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References Borrelli, R., and Coleman, C., “Computers, Lies, and the Fishing Season” in The College Mathematics Journal, November 1994, pp. 403-404
Devaney, R. L., An Introduction to Chaotic Dynamical Systems, (1986: Benjamin/Cummings), Section 2.3 “The Horseshoe Map”
Guckenheimer, J., and Holmes, P., Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields (1983: Springer-Verlag). The original classic in this field.
Hastings, S.P., and MacLeod, J.B., “Chaotic Motion of a Pendulum with Oscillatory Forcing” in The American Mathematical Monthly, June-July 1993
Hubbard, J.H., “What It Means to Understand a Differential Equation” in The College Mathematics Journal, November 1994, pp. 372-384
Hubbard, J.H., “The Forced Damped Pendulum: Chaos, Complexity, and Control” in C-ODE-E, Spring 95; (soon to be published in The American Mathematical Monthly)
Kapitaniak, T., Controlling Chaos (1996, Academic Press)
Nayfeh, A.H., and Balachandran, B., Applied Nonlinear Dynamics (1995, John Wiley & Sons, Inc.)
Ott, E., Sauer, T., Yorke, J.A., Coping with Chaos (1994, John Wiley & Sons, Inc.)
Sharp, J., “A Problem in Ship Stability.” Group project for interdisciplinary course in nonlinear dynamics and chaos, 1996-1997. Write to John Sharp, Department of Physics, Rose-Hulman University, Terre Haute, IN.
Strogatz, S., Nonlinear Dynamics and Chaos, (1994: Addison-Wesley). Nice treatment of many problems, including chaos induced in constant torque motion.
Answer questions in the space provided, or on
attached sheets with carefully labeled graphs. A
notepad report using the Architect is OK, too.
In each problem describe what you see and explain what the figures tell you about the behavior of the pendulum.
1. Choose a value for c = 0.1, take A = 1 in ODE (1), and produce graphs like those in the chapter cover figure and Figure 12.1.
2. Choose a value for A = 1 and c = 0.1 in ODE (1) and produce graphs like those in the chapter cover figure and Figure 12.1.
3. Choose a value for ø = 1 in the ODE
x' + 0.1/ + sin x = cos mt and produce graphs like those in the chapter cover figure and Figure 12.1.
4. Repeat Problems 1 and 2, but for the Duffing ODE,
x" + Û + x — x3 = A cos t
5. Repeat Problems 1 and 2, but for the ODE with a quadratic nonlinearity,
x" + cx/ + x — x2 = A cos t
13 Discrete Dynamical Systems
Supply and demand converge to a stable equilibrium.
Overview Processes such as population dynamics that evolve in discrete time steps are best
modeled using discrete dynamical systems. These take the form x„+\ = f(xn),
where the variable xn is the state ofthe system at "time" n and xn+1 is the state of the system at time n +1. Discrete dynamical systems are widely used in ecology, economics, physics and many other disciplines. In this section we present the basic techniques and phenomena associated with discrete dynamical systems.
Key words Iteration; fixed point; periodic point; cobweb and stairstep diagrams; stability;
sinks; sources; bifurcation diagrams; logisitic maps; chaos; sensitive dependence
on initial conditions; Julia sets; Mandelbrot sets
See also Chapter 6 for more on sinks and sources in differential equations, and Chapter 12 for Poincare sections and chaotic pendulum motion.
The function f (x) = Xx is denoted Lx, and so Lx(x) = Xx.
The function Xx(1 _ x) is denoted by gx (x).
The superscript ° reminds us that this is just the composition of f with itself; f is not being raised to a power.
Discrete dynamical systems arise in a large variety of applications. For example, the population of a species that reproduces on an annual basis is best modeled using discrete systems. Discrete systems also play an important role in understanding many continuous dynamical systems. For example, points calculated by a numerical ODE solver form a discrete dynamical system that approximates the solution of an initial value problem for an ODE. The Poincare section described in Chapter 12 is another example of a discrete dynamical system that gives information about a system of ODEs.
A discrete dynamical system is defined by the iteration of a function f, and takes the form
xn+1 = f (xn), n > 0, xo given (1)