Computer Algebra Recipes

96 CHAPTER 2. PHASE-PLANE ANALYSIS
and for (b) a period-two solution. Discuss how this interpretation can be made
from your plots.
Problem 2-24: Arti¯cial example
Consider the nonlinear equation
dy=dx = xy(y ¡ 2);
with y(0) = 1. Taking h =0:02 and 10-digit accuracy, solve for y(x) outto
x = 3 using the dial-up Euler's method and plot the result.
Problem 2-25: Modi¯ed Euler algorithm
If the rabbits{foxes system is written for brevity as _r = R(r; f )and_ f = F (r; f ),
the modi¯ed Euler algorithm for solving the equations is
rn+1 = rn + h
2 (R1[n]+R2[n]); fn+1 = fn + h
2 (F1[n]+F2[n]);
where
tn+1 = tn + h; R1[n] ´ R(rn;fn); F1[n] ´ F (rn;fn);
R2[n] ´ R(rn + hR1[n];fn + hF1[n]); F2[n] ´ F (rn + hR1[n];fn + hF1[n]):
Taking the same parameter values as in the text recipe, but N twice as large,
solve the rabbits{foxes equations using the modi¯ed Euler algorithm given
above. Rotate your plot to show the r{f phase plane and compare the result
with that for the Euler method.
Problem 2-26: Onset of numerical instability
Investigate the onset of numerical instability as h is increased in the modi¯ed
Euler algorithm of the previous problem.
2.3.3 Glycolytic Oscillator
Give me another horse!
William Shakespeare, King Richard III (1564{1616)
The Euler algorithm, which is the simplest example of an explicit ¯xed-step
method, is not a very accurate numerical procedure, being of order h accuracy
(error O(h2 )). A systematic approach to developing more accurate explicit numerical
schemes are the ¯xed-step Runge{Kutta (RK) methods [BF89], which
still use the FDA approximation for the ¯rst derivatives but create better approximations
to the functions on the right-hand side of the ODEs.
Consider an ODE system of the general form
_x = X(t; x; y); _y = Y (t; x; y); (2.17)
where X and Y are known functions of the arguments. Note that any secondorder
ODE of the general structure Äx = Y (t; x; _x) can be put into this standard
form by setting _x = y ´ X.
In the Euler method, the functions X and Y are evaluated only once on
each step. The general RK approach is to increase the accuracy by using more