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Transcript
Topology and Chaos
Dušan Repovš, University of Ljubljana
Maribor, July 1, 2008
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Hopf fibration
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"The Wiley.
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"The Topology of Chaos: Alice in Stretch and Squeezeland", a book
about topological analysis written by Robert Gilmore, Nonlinear
dynamics research group at the Physics department of Drexel
University, Philadelphia and Marc Lefranc, Laboratoire de Physique
des Lasers, Atomes, Molécules, Université des Sciences et
Technologies de Lille, France and published by Wiley.
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Topological analysis is about extracting from chaotic data the
topological signatures that determine the stretching and squeezing
mechanisms which act on flows in phase space and are responsible for
generating chaotic behavior. This book provides a detailed description
of the fundamental concepts and tools of topological analysis. For 3dimensional systems, the methodology is well established and relies on
sophisticated mathematical tools such as knot theory and templates
(i.e. branched manifolds).
The last chapters discuss how topological analysis could be extended to
handle higher-dimensional systems, and how it can be viewed as a key
part of a general program for dynamical systems theory. Topological
analysis has proved invaluable for: classification of strange attractors,
understanding of bifurcation sequences, extraction of symbolic
dynamical information and construction of symbolic codings. As such,
it has become a fundamental tool of nonlinear dynamics.
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Topology (topos =place and logos= study) is an extension of
geometry and analysis. Topology considers the nature of space,
investigating both its fine structure and its global structure.
The word topology is used both for the area of study and for a family
of sets with certain properties described below that are used to define
a topological space. Of particular importance in the study of
topology are functions or maps that are homeomorphisms - these
functions can be thought of as those that stretch space without
tearing it apart or sticking distinct parts together.
When the discipline was first properly founded, toward the end of the
19th century, it was called geometria situs (geometry of place) and
analysis situs (analysis of place). Since 1920’s it has been one of the
most important areas within mathematics.
Moebius band:
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Topology began with the investigation by Leonhard Euler in 1736 of
Seven Bridges of Königsberg. This was a famous problem.
Königsberg, Prussia (now Kaliningrad, Russia) is set on the Prege
River, and included two large islands which were connected to each
other and the mainland by seven bridges.
The problem was whether it is possible to walk a route that crosses
each bridge exactly once.
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Euler proved that it was not possible: The degree of a node is the
number of edges touching it; in the Königsberg bridge graph, three
nodes have degree 3 and one has degree 5.
Euler proved that such a walk is possible if and only if the graph is
connected, and there are exactly two or zero nodes of odd degree.
Such a walk is called an Eulerian path . Further, if there are two
nodes of odd degree, those must be the starting and ending points of
an Eulerian path.
Since the graph corresponding to Königsberg has four nodes of odd
degree, it cannot have an Eulerian path.
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Intuitively, two spaces are topologically equivalent if one can be
deformed into the other without cutting or gluing.
A traditional joke is that a topologist can't tell the coffee mug out of
which he is drinking from the doughnut he is eating, since a
sufficiently pliable doughnut could be reshaped to the form of a
coffee cup by creating a dimple and progressively enlarging it, while
shrinking the hole into a handle.
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Let X be any set and let T be a family of subsets of X. Then T is a
topology on X if both the empty set and X are elements of T.
Any union of arbitrarily many elements of T is an element of T. Any
intersection of finitely many elements of T is an element of T.
If T is a topology on X, then X together with T is called a
topological space.
All sets in T are called open; note that in general not all subsets of X
need be in T. A subset of X is said to be closed if its complement is
in T (i.e., it is open). A subset of X may be open, closed, both, or
neither.
A map from one topological space to another is called continuous if
the inverse image of any open set is open.
If the function maps the reals to the reals, then this definition of
continuous is equivalent to the definition of continuous in calculus.
If a continuous function is one-to-one and onto and if its inverse is
also continuous, then the function is called a homeomorphism.
If two spaces are homeomorphic, they have identical topological
properties, and are considered to be topologically the same.
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Formally, a topological manifold is a second countable Hausdorff
space that is locally homeomorphic to Euclidean space, which
means that every point has a neighborhood homeomorphic to an
open Euclidean n-ball
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The genus of a connected, orientable surface is an integer
representing the maximum number of cuttings along closed simple
curves without rendering the resultant manifold disconnected. It is
equal to the number of handles on it.
Alternatively, it can be defined in terms of the Euler characteristic χ,
via the relationship χ = 2 − 2g for closed surfaces, where g is the
genus. For surfaces with b boundary components, the equation reads
χ = 2 − 2g − b.
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Knot theory is the area of topology that studies embeddings of the
circle into 3-dimensional Euclidean space. Two knots are equivalent
if one can be transformed into the other via a deformation of R3 upon
itself (known as an ambient isotopy); these transformations
correspond to manipulations of a knotted string that do not involve
cutting the string or passing the string through itself.
