Introduction Knot Theory Nonlinear Dynamic Topology in chaos Open Questions Summary

Topology in ChaosA tangled tale about knot, link, template, and strange attractor

Wang XiongCentre for Chaos & Complex Networks

City University of Hong KongEmail: wangxiong8686@gmail.com

Oct 4 2013 Venue: B6605, CityU

Wang Xiong

Topology in Chaos

Introduction Knot Theory Nonlinear Dynamic Topology in chaos Open Questions Summary

Table of contents

1 Introduction

2 Knot Theory

3 Nonlinear Dynamic

4 Topology in chaos

5 Open Questions

6 Summary

Wang Xiong

Topology in Chaos

Introduction Knot Theory Nonlinear Dynamic Topology in chaos Open Questions Summary

Topology, and Dynamical Systems Theory

TopologyTopology is the mathematicalstudy of shapes and spaces,and focus on properties that arepreserved under continuousdeformations includingstretching and bending, but nottearing or gluing.One interesting branch is knottheory

Dynamical Systems TheoryDynamical Systems Theorydeals with the long-termqualitative behavior ofdynamical systems, and thestudies of the solutions to theequations of motion of systems.Much of modern research isfocused on the study of chaoticsystems.

This talk will show you the interesting and crucial connectionbetween these two important areas of mathematics...

Wang Xiong

Topology in Chaos

Introduction Knot Theory Nonlinear Dynamic Topology in chaos Open Questions Summary

Topology, and Dynamical Systems Theory

TopologyTopology is the mathematicalstudy of shapes and spaces,and focus on properties that arepreserved under continuousdeformations includingstretching and bending, but nottearing or gluing.One interesting branch is knottheory

Dynamical Systems TheoryDynamical Systems Theorydeals with the long-termqualitative behavior ofdynamical systems, and thestudies of the solutions to theequations of motion of systems.Much of modern research isfocused on the study of chaoticsystems.

This talk will show you the interesting and crucial connectionbetween these two important areas of mathematics...

Wang Xiong

Topology in Chaos

Introduction Knot Theory Nonlinear Dynamic Topology in chaos Open Questions Summary

1 Introduction

2 Knot Theory

3 Nonlinear Dynamic

4 Topology in chaos

5 Open Questions

6 Summary

Wang Xiong

Topology in Chaos

Introduction Knot Theory Nonlinear Dynamic Topology in chaos Open Questions Summary

What is a knot

Figure: Knot in daily life. It is saidthat before the invention ofcharacters, the ancient peoplekept records by tying knots.

Figure: Beautiful Chinese knot

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Topology in Chaos

Introduction Knot Theory Nonlinear Dynamic Topology in chaos Open Questions Summary

Mathematical knot

While inspired by knots which appear in daily life in shoelacesand rope, a mathematician’s knot differs in that the ends arejoined together so that it cannot be undone.

Mathematical definition of knotA knot is an embedding of a circle in 3-dimensional Euclideanspace e : S1 → R3. Two knots K ,K ? have the same type ifthere is a diffeomorphism of pairs (K ,R3)→ (K ?,R3)

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Topology in Chaos

Introduction Knot Theory Nonlinear Dynamic Topology in chaos Open Questions Summary

Knot equivalence

Two mathematical knots are equivalent if one can betransformed into the other via a deformation of R3 upon itself.Roughly, this means do not involve cutting the string.

Figure: On the left, the unknot, and a knot equivalent to it. It can bemore difficult to determine whether complex knots, such as the oneon the right, are equivalent to the unknot.

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Topology in Chaos

Introduction Knot Theory Nonlinear Dynamic Topology in chaos Open Questions Summary

Knot sum

Two knots can be added by cutting both knots and joiningthe pairs of ends.

Note also that if O is a picture of the ‘unknot’, and K is anyfixed picture of a knot K , then K#O just gives us a newpicture of K .Caution: there do not exist knots K ,K ′, both different fromthe unknot, with K#K ′ the unknot O. That is, knots forman addition semi-group, using the operation #.

Wang Xiong

Topology in Chaos

Introduction Knot Theory Nonlinear Dynamic Topology in chaos Open Questions Summary

Knot sum

Two knots can be added by cutting both knots and joiningthe pairs of ends.

Note also that if O is a picture of the ‘unknot’, and K is anyfixed picture of a knot K , then K#O just gives us a newpicture of K .

