Computer Algebra Recipes

8.3. THE BIFURCATION DIAGRAM 337
Problem 8-11: Interchanging signs
Use the power spectrum to determine the response of the forced Du±ng oscillator
for each F value when all numerical values are the same as in the text
recipe, but the signs of ® and ¯ are interchanged.
Problem 8-12: Steady-state solution?
Using the power spectrum approach, determine the nature of the steady-state
solution for the following forced oscillator equation:
Äx +0:7 _x + x 3 =0:75 cos t; x(0) = _x(0) = 0:
Problem 8-13: Solution?
Using the power spectrum approach, determine the nature of the steady-state
solution for the following forced Du±ng equation:
Äx +0:08 _x + x 3 =0:2cost; x(0) = 0:25; _x(0) = 0:
Problem 8-14: Solutions?
Using the power spectrum approach, determine the nature of the steady-state
solution for the following oscillator equation for F =0:357 and F =0:35797:
Äx +0:5 _x ¡ x + x 3 =0:357 cos(t +1); x(0) = 0:09; _x(0) = 0:
Problem 8-15: Forced glycolytic oscillator
The equations describing forced oscillations of the glycolytic oscillator are
_x = ¡x + ®y+ x 2 y; _y = ¯ ¡ ®y¡ x 2 y + A + F cos(!t):
Taking ® = ¯ =0,A =0:999, F =0:42, x(0) = 2, and y(0) = 1, determine the
periodicity of the response using the Poincare sectionapproachfor(a)! =2
and (b) ! =1:75. Explore the frequency range in between and identify any
interesting solutions.
8.3 The Bifurcation Diagram
Bifurcation diagrams can be generated for both nonlinear ODEs and nonlinear
di®erence equations by plotting the system \response" versus a \control" parameter
as the latter is varied. The word bifurcation is derived from the Latin
word furca for fork. When period doubling occurs from period one to period
two, the response curve resembles a two-pronged fork, the period-one portion
being the \handle" and the period-two portion looking like two \prongs." In a
typical period-doubling scenario, the two prongs then split into four and then
into eight and so on as the control parmeter is further increased. Period doubling
is not the only \route" to chaos, so bifurcation diagrams are useful in
revealing the nature of the route.