Problems

{ } 1.10.
Problems for section 1.2.
1.1.
For a period one orbit in the exact model show that

(a) .

(b) .

(c) the impact phase is exactly given by

 

1.2.
Assuming only that ,

(a) Show that the solution in Problem 1.1 can be generalized to an nth-order symmetric periodic (``equispaced'') orbit satisfying , for

(b) Show that the impact phase is exactly given by

 

(c) Draw pictures of a few of these orbits. Why are they called ``equispaced''?

(d) Find parameter values at which a period one and period two (n = 1, 2) equispaced orbit can coexist.

1.3.
If T= 0.01 s and , what is the smallest table amplitude for which a period one orbit can exist (use Prob. 1.1(c))? For these parameters, estimate the maximum height the ball bounces and express your answer in units of the average thickness of a human hair. Describe how you arrived at this thickness.

1.4.
Describe a numerical method for solving the exact bouncing ball map. How do you determine when the ball gets stuck? How do you propose to find the zeros of the phase map, equation (1.12)?

1.5.
Derive equation (1.14) from equation (1.13).

Section 1.3.
1.6.
Calculate in equation (1.25) for n = 1,2,3,4,5 when , , and .

1.7.
Confirm that the variables , , and given by equations (1.22-1.24) are dimensionless.

1.8.
Verify the derivation of equation (1.25), the standard map.

1.9.
Calculate the inverse of the standard map (eq. (1.25)).

1.10.
Write a computer program to iterate the model of the bouncing ball system given by equation (1.25), the high bounce approximation.

Section 1.4.
1.11.
(a) Calculate the stopping time (eq. (1.35)) for a first impact time , and a damping coefficient . Also calculate , , and .

(b) Calculate and the stopping time for an initial velocity of 10 m/s (1000 cm/s) and a damping coefficient of 0.5.

1.12.
For the high bounce approximation (eq. (1.25)) show that when ,

(a) .

(b) A trapping region is given by a strip bounded by , where . (Note: The reader may assume, as the book does at the end of section 1.4.1, that the cannot approach asymptotically, and that once inside the strip, the orbit cannot leave.)

1.13.
Calculate (eq. (1.37)) for A=0.1 cm and T = 0.01 s. What is the speed and acceleration of the table at this phase? Is the table on its way up or down? Are there table parameters for which the ball can become unstuck when the table is moving up? Are there table parameters for which the ball can become unstuck when the table is moving down?

1.14.
Show that the average table velocity between impacts equals the average ball velocity between impacts.

1.15.
These problems relate to the trapping region discussion in section 1.4.1.

(a) Prove that cannot approach the given by equation (1.29) asymptotically. (Hint: It is acceptable to increase by some small .)

(b) Prove that, for the trapping region D given by equation (1.31), once the orbit enters D, it can never escape D.

(c) Use the trapping region D in the impact map to find a trapping region in phase space. Hint: Use the maximum outgoing velocity to calculate a minimum incoming velocity and a maximum height.

(d) The trapping region found in the text is not unique; in fact, it is fairly ``loose.'' Try to obtain a smaller, tighter trapping region.

1.16.
Derive equation (1.33).

Section 1.5.
1.17.
How many period one orbits can exist according to Problem 1.1(c), and how many of these period one orbits are attractors?

Section 1.6.
1.18.
Write a computer program to generate a bifurcation diagram for the bouncing ball system.