Caution: there do not exist knots K ,K ′, both different fromthe unknot, with K#K ′ the unknot O. That is, knots forman addition semi-group, using the operation #.

Wang Xiong

Topology in Chaos

Introduction Knot Theory Nonlinear Dynamic Topology in chaos Open Questions Summary

Knot sum

Two knots can be added by cutting both knots and joiningthe pairs of ends.

Note also that if O is a picture of the ‘unknot’, and K is anyfixed picture of a knot K , then K#O just gives us a newpicture of K .Caution: there do not exist knots K ,K ′, both different fromthe unknot, with K#K ′ the unknot O. That is, knots forman addition semi-group, using the operation #.

Wang Xiong

Topology in Chaos

Introduction Knot Theory Nonlinear Dynamic Topology in chaos Open Questions Summary

Prime knot

A knot is prime if it is non-trivial and cannot be written asthe knot sum of two non-trivial knots.

A knot that can be written as such a sum composite.There is a prime decomposition for knots, analogous toprime and composite numbersIt can be a nontrivial problem to determine whether a givenknot is prime or not.

Schubert’s theoremEvery knot can be uniquely expressed as a connected sum ofprime knots.

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Topology in Chaos

Introduction Knot Theory Nonlinear Dynamic Topology in chaos Open Questions Summary

Prime knot

A knot is prime if it is non-trivial and cannot be written asthe knot sum of two non-trivial knots.A knot that can be written as such a sum composite.

There is a prime decomposition for knots, analogous toprime and composite numbersIt can be a nontrivial problem to determine whether a givenknot is prime or not.

Schubert’s theoremEvery knot can be uniquely expressed as a connected sum ofprime knots.

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Topology in Chaos

Introduction Knot Theory Nonlinear Dynamic Topology in chaos Open Questions Summary

Prime knot

A knot is prime if it is non-trivial and cannot be written asthe knot sum of two non-trivial knots.A knot that can be written as such a sum composite.There is a prime decomposition for knots, analogous toprime and composite numbers

It can be a nontrivial problem to determine whether a givenknot is prime or not.

Schubert’s theoremEvery knot can be uniquely expressed as a connected sum ofprime knots.

Wang Xiong

Topology in Chaos

Introduction Knot Theory Nonlinear Dynamic Topology in chaos Open Questions Summary

Prime knot

A knot is prime if it is non-trivial and cannot be written asthe knot sum of two non-trivial knots.A knot that can be written as such a sum composite.There is a prime decomposition for knots, analogous toprime and composite numbersIt can be a nontrivial problem to determine whether a givenknot is prime or not.

Schubert’s theoremEvery knot can be uniquely expressed as a connected sum ofprime knots.

Wang Xiong

Topology in Chaos

Introduction Knot Theory Nonlinear Dynamic Topology in chaos Open Questions Summary

Prime knot

A knot is prime if it is non-trivial and cannot be written asthe knot sum of two non-trivial knots.A knot that can be written as such a sum composite.There is a prime decomposition for knots, analogous toprime and composite numbersIt can be a nontrivial problem to determine whether a givenknot is prime or not.

Schubert’s theoremEvery knot can be uniquely expressed as a connected sum ofprime knots.

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Topology in Chaos

Introduction Knot Theory Nonlinear Dynamic Topology in chaos Open Questions Summary

Prime knots are the building blocks

Figure: Prime knots are the building blocks of all knots. A table ofprime knots up to seven crossings.

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Topology in Chaos

Introduction Knot Theory Nonlinear Dynamic Topology in chaos Open Questions Summary

Mathematical definition of Link

LinkA link L is the image under anembedding of N ≥ 1 disjointcircles in R3. If N = 1 it’s aknot.Two links L,L? have the sametype if there is adiffeomorphism of pairs(L,R3)→ (L?,R3).

Figure: A link with threecomponents each equivalent tothe unknot

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The problem of recognizing

The problem ofrecognizing when two knotdiagrams represent thesame knot type is a highlynon-trivial problem.That’s why we have a needfor computable invariantsthat are independent of theprojection.

Figure: The problem ofrecognizing is very difficult, evenfor those equivalent to the unknot

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Invariants

A knot (or link) invariant is a ”quantity” that is the same forequivalent knots (or links).Caution: generally not vise verse.

