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January 7
Predrag Cvitanović
1.
Trajectories
We start out by a recapitulation of the basic notions of
dynamics. Our aim is narrow; keep the exposition focused on
prerequisites to the applications to be developed in this text.
I assume that you are familiar with the dynamics on the level
of introductory texts such as Strogatz, and concentrate here on
developing intuition about what a dynamical system can do.
flows
Chapter 2
Flows
flowsOverh
lecture
overheads
intro
Chapter 1
Overture (optional)
Read quickly all of it - do not worry if there are stretches that you do not
understand yet.
appendHist
Appendix A
Brief history of chaos (optional)
A brief history of motion in time.
introOverheads
overture
overheads (optional)
Future's So Bright, I Gotta Wear Shades
[click right, open in new tab]
January 9
2.
Flow visualized as an iterated mapping
Discrete time
dynamical systems arise naturally by either strobing the flow at fixed time intervals
(we will not do that here),
or recording the coordinates of the flow
when a special event happens (the Poincare section method, key insight for
much that is to follow).
maps
Chapter 3
Discrete time dynamics
You probably want to print only a chapter at a time on paper
- the book is being edited concurrently with the course. You can see when a chapter was edited on the page footer
mapsOverh
lecture
overheads
HW1
exercises
(2.1) (2 pts), (2.7) (4 pts), and (2.8) (4 pts), bonus (3.5) (2 pts)
- due in class Tue
Jan 14
January 14
3.
There goes the neighborhood
So far
we have concentrated on description of the trajectory
of a single initial point.
Our next task is to define and determine the size of a
neighborhood, and describe the local geometry of
the neighborhood by studying the linearized flow.
What matters are the expanding directions. The repercussion
are far-reaching:
As long as the number of unstable directions is finite,
the same theory applies to finite-dimensional ODEs,
Hamiltonian flows, and dissipative, volume contracting
infinite-dimensional PDEs.
stability
Chapter 4
Local stability
stabilityOverh
lecture
overheads
January 16
4.
Cycle stability
If a flow is
smooth, in a sufficiently small neighborhood it is essentially
linear. Hence in this lecture, which might seem an embarrassment
(what is a lecture on linear flows doing in a book on
nonlinear dynamics?), offers a firm stepping stone on the way to
understanding nonlinear flows. Linear charts are the key tool of
differential geometry, general relativity, etc, so we are in good
company. If you know your eigenvalues and eigenvectors, you may
prefer to fast forward here.
invariants
Chapter 5
Cycle stability
Skip sect. 5.2.1.
Lyapunov
Chapter 6
Lyapunov exponents (optional)
Awkward stuff, much cited in the literature,
most of it safely ignored. Skipped in the lectures.
please take this quiz - we are testing the SOOC :)
HW2
exercises
(3.1) - 6 points,
(4.1) - 4 points, and
(4.3) - 6 points. Bonus exercises:
(3.2) - 6 points,
(3.7) - 6 points and
(4.6) - 4 points,
- due Tue
Jan 21
January 21
5.
Stability exponents are invariants of dynamics
We prove that (1) Floquet multipliers are the same everywhere
along a cycle, and (b) that they are invariant under any smooth
coordinate transformation.
January 23
6.
Pinball wizzard
The dynamics
that we have the best intuitive grasp on
is the dynamics of billiards.
For billiards, discrete time is altogether natural;
a particle moving through a billiard
suffers a sequence of instantaneous kicks,
and executes simple motion in between, so
there is no need to contrive a Poincare section.
billiards
Chapter 8
Billiards
Read all of it. The 3-disk pinball illustrates some of the key
concepts for what follows; invariance under discrete symmetries, symbolic dynamics.
Optional: download some simulations from ChaosBook.org/extras,
or write your own simulator.
billiardsOverh
Billiards
overheads
billiardsOverh
(optional)
a bit of history
While simulating weather patterns 50 years ago, Edward Lorenz
overthrew the idea of the clockwork universe with his
ground-breaking research on chaos.
HW3
exercises
(5.1) - 3 points,
(B.2) - 3 points,
and
(8.1) - 10 points.
Bonus exercise:
(8.6) - 6 points.
- due Tue
Jan 28
a hint: check out programs
ChaosBook.org/extras/
January 28
7.
Discrete symmetries of dynamics
The families of symmetry-related full state space cycles
are replaced by fewer and often much shorter
``relative" cycles, and
the notion of a prime periodic orbit has to be reexamined:
it is replaced by the notion of
a ``relative'' periodic orbit, the shortest segment
that tiles the cycle under the action of the group.
