing ball system (see Prob. 1.2) are born in a saddle-node bifurcation. Note that equation (1.41) gives two impact phases for each n. For n = 1 and for n = 2 in equation (1.41), which orbit is a saddle near birth, and which orbit is a node? What is the impact phase of the saddle? What is the impact phase of the node?

2.22.
Use Mathematica (or another computer program) to show that a pair of period three orbits are born in the quadratic map by a tangent bifurcation at .

2.23.
Show that the absolute value of the slope of evaluated at all points of a period n orbit have the same value at a bifurcation point.

2.24.
In Figure 2.17, the two sets of triangles (``open''--white interior and ``closed''--black interior) represent two different period three orbits. Show that the open triangles represent an unstable orbit.

Section 2.8.
2.25.
Using Table 2.1 and equations (2.26 and 2.27), estimate , c, and .

2.26.
Use the bifurcation diagram of the quadratic map (Fig. 2.22) and a ruler to measure the first few period doubling bifurcation values, . It may be helpful to use a photocopying machine to expand the figure before doing the measurements. Based on these measurements, estimate ``Feigenbaum's delta'' with equation (2.27). Do the same thing for the bifurcation diagram for the bouncing ball system found in Figure 1.16. How do these two values compare?

Section 2.9.
2.27.
Find a one-dimensional circle map for which Sarkovskii's ordering does not hold. Hint: Find a map that has no period one solutions by using a discontinuous map.

2.28.
Consider the periodic orbits of periods 1, 2, 3, 4, 5, 6, 7, 8, 9, and 10. Order these points according to Sarkovskii's ordering, equation (2.29). Show that Sarkovskii's ordering uniquely orders all the positive integers.

Section 2.11.
2.29.
For , determine the interval , as shown in Figure 2.23, as a function of . Verify that points in this interval leave the unit interval and never return.

2.30.
(a) Using the metric of equation (2.34), calculate the distance between the two period two points, and .

(b) Create a table showing the distances between all six period three points: , and .

2.31.
Establish an isomorphism between the unit interval and the sequence space of section 2.11.2. Hint: See Devaney [10], section 1.6.

2.32.
(a) Define by f(x) = mx + b and define by g(x) = mx•+ nb, where m, b, and . Show that f and g are topologically conjugate.

(b) Define by . Define by g(x)•= 1 - x. Show that f and g are topologically semiconjugate.

(c) Find a set of functions f, g, and h that satisfies the definition of topological conjugacy.

Section 2.12.
2.33.
Construct the binary tree  up to the fourth level where the nth level is defined by the rule



(a) Construct the sixteen symbolic coordinates at the fourth level of this binary tree, and show that the ordering from left to right at the nth level is given by



Why is it called the ``binary tree''?

(b) Show that the fractional ordering is given by



(c) Give an example of a one-dimensional map (not necessarily continuous) on the unit interval giving rise to the binary tree.

2.34.
Construct the alternating binary tree up to the fourth level and calculate the symbolic coordinate and position of each of the sixteen points ( ). Present this information in a table.

2.35.
Show that the order on the x-axis of two points and in the quadratic map with is determined by their itineraries and as follows: suppose , and that and (i.e., the itineraries of each initial condition are identical up to the kth iteration, and differ for the first time at the k+1st iteration). Then



Hint: See Appendix A of reference [13] and theorem 18.10 on page 145 of Devaney, reference [10].