An example of a link-type-invariant is the number ofcomponents in a link LSome invariants are indeed numbers, but invariants canrange from the simple, such as a yes/no answer, to thoseas complex as a homology theory .Research on invariants is not only motivated by the basicproblem of distinguishing one knot from another but also tounderstand fundamental properties of knots and theirrelations to other branches of mathematics.

Wang Xiong

Topology in Chaos

Introduction Knot Theory Nonlinear Dynamic Topology in chaos Open Questions Summary

Invariants

A knot (or link) invariant is a ”quantity” that is the same forequivalent knots (or links).Caution: generally not vise verse.An example of a link-type-invariant is the number ofcomponents in a link L

Some invariants are indeed numbers, but invariants canrange from the simple, such as a yes/no answer, to thoseas complex as a homology theory .Research on invariants is not only motivated by the basicproblem of distinguishing one knot from another but also tounderstand fundamental properties of knots and theirrelations to other branches of mathematics.

Wang Xiong

Topology in Chaos

Introduction Knot Theory Nonlinear Dynamic Topology in chaos Open Questions Summary

Invariants

A knot (or link) invariant is a ”quantity” that is the same forequivalent knots (or links).Caution: generally not vise verse.An example of a link-type-invariant is the number ofcomponents in a link LSome invariants are indeed numbers, but invariants canrange from the simple, such as a yes/no answer, to thoseas complex as a homology theory .Research on invariants is not only motivated by the basicproblem of distinguishing one knot from another but also tounderstand fundamental properties of knots and theirrelations to other branches of mathematics.

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Topology in Chaos

Introduction Knot Theory Nonlinear Dynamic Topology in chaos Open Questions Summary

Gauss linking number

The first subtle invariant is due to Gauss, it is a numericalinvariant that describes the linking of two closed curves inthree-dimensional space with an explicit formula as a doubleline integral, the Gauss linking integral:

∫K

∫K ′

(x′ − x)(dydz′ − dzdy′) + (y′ − y)(dzdx′ − dxdz′) + (z′ − z)(dxdy′ − dydx′)

((x′ − x)2 + (y′ − y)2 + (z′ − z)2)3/2

The Gauss integral turns out to be an integral multiple of 4π,and the integer is known as the linking number L(K ,K ′).

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Topology in Chaos

Introduction Knot Theory Nonlinear Dynamic Topology in chaos Open Questions Summary

Gauss linking number

Any two unlinked curveshave linking number zero.However, two curves withlinking number zero maystill be linked (e.g. theWhitehead link).Reversing the orientationof either of the curvesnegates the linkingnumber, while reversingthe orientation of bothcurves leaves itunchanged.

Figure: The two curves of theWhitehead link have linkingnumber zero.

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Topology in Chaos

Introduction Knot Theory Nonlinear Dynamic Topology in chaos Open Questions Summary

Some knot invariant

Turning to knots, the minimum crossing number cmin(K )over all possible projections K of a knot is, by definition, aknot or link type invariant, however (except for very lowcrossing number) it turns out to be astonishingly difficult tocompute, and to this day we do not have any realunderstanding of it in the general case

The Jones polynomial is a very sophisticated knot typeinvariant. It’s a Laurent polynomial with integer coefficients.Unlike the minimum crossing number, it can be calculatedfrom any projection, even though the calculation may belong and complicated if there are many crossings.

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Topology in Chaos

Introduction Knot Theory Nonlinear Dynamic Topology in chaos Open Questions Summary

Some knot invariant

Turning to knots, the minimum crossing number cmin(K )over all possible projections K of a knot is, by definition, aknot or link type invariant, however (except for very lowcrossing number) it turns out to be astonishingly difficult tocompute, and to this day we do not have any realunderstanding of it in the general caseThe Jones polynomial is a very sophisticated knot typeinvariant. It’s a Laurent polynomial with integer coefficients.Unlike the minimum crossing number, it can be calculatedfrom any projection, even though the calculation may belong and complicated if there are many crossings.

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Topology in Chaos

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Torus knots and links

If we restrict our attentionto special classes of knotsor links, then the search forinvariants can sometimesbe very much simpler thanin the general case.A torus knot is a specialkind of knot that lies on thesurface of an unknottedtorus in R3

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Torus knots and links

Torus knots are a well-understood class. They are classified bya pair of integers (p,q) (up to the indeterminacy(p,q) ≈ (q,p) ≈ (−p,−q) ≈ (−q,−p)).

Figure: Four examples of torus knots and links. Their classifyinginteger pairs, reading left to right, are (3,5), (5,5),5,8), (7,9)

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Summary: knots are complex...

