# Elementary Differential Equations and Boundary Value Problems - Boyce W.E.

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2. Show that if þ = an with a an irrational number, then no point on the circle is periodic.

252

Exploration 13.2

3. What is the long-term behavior of the orbit of a point on the circle if ø = an, where a is an irrational number?

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 13.3. Two-Dimensional Maps and the Discrete Tool

A two-dimensional discrete dynamical system looks like this:

x„+1 = f(xn, Óï, c) (3)

Óï+1 = g(xn, Óï , c)

where f and g are given functions and c is a "place holder” for parameters. For given values of c, x0, and y0, system (3) defines an orbit of points

( xo, Óî ), (x1, Ó1), (x2, Ó2 ),...

in the xy-plane. The two-dimensional tab in the Discrete Tool allows you to explore discrete systems of the form of system (3).

1. Open the Discrete Tool and explore the default system (a version of what is known as the Henon Map):

xn+1 = 1 + Óï — axn (4)

Óï+1 = bxn

where a and b are parameters. For fixed values of the parameters a and b find the fixed points. Are they sinks, sources, or neither? How sensitive is the long-term behavior of an orbit to small changes in the initial point (x0, y0)? What happens if you increment a through a range of values? If you increment b? Any period-doubling sequences? In your judgment, is there any long-term

chaotic wandering? [Suggestion: Keep the values of a and b within small

ranges of their default values to avoid instabilities.]

254

Exploration 13.3

2. Repeat Problem 1 with the following version of the Henon map:

xn+1 = a — xn + byn Óï+1 = xn

Start with a = 1.28, b = —0.3, x0 = 0, y0 = 0.

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 13.4. Julia and Mandelbrot Sets and the Discrete Tool

Note that the color schemes for the Julia and Mandelbrot sets in Module 13 differ from those in the discrete tool.

1. Use the Discrete Tool to explore the Mandelbrot set and Julia sets for the complex family fc = z2 + c. What happens to the filled Julia sets as you move c from inside the Mandelbrot set up toward the boundary, then across the boundary and out beyond the Mandelbrot set? Describe how the Julia sets change as you “walk” along the edge of the Mandelbrot set.

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2. Repeat Problem 1 for the complex family gc = c sin z.

3. Repeat Problem 1 for the family hc = cez.

Exploration 13.4

GLOSSARY

Acceleration The acceleration of a moving body whose position at time t is u( t) is given by

cPu

~dfi

Air resistance A body moving through air (or some other medium) is slowed down by a resistive force (also called a drag or damping force) that acts opposite to the body’s velocity. See also “Viscous damping” and “Newtonian damping.”

Amplitude The amplitude of a periodic oscillating function u( t) is half the difference between its maximum and minimum values.

Angular momentum The angular momentum vector of a body rotating about an axis is its moment of inertia about the axis times its angular velocity vector.

This is the analog in rotational mechanics of momentum (mass times velocity) in linear mechanics.

Angular velocity An angular velocity vector, rn(t), is the key to the relation between rotating body axes and a fixed coordinate system of the observer. The component rnj of the vector a(t) along the jth body axis describes the spin rate of the body about that axis.

Autocatalator This is a chemical reaction of several steps, at least one of which is autocatalytic.

Autocatalytic reaction In an autocatalytic reaction, a chemical species stimulates more of its own production than is destroyed in the process.

Autonomous ODE An autonomous ODE has no explicit mention of the independent variable (usually t) in the rate equations. For example, x = x2 is autonomous, but x = x2 + t is not.

Balance law The balance law states that the net rate of change of the amount of a substance in a compartment equals the net rate of flow in minus the net rate of flow out.

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