HorseshoeHorseshoeHorseshoeHorseshoe mapmapmapmap
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Not to be confused with Horseshoe theory.
In the mathematics of chaos theory, a horseshoehorseshoehorseshoehorseshoe mapmapmapmap is any member of a class of
chaotic maps of the square into itself. It is a core example in the study of dynamical
systems. The map was introduced by Stephen Smale while studying the behavior of
the orbits of the van der Pol oscillator. The action of the map is defined geometrically
by squishing the square, then stretching the result into a long strip, and finally folding
the strip into the shape of a horseshoe.
Mixing in a real ball of colored putty after consecutive iterations of Smale horseshoe
map
Most points eventually leave the square under the action of the map. They go to the
side caps where they will, under iteration, converge to a fixed point in one of the caps.
The points that remain in the square under repeated iteration form a fractal set and are
part of the invariant set of the map.
The squishing, stretching and folding of the horseshoe map are typical of chaotic
systems, but not necessary or even sufficient.[1]
In the horseshoe map the squeezing and stretching are uniform. They compensate
each other so that the area of the square does not change. The folding is done neatly,
so that the orbits that remain forever in the square can be simply described.
For a horseshoe map:
• there are an infinite number of periodic orbits;
• periodic orbits of arbitrarily long period exist;
• the number of periodic orbits grows exponentially with the period; and
• close to any point of the fractal invariant set there is a point of a periodic orbit.
[[[[editediteditedit]]]] TheTheTheThe horseshoehorseshoehorseshoehorseshoe mapmapmapmap
The horseshoe map is a diffeomorphism defined from a region of the plane
into itself. The region is a square capped by two semi-disks. The action of is
defined through the composition of three geometrically defined transformations. First
the square is contracted along the vertical direction by a factor . The caps
are contracted so as to remain semi-disks attached to the resulting rectangle.
Contracting by a factor smaller than one half assures that there will be a gap between
the branches of the horseshoe. Next the rectangle is stretched horizontally by a factor
of ; the caps remain unchanged. Finally the resulting strip is folded into a
horseshoe-shape and placed back into .
The interesting part of the dynamics is the image of the square into itself. Once that
part is defined, the map can be extended to a diffeomorphism by defining its action on
the caps. The caps are made to contract and eventually map inside one of the caps (the
left one in the figure). The extension of f to the caps adds a fixed point to the
non-wandering set of the map. To keep the class of horseshoe maps simple, the curved
region of the horseshoe should not map back into the square.
The horseshoe map is one-to-one, which means that an inverse f–1 exists when
restricted to the image of S under f.
By folding the contracted and stretched square in different ways, other types of
horseshoe maps are possible.
To ensure that the map remains one-to-one, the contracted square must not overlap
itself. When the action on the square is extended to a diffeomorphism, the extension
cannot always be done in the plane. For example, the map on the right needs to be
extended to a diffeomorphism of the sphere by using a “cap” that wraps around the
equator.
The horseshoe map is an Axiom A diffeomorphism that serves as a model for the
general behavior at a transverse homoclinic point, where the stable and unstable
manifolds of a periodic point intersect.
[[[[editediteditedit]]]] DynamicsDynamicsDynamicsDynamics ofofofof thethethethe mapmapmapmap
The horseshoe map was designed to reproduce the chaotic dynamics of a flow in the
neighborhood of a given periodic orbit. The neighborhood is chosen to be a small disk
perpendicular to the orbit . As the system evolves, points in this disk remain close to
the given periodic orbit, tracing out orbits that eventually intersect the disk once again.
Other orbits diverge.
The behavior of all the orbits in the disk can be determined by considering what
happens to the disk. The intersection of the disk with the given periodic orbit comes
back to itself every period of the orbit and so do points in its neighborhood. When this
neighborhood returns, its shape is transformed. Among the points back inside the disk
are some points that will leave the disk neighborhood and others that will continue to
return. The set of points that never leaves the neighborhood of the given periodic orbit
form a fractal.
A symbolic name can be given to all the orbits that remain in the neighborhood. The
initial neighborhood disk can be divided into a small number of regions. Knowing the
sequence in which the orbit visits these regions allows the orbit to be pinpointed
exactly. The visitation sequence of the orbits provide a symbolic representation of the
dynamics, known as symbolic dynamics.
[[[[editediteditedit]]]] OrbitsOrbitsOrbitsOrbits
It is possible to describe the behavior of all initial conditions of the horseshoe map.
