topology and chaos du šan repovš, university of ljubljana

Maribor, July 1, 2008 D. Repovš, Topology and Chaos 1

Topology and Chaos Topology and Chaos DuDušan Repovš, University of Ljubljanašan Repovš, University of Ljubljana

Maribor, July 1, 2008 2D. Repovš, Topology and Chaos


Hopf fibrationHopf fibration


"The Wiley."The Wiley.


..


"The Topology of Chaos: Alice in Stretch and "The Topology of Chaos: Alice in Stretch and Squeezeland", a book about topological analysis Squeezeland", a book about topological analysis written by Robert Gilmorewritten by Robert Gilmore, , Nonlinear dynamics Nonlinear dynamics research groupresearch group at the Physics department of Drexel at the Physics department of Drexel University, Philadelphia and Marc Lefranc,University, Philadelphia and Marc Lefranc, LLaboratoire de Physique des Lasers, Atomes, aboratoire de Physique des Lasers, Atomes, MoléculesMolécules,, Université des Sciences et Technologies Université des Sciences et Technologies de Lillede Lille, , France and published by Wiley. France and published by Wiley.


Topological analysis is about extracting from chaotic Topological analysis is about extracting from chaotic data the topological signatures that determine the data the topological signatures that determine the stretching and squeezing mechanisms which act on stretching and squeezing mechanisms which act on flows in phase space and are responsible for flows in phase space and are responsible for generating chaotic behavior. generating chaotic behavior. TThis book providehis book providess a a detailed description of the fundamental concepts and detailed description of the fundamental concepts and tools of topological analysis. For tools of topological analysis. For 33-dimensional -dimensional systems, the methodology is well established and systems, the methodology is well established and relies on sophisticated mathematical tools such as relies on sophisticated mathematical tools such as knot theory and templates (i.e. branched manifoldsknot theory and templates (i.e. branched manifolds).).

The last chapters discuss how topological analysis The last chapters discuss how topological analysis could be extended to handle higher-dimensional could be extended to handle higher-dimensional systems, and how it can be viewed as a key part of a systems, and how it can be viewed as a key part of a general program for dynamical systems theory. general program for dynamical systems theory. TTopological analysis has proved invaluable opological analysis has proved invaluable forfor: : classification of strange attractorclassification of strange attractorss, understanding of , understanding of bifurcation sequences, extraction of symbolic bifurcation sequences, extraction of symbolic dynamical information and construction of symbolic dynamical information and construction of symbolic codings. As such, it has become a fundamental tool of codings. As such, it has become a fundamental tool of nonlinear dynamics. nonlinear dynamics.


Topology (Topology (topostopos = =place and place and logoslogos== study) is an study) is an extension of extension of geometry and analysisgeometry and analysis. Topology . Topology considerconsiderss the nature of space, investigating both its the nature of space, investigating both its fine structure and its global structure. fine structure and its global structure.

The word topology is used both for the area of study The word topology is used both for the area of study and for a family of sets with certain properties and for a family of sets with certain properties described below that are used to define a described below that are used to define a topological space. Of particular importance in the topological space. Of particular importance in the study of topology are functions or maps that are study of topology are functions or maps that are homeomorphismshomeomorphisms - - these functions can be thought these functions can be thought of as those that stretch space without tearing it of as those that stretch space without tearing it apart or sticking distinct parts together.apart or sticking distinct parts together.

When the discipline was first properly founded, When the discipline was first properly founded, toward the end of the 19th centtoward the end of the 19th century,ury, it was called it was called geometria situs (geometry of place) and analysis geometria situs (geometry of place) and analysis situs (analysis of place). Since 192situs (analysis of place). Since 1920’s0’s it it hhasas been been one of the most one of the most important area important areass within within mathematics.mathematics.

Moebius band:Moebius band:


Topology began with the investigation Topology began with the investigation byby Leonhard Leonhard

EulerEuler in in 1736 o1736 off Seven Bridges of Königsberg. 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.


EulerEuler proved that it was not possibleproved that it was not possible:: The degree of The degree of a node is the number of edges touching it; in the a node is the number of edges touching it; in the Königsberg bridge graph, three nodes have degree Königsberg bridge graph, three nodes have degree 3 and one has degree 5. 3 and one has degree 5.

Euler proved that Euler proved that such a walksuch a walk is possible if and only is possible if and only if the graph is connected, and there are exactly two if the graph is connected, and there are exactly two or zero nodes of odd degree. Such a walk is called or zero nodes of odd degree. Such a walk is called an an Eulerian pathEulerian path . Further, if there are two nodes of . Further, if there are two nodes of odd degree, those must be the starting and ending odd degree, those must be the starting and ending points of an Eulerian path. points of an Eulerian path.

