Topology and Chaos Du

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Topology and Chaos Dušan Repovš, University of
Ljubljana

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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 3-dimensional 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 1920s 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 ?, via the relationship ?
  2 - 2g for closed surfaces, where g is the genus.
  For surfaces with b boundary components, the
  equation reads ? 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 3-space 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 arcwise-connected 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 Si1 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
• 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

• 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
• 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).