An initial point u0 = (x, y) gets mapped into the point u1 = f(u0). Its iterate is the point
u2 = f(u1) = f 2(u0), and repeated iteration generates the orbit u0, u1, u2, …
Under repeated iteration of the horseshoe map, most orbits end up at the fixed point in
the left cap. This is because the horseshoe maps the left cap into itself by an affine
transformation that has exactly one fixed point. Any orbit that lands on the left cap
never leaves it and converges to the fixed point in the left cap under iteration. Points
in the right cap get mapped into the left cap on the next iteration, and most points in
the square get mapped into the caps. Under iteration, most points will be part of orbits
that converge to the fixed point in the left cap, but some points of the square never
leave.[[[[editediteditedit]]]] IteratingIteratingIteratingIterating thethethethe squaresquaresquaresquare
Pre-images of the square region
Under forward iterations of the horseshoe map, the original square gets mapped into a
series of horizontal strips. The points in these horizontal strips come from vertical
strips in the original square. Let be the original square, map it forward n times,
and consider only the points that fall back into the square S0, which is a set of
horizontal stripes
∩ .
The points in the horizontal stripes came from the vertical stripes
,
which are the horizontal strips mapped backwards n times. That is, a point in
V
n
will, under n iterations of the horseshoe, end up in the set of vertical strips.
[[[[editediteditedit]]]] InvariantInvariantInvariantInvariant setsetsetset
Intersections that converge to the invariant set
Example of an invariant measure
If a point is to remain indefinitely in the square, then it must belong to a set that
maps to itself. Whether this set is empty or not has to be determined. The vertical
strips map into the horizontal strips , but not all points of map back
into . Only the points in the intersection of and may belong to , as can
be checked by following points outside the intersection for one more iteration.
The intersection of the horizontal and vertical stripes, ∩ , are squares that
converge in the limit → ∞ to the invariant set . The structure of this set can be
better understood by introducing a system of labels for all the intersections—a
symbolic dynamics.
[[[[editediteditedit]]]] SymbolicSymbolicSymbolicSymbolic dynamicsdynamicsdynamicsdynamics
The basic domains of the horseshoe map
The intersection ∩ is contained in . So any point that is in under
iteration must land in the left vertical strip A of , or on the right vertical strip B.
The lower horizontal strip of is the image of A and the upper horizontal strip is
the image of B, so H1 = f(A)∪ f(B). The strips A and B can be used to label the four
squares in the intersection of and :
Λ
A•A
= f(A) ∩ A Λ
A•B
= f(A) ∩ B
Λ
B•A
= f(B) ∩ A Λ
B•B
= f(B) ∩ B
The set ΛB•A consist of points from strip A that were in strip B in the previous iteration.
A dot is used to separate the region the point of an orbit is in from the region the point
came from.
The notation can be extended to higher iterates of the horseshoe map. The vertical
strips can be named according to the sequence of visits to strip A or strip B. For
example, the set ABB ⊂ V3 consists of the points from A that will all land in B in one
iteration and remain in B in the iteration after that:
ABB= { x∈A | f(x)∈ B and f 2(x)∈ B }
Working backwards from that trajectory determines a small region, the set ABB ,
within V3.
The horizontal strips are named from their vertical strip pre-images. In this notation,
the intersection of V2 and H2 consists of 16 squares, one of which is
ΛAB•BB = f 2(AB) ∩ BB.
All the points in ΛAB•BB are in B and will continue to be in B for at least one more
iteration. Their previous trajectory before landing in BB was A followed by B.
[[[[editediteditedit]]]] PeriodicPeriodicPeriodicPeriodic orbitsorbitsorbitsorbits
Any one of the intersections Λ
P•F
of a horizontal strip with a vertical strip, where P
and F are sequences of As and Bs, is an affine transformation of a small region in V
1
.
If P has k symbols in it, and if f –k(Λ
P•F
) and Λ
P•F
intersect, the region ΛP•F will have a
fixed point. This happens when the sequence P is the same as F. For example,
Λ
ABAB•ABAB
⊂ V
4
∩ H
4
has at least one fixed point. This point is also the same as the
fixed point in ΛAB•AB . By including more and more ABs in the P and F part of the
label of intersection, the area of the intersection can be made as small as needed. It
converges to a point that is part of a periodic orbit of the horseshoe map. The periodic
orbit can be labeled by the simplest sequence of As and Bs that labels one of the
regions the periodic orbit visits.
For every sequence of As and Bs there is a periodic orbit.
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