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back (“fold”) into the spiral unpredictably somewhere in a mixing mechanism that
has been called “displaced reinjection.” In the slow-fast oscillations of the forced van
der Pol in the chaotic regime, points in the slow phase (“repolarization”) jitter around
and step on each other’s heels, getting out of order while waiting on the ledge
before jumping (“depolarization”) to the next slow phase (“repolarization”) at some
unpredictable time, thus generating a variably irregular series of interspike intervals.

Stretching and folding are also responsible for getting points get out of order
in the single maximum map of the unit interval (studied for universal qualitative and
quantitative properties by May and Feigenbaum and others as described above).
With increases in parameter values, the parabolic hill function onto which the unit
line has been stretched gets steeper, more stretched. Mapping points on the hill
back onto the straight line of the unit interval results in what amounts to the line
folding back on itself. This stretching and folding eventually fills the line with points,
but their sequence, from end to end, gets shuffled like a deck of cards.

As described more generally above, points that start as neighbors may get
separated (“divergence along the attractor”) and those that start at a distance from
each other may be thrown together (“compression back onto the attractor”). These
expanding and folding motions that characterize the chaotic behavior on strange
attractors have been likened to the actions of a taffy puller (Rössler, 1976). It is in
this way that nearby points can separate without leaving the attractor. It is also the
case that once indistinguishably close but then separated points may be
compressed together again generating new, temporary (unstable) cycles of all
possible period lengths. These unstable fixed points may be the most important
feature of chaotic systems from the standpoint of new ideas about brain
mechanisms (Pei and Moss, 1996; So et al, 1997). This aggregation of unstable
loops can occur from points fluctuating away and back to the attractor as well as
during the crowding of points at the turns after their stretching out on more linear
parts of the flow. Under the mixing flow of a chaotic dynamics, it is also true that a
single point eventually explores the entire attractor, no attractor location is
inaccessible to it.

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