Figure 3.13 shows the Poincaré map
for the flow arising in the Duffing oscillator in
a parameter region where hysteresis exists.
We see that the inset to
consists of two different curves.
Similarly, the outset of also
consists of two different curves. One branch of the outset approaches
, while the other branch approaches 
.
The inset and the outset of the saddle
are not trajectories in the flow, so they can intersect
without violating a fundamental theorem of ordinary differential equations,
the unique dependence of an orbit with respect to initial conditions.
It is possible to find parameter values so that the inset and the
outset of the saddle point at do indeed cross.
A self-intersection of the inset of a saddle with
its outset is illustrated in Figure 3.14(b)
and, as originally observed by Poincaré, it always gives rise to wild
oscillations about the saddle (see section 4.6.2).
Figure 3.14: Schematic of a homoclinic tangle in the Duffing oscillator.
Such a self-intersection of the inset of a saddle with its
outset is called a homoclinic intersection,
and it is a fundamental mechanism by which chaos is created in
a nonlinear dynamical system.
The reason is roughly
the following. Consider a point at a crossing of the inset and
the outset indicated by the point I in Figure 3.14(b). By definition,
this point is part of an orbit that approaches the saddle by both its inset and
its outset; that is, it is doubly asymptotic.
Consider the next intersection point of I with the
cross section, 
. This point must lie on the
inset at a point closer to the saddle. Next, the second iterate of I
under the Poincaré map, 
, must be even closer to the saddle.
Similarly, the preimage of I approaches the saddle along the outset of
the saddle.
The outset and inset get bunched up near the
saddle, creating an image known as a homoclinic
tangle .
Homoclinic tangles beat at the heart of chaos because, in the region of a homoclinic tangle, initial conditions are
subject to a violent stretching and folding process, the two essential
ingredients for chaos.
A marvelous pictorial description of homoclinic tangles along
with an explanation as to their importance in dynamical systems
is presented by Abraham and Shaw [18].
Homoclinic tangles are often associated with the existence of
strange sets in a system. Indeed, it is thought that in many
instances a strange attractor is nothing but the
closure 
of the outset of some saddle when this outset
is bunched up in a homoclinic tangle. Figure 3.15(a) shows the cross
section for a strange attractor of the Duffing oscillator.
Figure 3.15: Comparison of a strange attractor and the outset of a period one
saddle in the Duffing oscillator.
Figure 3.15(b) shows the cross section of the outset of the period one saddle in this strange set for the exact same parameter values. The resemblance between these two structures is striking. Indeed, developing methods to dissect homoclinic tangles will be central to the study of chaos in low-dimensional nonlinear systems. In fact, one could call it problem of low-dimensional chaos.