Since the graph corresponding to Königsberg has Since the graph corresponding to Königsberg has four nodes of odd degree, it cannot have an four nodes of odd degree, it cannot have an Eulerian Eulerian path. path.


Intuitively, two spaces are topologically equivalent Intuitively, two spaces are topologically equivalent

if one can be deformed into the other without if one can be deformed into the other without cutting or gluing. cutting or gluing.

A traditional joke is that a topologist can't tell the A traditional joke is that a topologist can't tell the coffee mug out of which he is drinking from the coffee mug out of which he is drinking from the doughnut he is eating, since a sufficiently pliable doughnut he is eating, since a sufficiently pliable doughnut could be reshaped to the form of a coffee doughnut could be reshaped to the form of a coffee cup by creating a dimple and progressively cup by creating a dimple and progressively enlarging it, while shrinking the hole into a handle.enlarging it, while shrinking the hole into a handle.


Let X be any set and let Let X be any set and let TT be a family of subsets of X. be a family of subsets of X.

Then Then TT is a is a topology on X iftopology on X if b both the empty set and X oth the empty set and X are elements of are elements of TT. .

Any union of arbitrarily many elements of Any union of arbitrarily many elements of TT is an is an element of element of TT. . AAny intersection of finitely many elements ny intersection of finitely many elements of of TT is an element of is an element of TT. .

If If TT is a topology on X, then X together with is a topology on X, then X together with TT is called a is called a ttopological space.opological space.

All sets in All sets in TT are called open; note that in general not all are called open; note that in general not all subsets of X need be in subsets of X need be in TT. A subset of X is said to be . A subset of X is said to be closed if its complement is in closed if its complement is in TT (i.e., it is open). (i.e., it is open). A A subset ofsubset of X may be open, closed, both, or neither. X may be open, closed, both, or neither.

A map from one topological space to another is called A map from one topological space to another is called continuous if the inverse image of any open set is open. continuous if the inverse image of any open set is open.

If the function maps the reals to the reals, then this If the function maps the reals to the reals, then this definition of continuous is equivalent to the definition of definition of continuous is equivalent to the definition of continuous in calculus. continuous in calculus.

If a continuous function is one-to-one and onto and if If a continuous function is one-to-one and onto and if itsits inverseinverse is al is also continuous, then the function is called a so continuous, then the function is called a homeomorphism. homeomorphism.

If two spaces are homeomorphic, they have identical If two spaces are homeomorphic, they have identical topological properties, and are considered to be topological properties, and are considered to be topologically the same. topologically the same.


Formally, a topological manifold is a second Formally, a topological manifold is a second

countable Hausdorffcountable Hausdorff space that is locally space that is locally homeomorphic to Euclidean spacehomeomorphic to Euclidean space, which , which means means that every point has a neighborhood homeomorphic that every point has a neighborhood homeomorphic to an open Euclidean to an open Euclidean nn-ball-ball


The genus of a connected, orientable surface is an The genus of a connected, orientable surface is an

integer representing the maximum number of integer representing the maximum number of cuttings along closed simple curves without cuttings along closed simple curves without rendering the resultant manifold disconnected. It is rendering the resultant manifold disconnected. It is equal to the number of handles on it. equal to the number of handles on it.

Alternatively, it can be defined in terms of the Euler Alternatively, it can be defined in terms of the Euler characteristic χ, via the relationship characteristic χ, via the relationship χ = 2 − 2gχ = 2 − 2g for for closed surfaces, where closed surfaces, where gg is the genus. For surfaces is the genus. For surfaces with with bb boundary components, the equation reads boundary components, the equation reads χ χ = 2 − 2g − b= 2 − 2g − b..


KKnot theory is the area of topology that studnot theory is the area of topology that studies ies

embeddingembeddingss of of thethe circle in circle intoto 3-dimensional 3-dimensional Euclidean space. Two knots are equivalent if one Euclidean space. Two knots are equivalent if one can be transformed into the other via a can be transformed into the other via a deformation of R3 upon itself (known as an deformation of R3 upon itself (known as an ambient isotopy); these transformations ambient isotopy); these transformations correspond to manipulations of a knotted string correspond to manipulations of a knotted string that do not involve cutting the string or passing the that do not involve cutting the string or passing the string through itself.string through itself.


