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(c) Repeat the calculations in parts (a) and (b) for values of r slightly greater than 24. Try to estimate the value of r for which the duration of the chaotic transient approaches infinity.
? 9. For certain r intervals, or windows, the Lorenz equations exhibit a period-doubling prop-
erty similar to that of the logistic difference equation discussed in Section 2.9. Careful calculations may reveal this phenomenon.
(a) One period-doubling window contains the value r — 100. Let r — 100 and plot the trajectory starting at (5, 5, 5) or some other initial point of your choice. Does the solution appear to be periodic? What is the period?
(b) Repeat the calculation in part (a) for slightly smaller values of r. When r = 99.98, you may be able to observe that the period of the solution doubles. Try to observe this result by performing calculations with nearby values of r.
(c) As r decreases further, the period of the solution doubles repeatedly. The next period doubling occurs at about r — 99.629. Try to observe this by plotting trajectories for nearby values of r.
? 10. Now consider values of r slightly larger than those in Problem 9.
(a) Plot trajectories of the Lorenz equations for values of r between 100 and 100.78. You should observe a steady periodic solution for this range of r values.
(b) Plot trajectories for values of r between 100.78 and 100.8. Determine as best you can how and when the periodic trajectory breaks up.
REFERENCES There are many recent books that expand on the material treated in this chapter, for example:
Drazin, P. G., Nonlinear Systems (Cambridge: Cambridge University Press, 1992).
Glendinning, P., Stability, Instability, and Chaos (Cambridge: Cambridge University Press, 1994). Grimshaw, R., Nonlinear Ordinary Differential Equations (Oxford: Blackwell Scientific Publications, 1990).
Two books that are especially notable from the point of view of applications are:
Danby, J. M. A., Computer Applications to Differential Equations (Englewood Cliffs, NJ: Prentice Hall, 1985).
Strogatz, S. H., Nonlinear Dynamics and Chaos (Reading, MA: Addison-Wesley, 1994).
A good reference on Liapunov's second method is:
LaSalle, J., and Lefschetz, S., Stability by Liapunov's Direct Method with Applications (New York: Academic Press, 1961).
Among the large number of more comprehensive books on differential equations are:
Arnold, V. I., Ordinary Differential Equations (New York/Berlin: Springer-Verlag, 1992). Translation of the third Russian edition by Roger Cooke.
Brauer, F., andNohel, J., Qualitative Theory of Ordinary Differential Equations (New York: Benjamin, 1969; New York: Dover, 1989).
Guckenheimer, J. C., and Holmes, P., Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields (New York/Berlin: Springer-Verlag, 1983).
A classic reference on ecology is:
Odum, E. P., Fundamentals of Ecology (3rd ed.) (Philadelphia: Saunders, 1971).
Two books dealing with ecology and population dynamics on a more mathematical level are:
May, R. M., Stability and Complexity in Model Ecosystems (Princeton, NJ: Princeton University Press, 1973).
Chapter 9. Nonlinear Differential Equations and Stability
Pielou, E. C., Mathematical Ecology (New York: Wiley, 1977).
The original paper on the Lorenz equations is:
Lorenz, E. N., “Deterministic Nonperiodic Flow,” Journal of the Atmospheric Sciences 20 (1963), pp. 130-141.
A very detailed treatment of the Lorenz equations is:
Sparrow, C., The Lorenz Equations: Bifurcations, Chaos, and Strange Attractors (New York/Berlin: Springer-Verlag, 1982).
Partial Differential Equations and Fourier Series
In many important physical problems there are two or more independent variables, so that the corresponding mathematical models involve partial, rather than ordinary, differential equations. This chapter treats one important method for solving partial differential equations, a method known as separation of variables. Its essential feature is the replacement of the partial differential equation by a set of ordinary differential equations, which must be solved subject to given initial or boundary conditions. The first section of this chapter deals with some basic properties of boundary value problems for ordinary differential equations. The desired solution of the partial differential equation is then expressed as a sum, usually an infinite series, formed from solutions of the ordinary differential equations. In many cases we ultimately need to deal with a series of sines and/or cosines, so part of the chapter is devoted to a discussion of such series, which are known as Fourier series. With the necessary mathematical background in place, we then illustrate the use of separation of variables on a variety of problems arising from heat conduction, wave propagation, and potential theory.
10.1 Two-Point Boundary Value Problems
Up to this point in the book we have dealt with initial value problems, consisting of a differential equation together with suitable initial conditions at a given point. A typical example, which was discussed at length in Chapter 3, is the differential equation