Figure: Knots with 7 or fewer crossings From http://katlas.math.toronto.edu/wiki/The_Rolfsen_Knot_Table

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Topology in Chaos

Introduction Knot Theory Nonlinear Dynamic Topology in chaos Open Questions Summary

Summary: knots are complex...

Figure: Knots with 8 crossings, the number of inequivalent knotsincreases...

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Topology in Chaos

Introduction Knot Theory Nonlinear Dynamic Topology in chaos Open Questions Summary

Summary: knots are complex...

Figure: Knots with 9 crossings, the number of inequivalent knotsincreases dramatically... more and more complicated...Wang Xiong

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Introduction Knot Theory Nonlinear Dynamic Topology in chaos Open Questions Summary

Summary: links are even more complex...

Figure: Menasco’s Knot Theory Hot List. Fromhttp://www.math.buffalo.edu/ menasco/knot-theory.html

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Summary: complex knots and links ...

Knot theory is to understand all possible knot...too bigproblem

Prime knots are building block... still hardEven for same type of knot... hard to recognize, we needcomputable invariantsHave to restrict our attention to special classes ofknots...only very special classes are well understoodLinks are collection of knots...

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Topology in Chaos

Introduction Knot Theory Nonlinear Dynamic Topology in chaos Open Questions Summary

Summary: complex knots and links ...

Knot theory is to understand all possible knot...too bigproblemPrime knots are building block... still hard

Even for same type of knot... hard to recognize, we needcomputable invariantsHave to restrict our attention to special classes ofknots...only very special classes are well understoodLinks are collection of knots...

Wang Xiong

Topology in Chaos

Introduction Knot Theory Nonlinear Dynamic Topology in chaos Open Questions Summary

Summary: complex knots and links ...

Knot theory is to understand all possible knot...too bigproblemPrime knots are building block... still hardEven for same type of knot... hard to recognize, we needcomputable invariants

Have to restrict our attention to special classes ofknots...only very special classes are well understoodLinks are collection of knots...

Wang Xiong

Topology in Chaos

Introduction Knot Theory Nonlinear Dynamic Topology in chaos Open Questions Summary

Summary: complex knots and links ...

Knot theory is to understand all possible knot...too bigproblemPrime knots are building block... still hardEven for same type of knot... hard to recognize, we needcomputable invariantsHave to restrict our attention to special classes ofknots...only very special classes are well understood

Links are collection of knots...

Wang Xiong

Topology in Chaos

Introduction Knot Theory Nonlinear Dynamic Topology in chaos Open Questions Summary

Summary: complex knots and links ...

Knot theory is to understand all possible knot...too bigproblemPrime knots are building block... still hardEven for same type of knot... hard to recognize, we needcomputable invariantsHave to restrict our attention to special classes ofknots...only very special classes are well understoodLinks are collection of knots...

Wang Xiong

Topology in Chaos

Introduction Knot Theory Nonlinear Dynamic Topology in chaos Open Questions Summary

1 Introduction

2 Knot Theory

3 Nonlinear Dynamic

4 Topology in chaos

5 Open Questions

6 Summary

Wang Xiong

Topology in Chaos

Introduction Knot Theory Nonlinear Dynamic Topology in chaos Open Questions Summary

Parametric Lorenz System

In 1963, the meteorologistEdward Lorenz was studying avery simplified numerical modelfor the atmosphere, which ledhim to the now-classicParametric Lorenz systemwhich is described by:

x = σ (y − x)y = rx − y − xzz = −bz + xy ,

(1)

with three real parametersσ, r , b.

Figure: Edward Lorenz

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Lorenz system

The dynamic behaviors are highly depended on the threeparameters and will differ from region to region, or evenpoint to point in the 3-dimensional parameter space.

When σ = 10, r = 28, b = 83 , this is the so-called Lorenz

system and it’s chaotic with a strange attractorx = 10 (y − x)y = 28x − y − xzz = −8

3z + xy ,(2)

Lorenz system has a strange attractor, that is what weusually called Lorenz attractor

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Lorenz system

The dynamic behaviors are highly depended on the threeparameters and will differ from region to region, or evenpoint to point in the 3-dimensional parameter space.When σ = 10, r = 28, b = 8

3 , this is the so-called Lorenzsystem and it’s chaotic with a strange attractor

x = 10 (y − x)y = 28x − y − xzz = −8

3z + xy ,(2)

Lorenz system has a strange attractor, that is what weusually called Lorenz attractor

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Lorenz attractor

Figure: Lorenz attractor, which vividly resembles a butterfly’s wings,has become an emblem of chaos; Edward N. Lorenz himself, hasbeen marked by the history as an icon of chaos.

