# Elementary differential equations 7th edition - Boyce W.E

ISBN 0-471-31999-6

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Answer questions in the space provided, or on

attached sheets with carefully labeled graphs. A

notepad report using the Architect is OK, too.

Name/Date______

Course/Section

Exploration 11.3. Aging Spring Models

1. Check out the Library file “Modeling an Aging Spring” in the “Physical Models” folder (see Figure 11.4). The ODE in the file models the motion of a vertically suspended damped and aging spring that is subject to gravity. Carry out the suggested explorations.

2. Show that

ln(?+ l)y — Vt + 1 cos — ln(f + l)y is an analytic solution of the initial value problem

+ rrrhi = °> ^(°) = -1> AQ) = o

(t + 1)2

Explain why this IVP provides another model for the motion of an aging spring that is sliding back and forth (without damping) on a support table. [Suggestion: let s = t + 1, u(s) = x(t). Then u(s) satisifes the Euler equation s2u"(s) + u(s) = 0. See the references on page 211 for solution formulas for Euler ODEs.]

x(t ) =

t + 1

? sin

V3

218

Exploration 11.3

3. Graph the solution x(t) from Problem 2 over the interval o < t < 3oo and compare the graph to the one for the ODE, x" + e—atx = o, x = o, x (o) = o, a = o.o4 (see Screen 2.6 in Module ii). Repeat for other values of a.

Answer questions in the space provided, or on

attached sheets with carefully labeled graphs. A

notepad report using the Architect is OK, too.

Name/Date______

Course/Section

Exploration 11.4. The Incredible Lengthening Pendulum

Suppose that we have an undamped pendulum whose length L = a + bt increases linearly over time. Then the ODE that models the motion of this pendulum is

(a + bt)6" (t) + 2b0'(t) + g0(t) = 0 (i3)

where 0 is small enough that sin0 ^ 0, the mass of the pendulum bob is 1, and the value of the acceleration due to gravity is g = 32.!

1. With a = b = i and initial conditions 0(0) = i and 0'(0) = 0, use ODE Architect to solve ODE (i3) numerically. What happens to 0(t) as t ^ +??

2. Under the same conditions, what happens to the oscillation time of the pendulum as t ^ +?? (The oscillation time is the time between successive maxima of 0(t).)

The ODE for a pendulum of varying length is derived in Chapter 10 (see equation (15)).

1See the article “Poe's Pendulum” by Borrelli, Coleman, and Hobson in Mathematics Magazine, Vol. 58 (1985) No. 2, pp. 78-83. See also “Child on a Swing” in Module io.

220

Exploration 11.4

3. Show that the change of variables

s = (2/6)-v/(a+ bt)g, x=ey a + bt

transforms Bessel’s equation of order 1

+ + D,=0

ds2 ds

into ODE (13) for the lengthening pendulum. [Suggestion: Take a look at the section “Transforming Bessel’s Equation to the Aging Spring Equation” in this chapter to help you get started. Use the change of variables given above to express the solution of the IVP in Problem 1 using Bessel functions.]

12 Chaos and Control

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Overview

Key words See also

X

Poincare map of a forced damped pendulum superimposed on a trajectory.

In this chapter we'll look at solutions of a forced damped pendulum ODE. In the linear approximation of small oscillations, this ODE becomes the standard constant-coefficient ODE x" + cX + kx = F(t), which can be solved explicitly in all cases. Without the linear approximation, the pendulum ODE contains the term ksinx instead of kx. Now the study becomes much more complicated. We'll focus on the special case ofthe nonlinear pendulum ODE

x" + ai + sin x = A cos t (1)

but our results leave a world of further things to be discovered. We'll show that appropriate initial conditions will send the pendulum on any desired sequence of gyrations, and hint at howto control the chaos by finding such an initial condition.

Forced damped pendulum; sensitivity to initial conditions; chaos; control; Poincare sections; discrete dynamical systems; Lakes of Wada; control

Chapter 10 for background on the pendulum. Chapter 13 for more on discrete dynamical systems and other instances of chaos and sensitivity to initial conditions.

222

Chapter 12

? Introduction

See the glossary for one definition of chaos.

How might chaos and control possibly be related? These concepts appear at first to be opposites, but in fact they are two faces of the same coin!

A good way to start discussing this apparent paradox is to think about learning to ski. The beginning skier tries to be as stable as possible, with feet firmly planted far enough apart to give confidence that she or he will not topple over. If you try to ski in such a position, you cannot turn, and the only way to stop, short of running into a tree, is to fall down. Learning to ski is largely a matter of giving up on “stability,” bringing your feet together so as to acquire controllability! You need to allow chaos in order to gain control.

Another example of the relation between chaos and control is the early aircraft available at the beginning of World War I, carefully designed for greatest stability. The result was that their course was highly predictable, an easy target for anti-aircraft fire. Very soon the airplane manufacturers started to build in enough instability to allow maneuverability!

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