Furthermore, the group operations that relate
distinct tiles do double duty as letters of an
alphabet which
assigns symbolic itineraries to trajectories.
discrete
Chapter 9
World in a mirror
Read all of it. Ask tons of questions in the class.
discreteOverh
lecture
E. Siminos notes (optional)
discreteOverh
lecture
overheads (optional)
January 30
8.
Discrete symmetry reduction of dynamics to a fundamental domain
discrete
Chapter 9
World in a mirror
Read sects. 9.2 - 9.4 Dynamics for Fundamentalists. Skip
sect. 9.5 Invariant polynomials.
HW4
ChaosBook ver. 14.5: Exercise 1.1 3-disk symbolic dynamics - 2 points;
Exercise 8.3 Stability of billiard cycles - 4 bonus points;
Exercise 9.5 Symmetries of an equilateral triangle - 4 points;
Exercise 9.6 Reduction of 3-disk symbolic dynamics to binary - 3 points;
Exercise 9.7 C2-equivariance of Lorenz system - 3 bonus points;
- due Tue
February 4
February 4
9.
Continuous symmetries of dynamics
If the symmetry is continuous, the interesting dynamics unfolds on a
lower-dimensional ``quotiented'' system, with
``ignorable" coordinates eliminated (but not forgotten).
The families of symmetry-related full state space cycles
are replaced by fewer and often much shorter
``relative" cycles, and
the notion of a prime periodic orbit has to be reexamined:
it is replaced by the notion of
a ``relative'' periodic orbit, the shortest segment
that tiles the cycle under the action of the group.
continuous
Chapter 10
Relativity for cyclists
Read Sects. 10.1 to 10.3. Ask tons of questions in the class.
continuousOverh
lecture
overheads
February 6
10.
Got a continuous symmetry? Freedom and its challenges
HW5 = click here
posted here as a pdf file. To get correct cross references, you have to open today's (imperfect) version 14.5.1 of the ChaosBook.
- due Tue
Feb 11
HW5 = Howey 5 click here
exercise (10A.X), mandatory for Howey 5th and 3rd floor folks whose systems have continuous symmetries, optional for everyone else:
Is slicing the best thing since invention of sliced bread? It is not clear - it reveals the dynamics hidden behind drifts along symmetry directions, but it gets dicey as one approaches a chart border. This is a problem set: propose a coordinate change (perhaps rescaling of time)
that regularizes close passages to the chart border
without a need to refine the time steps of your integrator. A suggestion - close to the chart border al trajectories are straight lines, so some kind of simple
rescaling should work.
February 11
11.
Slice and dice
Actions of a Lie group on a state trace out a manifold of equivalent
states, or its group orbit.
Symmetry reduction is the identification of a
unique point on a group orbit as the representative
of this equivalence class.
Thus, if the symmetry is continuous, the interesting dynamics unfolds on a
lower-dimensional `quotiented', or `reduced' state space M/G, with
`ignorable' coordinates eliminated (but not forgotten).
In the method of slices the symmetry reduction is achieved by cutting the group orbits
with a set of hyperplanes, one for each continuous group parameter, with each
group orbit of symmetry-equivalent points represented by a single point, its intersection
with the slice.
continuous
Chapter 10
Relativity for cyclists
Read Sect. 10.4 Reduced state space.
continuousOverh
lecture
overheads
February 13
12.
Qualitative dynamics, for pedestrians
Qualitative properties of
a flow partition the state space in a topologically invariant way.
knead
Chapter 11
Charting the state space
Sects 11.1 and 11.2
HW6 = click here
posted here as a pdf file. To get correct cross references, you have to open today's (imperfect) version 14.5.2 of the ChaosBook.
- due Tue
Feb 25
Nazmi Burak Budanur's solution set: a Mathematica notebook.
February 18
13.
The spatial ordering of trajectories from the time ordered itineraries
of qualitative dynamics: (1) temporal ordering, or itinerary with
which a trajectory visits state space regions and (2) the spatial ordering
between trajectory points, the key to determining the admissibility
of an orbit with a prescribed itinerary. Kneading theory.
knead
Chapter 11
Charting the state space
Sects 11.3 - 11.6
Eye candy: Chaos in Heavens
(and a lesson: you will never understand anything by staring at
projections of chaotic trajectories onto configuration coordinates -
you need to take care of symmetries,
look for equilibria, construct Poincare
sections, ...)
February 20
14.
Qualitative dynamics, for cyclists
Dynamical partitioning of a plane
smale
Chapter 12
Stretch, fold, prune
Sects 12.1 - 12.3
February 25
15.
Finding cycles
cycles
Chapter 13
Fixed points, and how to get them
Read all of it.
J. Newman: Mathematica periodic orbits routines
A. Basu: Matlab periodic orbits routines
Ring of Fire
Visualize O(2) equivariance of Kuramoto-Sivashinsky (AKA "Ring of Fire")
February 27
16.
