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Topology and Chaos Dušan Repovš, University of Ljubljana Maribor, July 1, 2008 D. Repovš, Topology and Chaos 1 Maribor, July 1, 2008 D. Repovš, Topology and Chaos 2 Maribor, July 1, 2008 D. Repovš, Topology and Chaos 3 Maribor, July 1, 2008 D. Repovš, Topology and Chaos 4 Hopf fibration Maribor, July 1, 2008 D. Repovš, Topology and Chaos 5 Maribor, July 1, 2008 D. Repovš, Topology and Chaos 6 Maribor, July 1, 2008 D. Repovš, Topology and Chaos 7 Maribor, July 1, 2008 D. Repovš, Topology and Chaos 8 Maribor, July 1, 2008 D. Repovš, Topology and Chaos 9 Maribor, July 1, 2008 D. Repovš, Topology and Chaos 10 "The Wiley. Maribor, July 1, 2008 D. Repovš, Topology and Chaos 11 . Maribor, July 1, 2008 D. Repovš, Topology and Chaos 12 . Maribor, July 1, 2008 D. Repovš, Topology and Chaos 13 . Maribor, July 1, 2008 D. Repovš, Topology and Chaos 14 . Maribor, July 1, 2008 D. Repovš, Topology and Chaos 15 . Maribor, July 1, 2008 D. Repovš, Topology and Chaos 16 . Maribor, July 1, 2008 D. Repovš, Topology and Chaos 17 . Maribor, July 1, 2008 D. Repovš, Topology and Chaos 18 . Maribor, July 1, 2008 D. Repovš, Topology and Chaos 19 . Maribor, July 1, 2008 D. Repovš, Topology and Chaos 20 Maribor, July 1, 2008 D. Repovš, Topology and Chaos 21 Maribor, July 1, 2008 D. Repovš, Topology and Chaos 22 Maribor, July 1, 2008 D. Repovš, Topology and Chaos 23 Maribor, July 1, 2008 D. Repovš, Topology and Chaos 24 Maribor, July 1, 2008 D. Repovš, Topology and Chaos 25 "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. Maribor, July 1, 2008 D. Repovš, Topology and Chaos 26 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. Maribor, July 1, 2008 D. Repovš, Topology and Chaos 27 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: Maribor, July 1, 2008 D. Repovš, Topology and Chaos 28 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. Maribor, July 1, 2008 D. Repovš, Topology and Chaos 29 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. Maribor, July 1, 2008 D. Repovš, Topology and Chaos 30 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. Maribor, July 1, 2008 D. Repovš, Topology and Chaos 31 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. Maribor, July 1, 2008 D. Repovš, Topology and Chaos 32 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 Maribor, July 1, 2008 D. Repovš, Topology and Chaos 33 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. Maribor, July 1, 2008 D. Repovš, Topology and Chaos 34 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. Maribor, July 1, 2008 D. Repovš, Topology and Chaos 35 Knot Theory Puzzle: Separate the rope from the carabiners without cutting the rope and/or unlocking the carabiners! Maribor, July 1, 2008 D. Repovš, Topology and Chaos 36 Reideister moves I, II and III: Maribor, July 1, 2008 D. Repovš, Topology and Chaos 37 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). Maribor, July 1, 2008 D. Repovš, Topology and Chaos 38 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. Maribor, July 1, 2008 D. Repovš, Topology and Chaos 39 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. Maribor, July 1, 2008 D. Repovš, Topology and Chaos 40 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. Maribor, July 1, 2008 D. Repovš, Topology and Chaos 41 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. Maribor, July 1, 2008 D. Repovš, Topology and Chaos 42 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. Maribor, July 1, 2008 D. Repovš, Topology and Chaos 43 Eversion of Sphere Maribor, July 1, 2008 D. Repovš, Topology and Chaos 44 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. Maribor, July 1, 2008 D. Repovš, Topology and Chaos 45 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 Maribor, July 1, 2008 D. Repovš, Topology and Chaos 46 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). Maribor, July 1, 2008 D. Repovš, Topology and Chaos 47