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Unstable periodic orbits

Lorenz system is notable for having chaotic solutions forthe most initial conditions.

Also for some very special initial conditions you can getperiodic orbits solutions. There are an infinite number ofunstable periodic orbits(UPOs) embedded in a chaoticsystem.These UPOs play important roles in characterizing andanalyzing the system, however even numerically, it is noteasy to detect many UPOs from a continuous-time chaoticsystem, because they cannot be found by the forward timeintegration of the system.

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Unstable periodic orbits

Lorenz system is notable for having chaotic solutions forthe most initial conditions.Also for some very special initial conditions you can getperiodic orbits solutions. There are an infinite number ofunstable periodic orbits(UPOs) embedded in a chaoticsystem.

These UPOs play important roles in characterizing andanalyzing the system, however even numerically, it is noteasy to detect many UPOs from a continuous-time chaoticsystem, because they cannot be found by the forward timeintegration of the system.

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Unstable periodic orbits

Lorenz system is notable for having chaotic solutions forthe most initial conditions.Also for some very special initial conditions you can getperiodic orbits solutions. There are an infinite number ofunstable periodic orbits(UPOs) embedded in a chaoticsystem.These UPOs play important roles in characterizing andanalyzing the system, however even numerically, it is noteasy to detect many UPOs from a continuous-time chaoticsystem, because they cannot be found by the forward timeintegration of the system.

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Attractor and UPOs Animation

This video shows Lorenz system have chaotic solutions formost initial conditions, while for some very special initialconditions you can get periodic orbits solutions.

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Connection between knot topology and dynamicsystem

The simple but crucial observation is that such a periodicorbit is a closed embedded curve which therefore defines aknot.So the study of the topology of the knots appearing in theLorenz system could yield some understanding of thisimportant dynamical systemAny knot could appear as a UPO in Lorenz system iscalled a Lorenz knot.

Wang Xiong

Topology in Chaos

Introduction Knot Theory Nonlinear Dynamic Topology in chaos Open Questions Summary

1 Introduction

2 Knot Theory

3 Nonlinear Dynamic

4 Topology in chaos

5 Open Questions

6 Summary

Wang Xiong

Topology in Chaos

Introduction Knot Theory Nonlinear Dynamic Topology in chaos Open Questions Summary

Lorenz attractor and Lorenz knot

Figure: Lorenz attractor and Lorenz knot

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Some Lorenz knots are non-trivial

Some of these knots look topologically trivial, but some are nontrivial, like the red orbit which turns out to be a trefoil knot.

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Some example and counter-example of Lorenz knots

Figure: A trefoil knot

Figure: As a samplecounter-example, the figure eightknot is not a Lorenz knot.

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Lorenz knots and links are very peculiar

There are 250 (prime) knots with 10 crossings or fewer. Itfollows from the work of Birman and Williams that among those250, only the following 8 knots appear as periodic orbits of theLorenz attractor.

Figure: Among the 1701936 (prime) knots with 16 crossings or less ,only 21 appear as Lorenz knots.[1]

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How to understand Lorenz attractor

Observation: Lorenz attractor appears to be a two-sheetedsurface on which there is a unidirectional flow about each strip.

Figure: Lorenz attractor Figure: geometric Lorenz attractor

See here for animations Lorenz Template1, 2, 3

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Template

TemplateA template (also known as a knot holder) is a compactbranched two-manifold fitted with a smooth expansive semifl owand built from a finite number of joining and splitting charts

Figure: (a) joining and (b) splitting charts

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Lorenz Template

How to construct this Lorenz Template?

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Why such simplification possible?

Assume a dissipative dynamical system in R3 has Lyapunovexponents λ1 > 0 (stretching direction), λ2 = 0 (flow direction),and λ3 < 0 (squeezing direction) that satisfy λ1 + λ2 + λ3 < 0(dissipative condition) and produces a hyperbolic chaoticattractor.

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Why such simplification possible?

The identification of all points with the same future

x ∼ y if limt→∞|x(t)− y(t)| = 0

maps the chaotic attractor onto a branched manifold and theflow in the chaotic attractor to a semi-flow on the branchedmanifold.