Finding cycles; long cycles, continuous time cycles
HW7 = click here
posted here as a pdf file. To get correct cross references, you have to open today's (imperfect) version 14.5.3 of the ChaosBook.
- due Tue
Mar 4
part II
chaos rules
Predrag
March 4
17.
Markov graphs
The topological dynamics is encoded
by means of transition matrices/Markov graphs.
Markov
Chapter 14
Walkabout: Transition graphs
Read all of it.
Predrag
March 6
18.
Learning hoow to count
count
Chapter 15
Counting
Read sects. 15.1 - 15.4; 15.6 - 15.7,
ChaosBook vers. 14.5.3.
Please derive yourself the trace formula
the determinant
and the topological zeta function.
If you do not understand how to derive these, you'll be lost for the rest of the semester, and what fun is that?
The lecturer seems to flounder while attempting to derive these,
totally essential formulas, let him know if you have simpler derivations.
HW8 = click here
posted here as a pdf file
- due Tue
March 11
March 11
19.
Transporting densities
measure
Chapter 16
Transporting densities
Skip sects. 16.3 and 16.6.
March 13
20.
Averaging,
trace formulas
average
Chapter 17
Averaging
Read sects. 17.1 and 17.2.
Skip sect. 17.1.3 "Moments, cumulants",
trace
Chapter 18
Trace formulas
Read all of it.
HW9 = click here
posted here as a pdf file
- due Tue
Mar 25
March 17-21
spring break
March 25
21.
Spectral determinants
det
Chapter 19
Spectral determinants
Skip sects. 17.5 and 17.6.
Sleep deprivation may cause brain damage
The leading cause of migranes
lecturer TBA
March 27
22.
Cycle expansions
recycle
Chapter 20
Cycle expansions
Skip sects. 20.3.1 "Newton algorithm for determining the evolution operator eigenvalues",
and
20.6 "Stability ordering of cycle expansions".
(version 14.5.4, Mar 25 2014)
HW10
posted here as a pdf file. To get correct cross references, you have to open today's (imperfect) version 14.5.4 of the ChaosBook.
- due Tue
Apr 1
April 1
23.
Deterministic diffusion
average
Chapter 17
Averaging
Read sects. 17.1, including sect. 17.1.3 "Moments, cumulants",
diffusion
Chapter 24
Deterministic diffusion
Read sect. 25.1:
Foundations of statistical mechanics illuminated
by 2-dimensional
Lorentz gas.
Overheads for the day's lecture
April 3
24.
Deterministic diffusion
diffusion
Chapter 24
Deterministic diffusion
Read sect. 25.2:
A class of simple
1-dimensional dynamical systems where all transport coefficients can be evaluated analytically.
project, step #1:
For a project, pick a dynamical systems topic (or a paper to read)
related to your research - the idea is that the project in this course
can be included into your thesis, perhaps as an appendix.
If there are symmetries in the formulation of the problem, do explain them
and classify all types of relevant solutions by their symmetry. Follow
the literature on the notation and terminology, include detailed
bibliography, with proper credits given; this term paper
is not meant to be original research (though that would be sweet :).
If you are going to write up the project in LaTeX
(and not as a part of your CNS subversion blog),
download the template from ChaosBook.org/projects/
project, step #2:
Append to HW11 pdf pages (built from the template)
a brief skeleton of your project: title,
your name, names of advisors (professors, other students) who
might help you with their advice, an abstract (of any length), perhaps also a paper that you will base your project on.
Next homework append the draft of updated project, and so on - it will
be refereed by another participant. And so on...
If good, the project will be ethernalized
on the ChaosBook.org/projects homepage, where you can
see descriptions of earlier projects.
project refereeing:
When you receive a homework to grade, comment on the appended
draft project as you would referee a paper submission -
annotate, ask questions,
suggest improvements, etc. (no numerical grade).
HW11
- due Tue
Apr 8. Exercise 26.8 "Diffusion reduced to the fundamental domain" is
essential for most CNS PhD projects (a baby version of how to reduce
Euclidean symmetry), so please make an attempt at solving this
exercise.
April 8
25.
Discrete symmetry factorization of spectral determinants
A triple home run: simpler symbolic dynamics,
fewer cycles needed, much better convergence of cycle expansions. Once you
master this, going back is unthinkable.
symm
Chapter 21
Discrete factorization
Symmetries simplify and improve the cycle expansions in a rather beautiful,
not entirely obvious way, by factorizing cycle expansions.
Read sect. 21.1
April 10
26.
Discrete symmetry factorization of spectral determinants
3 disk pinball symmetries suffice to illustrate all that is
needed to factorized spectral determinants for any system
with a discrete symmetry: character.
symm2
Chapter 21
Discrete factorization
Read sects. 21.2 - 21.6
HW12
- due Tue
Apr 15.
