Computer Algebra Recipes

1.2. THREE-DIMENSIONAL AUTONOMOUS SYSTEMS 37
y y
0 x 1
0 x 1
y
0 x 1
Figure 1.14: Phase-plane portraits for F1 =0:325 (top left), F2 =0:35 (top
right), F3 =0:356 (bottom left), and F4 =0:42 (bottom right).
To produce a deeper understanding, Jennifer reruns the ¯le with the scene
option in the ith graph, gr[i], replaced with scene=[t,x] and the time range
shortened to t = 100 to 160. The four displacement (x(t)) curves in Figure 1.15
result, each corresponding to the matching phase-plane portrait in Figure 1.14.
For F1 =0:325, the inverted spring responds periodically at exactly the
driving frequency, the period being T =2¼=! =6:28. If the period is written
as T = n (2 ¼=!), then n = 1 for this case, and the motion is referred to as a
period-one response.
For F2 =0:35, the spring has a repeat period that is twice that of the driving
term, i.e., one has n =2andthereforeaperiod-two response. Notice that now
the system alternates each half-cycle between di®erent maximum values of the
displacement.
For F3 =0:356, the repeat period is four times as large, corresponding to
period four. AsF was increased, the period doubled from period one to period
twotoperiodfour. AsF is further increased, this period doubling will continue
–1
y
x
1