Knot Theory Puzzle: Knot Theory Puzzle: Separate the rope from the carabiners without Separate the rope from the carabiners without

cutting the rope cutting the rope and/or unlocking the carabiners!and/or unlocking the carabiners!


Reideister moves I, II and III:Reideister moves I, II and III:


A knot invariant is a "quantity" that is the same for A knot invariant is a "quantity" that is the same for

equivalent knotsequivalent knots.. An invariant may take the same An invariant may take the same value on two different knots, so by itself may be value on two different knots, so by itself may be incapable of distinguishing all knots.incapable of distinguishing all knots.

"Classical" knot invariants include the knot group, "Classical" knot invariants include the knot group, which is the fundamental groupwhich is the fundamental group of the knot of the knot complement, and the Alexander polynomial. complement, and the Alexander polynomial.

Actually, there are two trefoil knots, called the Actually, there are two trefoil knots, called the right and left-handed trefoils, which are mirror right and left-handed trefoils, which are mirror images of each other. These are not equivalent to images of each other. These are not equivalent to each othereach other.. This was shown by Max Dehn (Dehn This was shown by Max Dehn (Dehn 1914). 1914).


LLet et LL be a tame oriented be a tame oriented k knot or link in Euclidean not or link in Euclidean 3-space. A Seifert surface is a compact, connected, 3-space. A Seifert surface is a compact, connected, oriented surface oriented surface SS embedded in 3-space whose embedded in 3-space whose boundary is boundary is LL such that the orientation on such that the orientation on LL is just is just the induced orientation from the induced orientation from SS, and every , and every connected component of connected component of SS has non-empty has non-empty boundary.boundary.

AAny cny closedlosed oriented surface with boundary in 3- oriented surface with boundary in 3-space is the Seifert surface associated to its space is the Seifert surface associated to its boundary link. A single knot or link can have many boundary link. A single knot or link can have many different inequivalent Seifert surfaces. It is different inequivalent Seifert surfaces. It is important to note that a Seifert surface must be important to note that a Seifert surface must be oriented. It is possible to associate unoriented (and oriented. It is possible to associate unoriented (and not necessarily orientable) surfaces to knots as not necessarily orientable) surfaces to knots as

well.well.


The fundamental group The fundamental group ((introduced by Poincaréintroduced by Poincaré)) of of an arcwise-connected set an arcwise-connected set XX is the group formed by is the group formed by the sets of equivalence classes of the set of all the sets of equivalence classes of the set of all loops, i.e., paths with initial and final points at a loops, i.e., paths with initial and final points at a given basepointgiven basepoint p p, under the equivalence relation , under the equivalence relation of homotopy. of homotopy.

The identity element of this group is the set of all The identity element of this group is the set of all paths homotopic to the degenerate path consisting paths homotopic to the degenerate path consisting of the point of the point pp. The fundamental groups of . The fundamental groups of homeomorphic spaces are isomorphic. In fact, the homeomorphic spaces are isomorphic. In fact, the fundamental group only depends on the homotopy fundamental group only depends on the homotopy type of type of X. X.


SSingular homology refers to the study of a certain set ingular homology refers to the study of a certain set of topological invariants of a topological space of topological invariants of a topological space XX, the , the so-called homology groups so-called homology groups HnHn((XX). ).

Singular homology is a particular example of a Singular homology is a particular example of a homology theory, which has now grown to be a rather homology theory, which has now grown to be a rather broad collection of theories.broad collection of theories.

Of the various theories, it is perhaps one of the simpler Of the various theories, it is perhaps one of the simpler ones to understand, being built on fairly concrete ones to understand, being built on fairly concrete constructions.constructions.

In brief, singular homology is constructed by taking In brief, singular homology is constructed by taking maps of the standard maps of the standard nn-simplex-simplex to a topological space, to a topological space, and composing them into formal sums, called singular and composing them into formal sums, called singular chains. chains.

The boundary operation on a simplex induces a The boundary operation on a simplex induces a singular chain complex. singular chain complex.

The singular homology is then the homology of the The singular homology is then the homology of the chain complex.chain complex.

The resulting homology groups are the same for all The resulting homology groups are the same for all homotopically equivalent spaces, which is the reason homotopically equivalent spaces, which is the reason for their study. for their study.


TThe Whitehead manifold is an open 3-manifold that he Whitehead manifold is an open 3-manifold that is contractible, but not homeomorphic to R3.is contractible, but not homeomorphic to R3. Whitehead discovered this puzzling object while he Whitehead discovered this puzzling object while he was trying to prove the Poincaré conjecture.was trying to prove the Poincaré conjecture.

