By S. (Ed.) Flugge
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Recall that c(x) = +1 for l x i > 1 and I(x) = -1, if l x l < 1. He then showed that this is an unstable solution (see Appendix J for more details). 2). 2), C1 = TC° (Fig. 20). Proceeding with higher iterates Ck+ 1 = TCk, Fig. 20 and the interior of Ck+1 contains the interior of Ck. 3a) M=0 which is some open connected set of points, of unknown extent. He next showed (Fig. 21) that there is a large curve, CO (note the superscript zero), which after a sufficient number (N) of iterates, lies entirely inside C°; that is, Interior T"C° c Interior C° (n > N).
26. The `flap' formed on the left of Fig. 26(a) moves down and is squeezed into the attractor (t = (n + 4)T), while a new flap forms on the right (t = (n + and then squeezes into the attractor Z)T) on the upper side (t = (n + 4)T). Such numerically computed maps of strange attractors are, of course, subject to the usual caveats concerning numerical accuracy, Models based on second order difference equations 22 Fig. 26 ti (a) (b) t=nT t=(n+ 4)T (d) (c) (n+ 2)T t=(n+ 4)T pseudo-orbits, and the like.
This is because any dissipative effect can only be compensated by an external periodic force for a discrete (enumerable) set of periodic orbits - otherwise the force and the dynamics will not stay in a'compensating phase'. These two physical systems are illustrated in Fig. 19, where p = 1 for the dissipative case. Note that, in the case of an attracting periodic orbit (stable limit Models based on second order difference equations 16 Fig. 19 p=1 cycle), there must also be an unstable limit cycle on the torus.