April 15
27.
Projects presentations session
mugshots
gallery
Who's who?
projects
project
TechBurst 2011
Can we do better than TechBurst 2011?
projects:
discussion - for videos, see ChaosBook.org project homepage
12:05-12:20 Marc Fleury: "Quantum" walkers.
12:20-12:35 Mikel Jon De Viana: Terrible allergies due to pollen.
12:35-12:50 Kimberly Y. Short: The role of time scales in non-linear systems.
12:55-1:10 Benjamin McInroe: Periodic orbit theory of linear response.
1:10-1:25 Michael S. Dimitriyev: A continuum elastic model of thermal fluctuation allosteric regulation.
April 17
28.
Continuous symmetry factorization of spectral determinants
Continuous symmetries simplify and improve the cycle expansions in a rather beautiful,
not entirely obvious way, by factorizing cycle expansions. The lecture, however, was not good, and will not be posted as a video. See Apr 22 instead.
rpos
(To be published)
Continuous symmetry reduced trace formulas
Why is this paper not published yet? I would like at least one person out there in the universe (a graduate student, per chance?) to understand it before submitting it. I strongly recommend going through this paper and checking it as
a term project.
HW13
focus on this week's draft of your term project,
submit it as the homework
- due Tue
Apr 22.
1:10-1:25 Jeffrey M. Heninger: Noise is your friend.
April 22
29.
Symmetry factorization of spectral determinants - attempt 2
Dresselhaus
MIT course 6.734
Group Theory:
Application to the Physics of Condensed Matter
Chapters 1 to 4 of Dresselhaus lecture notes (or the textbook) are
perhaps student-friendlier than Tinkham textbook.
I recommend learning about the "Wonderful Orthogonality Theorem" on
your own - many of you will find that useful in your research.
1:10-1:25 Alexandre Damião: Thermoacoustic Instatilities using Rijke Tube Model.
April 24
30.
Turbulence
The last lecture of the semester: whence from here?
tutorialSD
project
plane Couette movies
Can we do better with a "Slice and Dice" tutorial than plane Couette movies?
[the project, and what you should do about it]
12:55-1:25 Pavel M. Svetlichnyy and Tingnan Zhang: Cycle averaging formulas applied to a periodic Lorentz gas
(for videos, see ChaosBook.org project homepage).
April 25
GT classes end
May 1
11:30am - 2:20pm term project due
Please upload the project, as you would upload a homework.
If you have a paper copy, you can stop by Predrag's office,
or put it into Predrag's mailbox.
to May 5
Course opinion survey
We would very much appreciate your input on how to
improve the video aspects of the course, and what one could do to
make it a viable MOOC course - or whether we should attempt to
go online at all. If the format of this survay is not helpful for that,
maybe you can discuss that on piazza.
CETL web link
May 5
GT grades due at noon
May 5
have good holidays!
some good stuff we did not have time to cover:
The rest has yet to be worked out.

?? ??
??.
Much noise about nothing
noise
Chapter 26
Noise
We derive the continuity equation for purely deterministic, noiseless
flow, and then incorporate noise in stages: diffusion equation, Langevin equation,
Fokker-Planck equation, Hamilton-Jacobi formulation, stochastic path integrals.
homework HW1?:
exercises
26.1, 26.2 and 26.3
- not due in this course [work them out anyway, Gaussians will serve you well later on]
?? ??
??.
How good is your Poincare section?
Deconstruct exercise
(3.7) "Poincare section border". The gang is right - as Roessler equatins are
quadratic, the borders are conic sections (line, circle, ellipse, parabola, hyperbola). Dr. C. is right - sections not going through equilibria are no good, as they do not intersect all trajectories winding around their real (un)stable eigen-vectors.
??
??.
Cycle expansions - heuristscs
getused
Chapter 20
Why cycle?
Skip sects. 20.4 and 20.5.
(version 12.3.3, Nov 10 2012)
??
??.
Why does it work?
converg
Chapter 21
Why does it work?
Some of the mathematical ideas that underpin trace formulas.
Read only sect. 21.1, skim the rest.
??
??.
Why doesn't it work?
inter
Chapter 23
Intermittency
Everything that we have done so far hinges on exponential
separation of nearby trajectories. What happens if we get stuck
close to the border of integrable, regular motion?
Read sects. 23.1 to 23.2.3, skim the rest.
homework HW9:
exercises
18.14, 20.2, 23.3; optional 21.3
- due Tue
??? ??
??? ??
??.
Turbulence
tutorialSD
project
plane Couette movies
Can we do better with a "Slice and Dice" tutorial than plane Couette movies?
projects update:
discussion session