A contractible manifold is one that can A contractible manifold is one that can continuously be shrunk to a point inside the continuously be shrunk to a point inside the manifold itself. For example, an open ball is a manifold itself. For example, an open ball is a contractible manifold. All manifolds homeomorphic contractible manifold. All manifolds homeomorphic to the ball are contractible, too. One can ask to the ball are contractible, too. One can ask whether whether allall contractible manifolds are contractible manifolds are homeomorphic to a ball. For dimensions 1 and 2, homeomorphic to a ball. For dimensions 1 and 2, the answer is classical and it is "yes". the answer is classical and it is "yes". DDension 3 ension 3 presents the first counterexamplepresents the first counterexample..


FFor a given prime number or a given prime number pp, the , the pp-adic solenoid is -adic solenoid is the topological group defined as inverse limit of the the topological group defined as inverse limit of the inverse systeminverse system ((SiSi, , qiqi) ) , , where where ii runs over natural runs over natural numbers, and each numbers, and each SiSi is a circle, and is a circle, and qiqi wraps the wraps the circle circle SiSi+1 +1 pp times around the circle times around the circle SiSi..

The solenoid is the standard example of a space The solenoid is the standard example of a space with bad behaviour with respect to various with bad behaviour with respect to various homology theories, not seen for simplicial homology theories, not seen for simplicial complexes. For example, in , one can construct a complexes. For example, in , one can construct a non-exact long homology sequence using the non-exact long homology sequence using the solenoid. solenoid.


Eversion of SphereEversion of Sphere


TOPOLOGY TOPOLOGY AND CHAOSAND CHAOS

Poincaré developed topology and exploited this Poincaré developed topology and exploited this new branch of mathematics in ingenious ways to new branch of mathematics in ingenious ways to study the properties of differential equations. study the properties of differential equations. Ideas and tools from this branch of mathematics Ideas and tools from this branch of mathematics are particularly well suited to describe and to are particularly well suited to describe and to classify a restricted but enormously rich class of classify a restricted but enormously rich class of chaotic dynamical systems, and thus the term chaotic dynamical systems, and thus the term chaos topology refers to the description of such chaos topology refers to the description of such systems. These systems are restricted to flows in systems. These systems are restricted to flows in 33-dimensional spaces, but they are very rich -dimensional spaces, but they are very rich because these are the only chaotic flows that can because these are the only chaotic flows that can easily be visualized at present. easily be visualized at present.

In this description there is a hierarchy of In this description there is a hierarchy of structures that we study. This hierarchy can be structures that we study. This hierarchy can be expressed in biological terms. The skeleton of expressed in biological terms. The skeleton of the attractor is its set of unstable periodic the attractor is its set of unstable periodic orbits, the body is the branched manifold that orbits, the body is the branched manifold that describes the attractor, and the skin that describes the attractor, and the skin that surrounds the attractor is the surface of its surrounds the attractor is the surface of its bounding torus.bounding torus.


Periodic orbits & topological invariantsPeriodic orbits & topological invariants A deterministic trajectory from a prescribed initial A deterministic trajectory from a prescribed initial

condition can exhibit bizarre behavior. condition can exhibit bizarre behavior. Plots of such trajectories in the phase space are Plots of such trajectories in the phase space are

called strange attractors or chaotic attractors. called strange attractors or chaotic attractors. A useful working definition of chaotic motion is A useful working definition of chaotic motion is

motion that is: motion that is: deterministic deterministic bounded bounded recurrent but not periodicrecurrent but not periodic

sensitive to initial conditionssensitive to initial conditions


Relationship with topology: In three dimensional Relationship with topology: In three dimensional

space an integer invariant can be associated to space an integer invariant can be associated to each pair of closed orbits. This invariant is the each pair of closed orbits. This invariant is the Gauss linking number. It can be defined by an Gauss linking number. It can be defined by an integraintegral.l.

TThis integralhis integral always has integer values - which is a always has integer values - which is a signature of topological origins. This can be signature of topological origins. This can be explained many ways, all equivalent, e.g. take one explained many ways, all equivalent, e.g. take one of the orbits, say , dip it into soapy water, then pull of the orbits, say , dip it into soapy water, then pull it out. A soap film will form whose boundary is the it out. A soap film will form whose boundary is the closed orbit (this is a difficult theorem and the closed orbit (this is a difficult theorem and the

surface is called a Seifert surface).surface is called a Seifert surface).

topology and chaos du šan repovš, university of ljubljana

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