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Is the geometric Lorenz attractor arcuate?

Birman-Williams Theorem(1983)[4]

Given a flow φt on a three-dimensional manifold M3 having ahyperbolic structure on its chain recurrent set there is a knotholder (H, φt) , H ⊂ M3 , such that with one or two specifiedexceptions the periodic orbits under φt correspond one-to-oneto those under φt . On any finite subset of the periodic orbits thecorrespondence can be taken to be via isotopy.

This guarantee that the organization of the unstable periodicorbits in the chaotic attractor remains unchanged under thisprojection.

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Lorenz template and Lorenz attractor

Figure: The periodic orbits in the chaotic attractor correspond in aone to one way with the periodic orbits on the branched manifold withone or two exceptions.

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Lorenz template and Lorenz attractor

Using their results, a Lorenz knot can be defined to be asimple closed curve in 3-space that embeds in the Lorenztemplate, and a Lorenz link is a finite collection of disjointsimple closed curves carried by the template.

Using their results, a Lorenz knot can be defined to be asimple closed curve in 3-space that embeds in the Lorenztemplate, and a Lorenz link is a finite collection of disjointsimple closed curves carried by the template.

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Lorenz template and Lorenz attractor

Using their results, a Lorenz knot can be defined to be asimple closed curve in 3-space that embeds in the Lorenztemplate, and a Lorenz link is a finite collection of disjointsimple closed curves carried by the template.Using their results, a Lorenz knot can be defined to be asimple closed curve in 3-space that embeds in the Lorenztemplate, and a Lorenz link is a finite collection of disjointsimple closed curves carried by the template.

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Some conclusions of Lorenz knot and link [5, 6]

1 Lorenz knots are prime2 All knots ⊃ Fibered knots ⊃ Closed + braids ⊃ Lorenz

knots3 Every torus link is a Lorenz link4 Non trivial Lorenz links have positive signature

So Lorenz knots and links are quite special, because Lorenztemplate is special.Seems each special template has its own special class of knotsand links...

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The Universal Knot-Holder

Could it possible, to have a template that contains all knots?

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The Universal Knot-Holder

Could it possible, to have a template that contains all knots?

Figure: Ghrist’s universal knot-holder[7]

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An ODE whose solutions contain all knots and links

There is a corresponding flow governed by the equations -derived from a Chua circuit:

Theorem [8]There exists an open set of parameters β ∈ [6.5,10.5] for whichperiodic solutions to the differential equation

x = 7(y − 27x − 3

14(|x + 1| − |x − 1|))y = x − y + zz = −βy

(3)

contain representatives from every knot and link equivalenceclass.

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All knots and links live here

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Introduction Knot Theory Nonlinear Dynamic Topology in chaos Open Questions Summary

1 Introduction

2 Knot Theory

3 Nonlinear Dynamic

4 Topology in chaos

5 Open Questions

6 Summary

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Chen attractor

In 1999, from an engineeringfeedback anti-control approach,Chen coined a new chaoticsystem [9], lately referred to asthe Chen system by others.

x = 35(y − x)y = −7x + 28y − xzz = −3z + xy ,

(4)

Figure: Chen Attractor

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Chen knots and links?

What is the template of Chen attractor?

What are the special properties of Chen knots and links?What is the relation between Lorenz attractor and Chenattractor, from this topology point of view?

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Chen knots and links?

What is the template of Chen attractor?What are the special properties of Chen knots and links?

What is the relation between Lorenz attractor and Chenattractor, from this topology point of view?

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Chen knots and links?

What is the template of Chen attractor?What are the special properties of Chen knots and links?What is the relation between Lorenz attractor and Chenattractor, from this topology point of view?

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From Lorenz-like to Chen-like

Figure: Generalized Lorenz system familyWang Xiong

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From Lorenz-like to Chen-like

What are the templates of all the attractors in thisgeneralized Lorenz system family?

How the templates evolve as the parameter changes?What are the special properties of these Lorenz-like knotsand links?How the knots and links evolve as the parameter changes?

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From Lorenz-like to Chen-like

What are the templates of all the attractors in thisgeneralized Lorenz system family?How the templates evolve as the parameter changes?

What are the special properties of these Lorenz-like knotsand links?How the knots and links evolve as the parameter changes?