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Knot Theory Puzzle:
Separate the rope from the carabiners without cutting the rope
and/or unlocking the carabiners!
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Reideister moves I, II and III:
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A knot invariant is a "quantity" that is the same for equivalent knots.
An invariant may take the same value on two different knots, so by
itself may be incapable of distinguishing all knots.
"Classical" knot invariants include the knot group, which is the
fundamental group of the knot complement, and the Alexander
polynomial.
Actually, there are two trefoil knots, called the right and left-handed
trefoils, which are mirror images of each other. These are not
equivalent to each other. This was shown by Max Dehn (Dehn 1914).
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Let L be a tame oriented knot or link in Euclidean 3-space. A Seifert
surface is a compact, connected, oriented surface S embedded in 3space whose boundary is L such that the orientation on L is just the
induced orientation from S, and every connected component of S has
non-empty boundary.
Any closed oriented surface with boundary in 3-space is the Seifert
surface associated to its boundary link. A single knot or link can have
many different inequivalent Seifert surfaces. It is important to note
that a Seifert surface must be oriented. It is possible to associate
unoriented (and not necessarily orientable) surfaces to knots as well.
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The fundamental group (introduced by Poincaré) of an arcwiseconnected set X is the group formed by the sets of equivalence
classes of the set of all loops, i.e., paths with initial and final points at
a given basepoint p, under the equivalence relation of homotopy.
The identity element of this group is the set of all paths homotopic to
the degenerate path consisting of the point p. The fundamental
groups of homeomorphic spaces are isomorphic. In fact, the
fundamental group only depends on the homotopy type of X.
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Singular homology refers to the study of a certain set of topological
invariants of a topological space X, the so-called homology groups
Hn(X).
Singular homology is a particular example of a homology theory,
which has now grown to be a rather broad collection of theories.
Of the various theories, it is perhaps one of the simpler ones to
understand, being built on fairly concrete constructions.
In brief, singular homology is constructed by taking maps of the
standard n-simplex to a topological space, and composing them into
formal sums, called singular chains.
The boundary operation on a simplex induces a singular chain
complex.
The singular homology is then the homology of the chain complex.
The resulting homology groups are the same for all homotopically
equivalent spaces, which is the reason for their study.
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The Whitehead manifold is an open 3-manifold that is contractible,
but not homeomorphic to R3. Whitehead discovered this puzzling
object while he was trying to prove the Poincaré conjecture.
A contractible manifold is one that can continuously be shrunk to a
point inside the manifold itself. For example, an open ball is a
contractible manifold. All manifolds homeomorphic to the ball are
contractible, too. One can ask whether all contractible manifolds are
homeomorphic to a ball. For dimensions 1 and 2, the answer is
classical and it is "yes". Dension 3 presents the first counterexample.
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For a given prime number p, the p-adic solenoid is the topological
group defined as inverse limit of the inverse system (Si, qi) , where i
runs over natural numbers, and each Si is a circle, and qi wraps the
circle Si+1 p times around the circle Si.
The solenoid is the standard example of a space with bad behaviour
with respect to various homology theories, not seen for simplicial
complexes. For example, in , one can construct a non-exact long
homology sequence using the solenoid.
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Eversion of Sphere
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TOPOLOGY AND CHAOS
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Poincaré developed topology and exploited this new
branch of mathematics in ingenious ways to study the
properties of differential equations. Ideas and tools from
this branch of mathematics are particularly well suited to
describe and to classify a restricted but enormously rich
class of chaotic dynamical systems, and thus the term
chaos topology refers to the description of such systems.
These systems are restricted to flows in 3-dimensional
spaces, but they are very rich because these are the only
chaotic flows that can easily be visualized at present.
In this description there is a hierarchy of structures that
we study. This hierarchy can be expressed in biological
terms. The skeleton of the attractor is its set of unstable
periodic orbits, the body is the branched manifold that
describes the attractor, and the skin that surrounds the
attractor is the surface of its bounding torus.
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Periodic orbits & topological invariants
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A deterministic trajectory from a prescribed initial condition can exhibit
bizarre behavior.
Plots of such trajectories in the phase space are called strange attractors
or chaotic attractors.
A useful working definition of chaotic motion is motion that is:
deterministic
bounded
recurrent but not periodic
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sensitive to initial conditions
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Relationship with topology: In three dimensional space an integer
invariant can be associated to each pair of closed orbits. This
invariant is the Gauss linking number. It can be defined by an
integral.
This integral always has integer values - which is a signature of
topological origins. This can be explained many ways, all equivalent,
e.g. take one of the orbits, say , dip it into soapy water, then pull it
out. A soap film will form whose boundary is the closed orbit (this is
a difficult theorem and the surface is called a Seifert surface).
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