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From Lorenz-like to Chen-like

What are the templates of all the attractors in thisgeneralized Lorenz system family?How the templates evolve as the parameter changes?What are the special properties of these Lorenz-like knotsand links?

How the knots and links evolve as the parameter changes?

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From Lorenz-like to Chen-like

What are the templates of all the attractors in thisgeneralized Lorenz system family?How the templates evolve as the parameter changes?What are the special properties of these Lorenz-like knotsand links?How the knots and links evolve as the parameter changes?

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Yet More Lorenz-like to Chen-like [10]

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Yet More Lorenz-like to Chen-like [10]

Wang Xiong

Topology in Chaos

Introduction Knot Theory Nonlinear Dynamic Topology in chaos Open Questions Summary

Yet More Lorenz-like to Chen-like [10]

Wang Xiong

Topology in Chaos

Introduction Knot Theory Nonlinear Dynamic Topology in chaos Open Questions Summary

Yet More Lorenz-like to Chen-like [10]

Wang Xiong

Topology in Chaos

Introduction Knot Theory Nonlinear Dynamic Topology in chaos Open Questions Summary

Yet More Lorenz-like to Chen-like [10]

Wang Xiong

Topology in Chaos

Introduction Knot Theory Nonlinear Dynamic Topology in chaos Open Questions Summary

Yet More Lorenz-like to Chen-like [10]

Wang Xiong

Topology in Chaos

Introduction Knot Theory Nonlinear Dynamic Topology in chaos Open Questions Summary

Yet More Lorenz-like to Chen-like [10]

Wang Xiong

Topology in Chaos

Introduction Knot Theory Nonlinear Dynamic Topology in chaos Open Questions Summary

1 Introduction

2 Knot Theory

3 Nonlinear Dynamic

4 Topology in chaos

5 Open Questions

6 Summary

Wang Xiong

Topology in Chaos

Introduction Knot Theory Nonlinear Dynamic Topology in chaos Open Questions Summary

Summary

1 Topology is the mathematical study of shapes and spaces.2 Dynamical Systems Theory deals with the long-term

qualitative behavior of dynamical systems.3 Periodic orbits of a third order ODE are topological knots.

This simple but crucial observation connects these twoimportant areas of math.

4 The study of the topology of the knots appearing indynamical system could yield some understanding of thisdynamical system.

5 Still many many open problems...

Wang Xiong

Topology in Chaos

Introduction Knot Theory Nonlinear Dynamic Topology in chaos Open Questions Summary

THE KNOT ATLAS.:http://katlas.math.toronto.edu/wiki/Main Page

STEWART, I : The Lorenz Attractor Exists. Nature 406(2000), 948-949.

TUCKER, W.: A rigorous ODE solver and Smales 14thproblem. Found.Comput. Math. 2 (2002), no. 1, 53C117.

J. S. Birman & R. F. Williams (1983), Knotted periodic orbitsin dynamical systems I: Lorenz’s equations, Topology, 22(1), 47-82. J. S. Birman & R. F. Williams (1983), Knottedperiodic orbits in dynamical systems II: Knot holders forfibered knots, Contemporary Mathematics, 20, 1-60

R. Williams, Lorenz knots are prime, Ergodic TheoryDynamical Systems 4, 147-163 (1983).

Wang Xiong

Topology in Chaos

Introduction Knot Theory Nonlinear Dynamic Topology in chaos Open Questions Summary

JS Birman 2011 Lorenz knots,http://arxiv.org/abs/1201.0214

R. W. Ghrist (1995), Flows on S3 supporting all links asorbits, Electronic Research Announcements of theAmerican Mathematical Society, 1 (2), 91-97.

RW GHRIST An ODE whose solutions contain all knotsand links www.math.upenn.edu/ ghrist/preprints/silnikov.pdf

Chen, G. & Ueta, T. [1999] Yet another chaotic attractor, Int.J. Bifur. Chaos 9, 1465–1466.

XIONG WANG and GUANRONG CHEN, A GALLERY OFLORENZ-LIKE AND CHEN-LIKE ATTRACTORS Int. J.Bifur. Chaos 23, 1330011 (2013)http://www.worldscientific.com/doi/abs/10.1142/S0218127413300115?journalCode=ijbc

Wang Xiong

Topology in Chaos

Introduction Knot Theory Nonlinear Dynamic Topology in chaos Open Questions Summary

Thank You!! Q&A

Wang Xiong

Topology in Chaos