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August 19
Predrag CvitanoviÄ‡
1.
Things fall apart
A brief history of motion in time.
intro
Chapter 1
Overture
Read quickly all of it - do not worry if there are stretches that you do not
understand yet.
The rest is optional reading:
appendHist
Appendix A
Brief history of chaos
ChaosBook.org version12.1, Aug 22 2008:
iPaper version (let me know if you prefer PDF files)
introOverheads
lecture
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August 21
2.
Trajectories
We start out by a recapitulation of the basic notions of
dynamics. Our aim is narrow; keep the exposition focused on
prerequsites 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
homework #1:
exercises
(1.1),
(2.1), (2.7), and (2.8), optional (2.10)
- due Tue
Aug 26
[solutions to chap. 1 exercises]
[solutions to chap. 2 exercises]
August 26
3.
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
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lecture
overheads
August 28
4.
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
ChaosBook.org version12.1, Aug 22 2008: skip sect. 4.5.1
stabilityOverh
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invariants
Chapter 5
Cycle stability
Read quickly through it, skip sect. 5.3. Skipped in the lectures, but will
need some of the definitions in what follows.
conjug
Chapter 6
Get straight
Advanced material, most of it safely skipped. Read at least sect. 6.6, and,
if you have had trouble with integrating helium dynamics, sect 6.3. Skipped in the lectures.
homework #2:
exercises
(3.1), (3.5), (4.1), and (4.3), optional (3.6) and (4.4)
- due Tue
Sep 2
[solutions to chap. 4 exercises]
September 1
Labor Day
September 2
5.
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.
newton
Chapter 7
Hamiltonian dynamics
Read cursorily through sects. 7.1 and 7.2. In this course we will focus on far-from-equilibrium dissipative systems
(rather than energy-conserving systems typical of quantum-mechanical applications), so this
is not material of importance for what follows.
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.
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September 4
Evangelos Siminos
6.
Group theory - a brief introduction
Combinatorics cannot be tought. But we will make a brave attempt.
discrete
Chapter 9
World in a mirror
Study section 9.1. (make sure the date in the page footers is Sep 3 2008
or later, otherwise download from here the current version of the chapter)
discreteOverh
lecture
E. Siminos notes
homework #3:
exercises
9.2, 9.3, 9.4, 9.5 (a)-(e)
- due Tue
Sep 9
[solutions to chap. 9 exercises]
September 9
Evangelos Siminos
7.
Discrete symmetries of dynamics
Dynamical systems
often come equipped with discrete symmetries, such as the
reflection symmetries of various potentials, and
they simplify the dynamics in a
rather beautiful way:
If dynamics is invariant under a set of
discrete symmetries, the state space is
tiled by a set of symmetry-related tiles,
and the dynamics can be reduced to dynamics
within one such tile, the fundamental domain.
Read sections 9.2 and 9.3.
September 11
Evangelos Siminos
8.
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.
Furthermore, the group operations that relate
distinct tiles do double duty as letters of an
alphabet which
assigns symbolic itineraries to trajectories.
Read sections 9.4 and 9.5 if Vaggelis covers the continuous
symmetries, skip if the lecture stops at
the fundamental domain discussion.
discreteOverh
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homework #4:
exercises
(9.7), (9.8) and (9.12), optional (9.13)
- due Tue
Sep 16
[solutions to chap. 9 exercises]
September 16
9.
Symbolic dynamics
knead
Chapter 10
Qualitative dynamics, for pedestrians
(version 12.3, Sep 30 2008)
Qualitative properties of
a flow partition the state space in a topologically invariant way.
The topological dynamics is incoded
by means of transition matrices/Markov graphs.
September 18
10.
Kneading theory
smale
Chapter 11
Qualitative dynamics, for cyclists
Skip section 11.5.
homework #5:
exercises
10.4, 10.6, 3.2; optional 10.5, 11.7
- due Tue
Sep 23
[solutions to chap. 10 exercises]
[solutions to chap. 11 exercises]
September 23
11.
Finding cycles
cycles
Chapter 12
Fixed points, and how to get them
(version 12.3.1, Oct 6 2008; includes a sketch of chapter `12a' on transition graphs)
September 25
12.
Finding cycles
September 30
13.
Finding cycles
homework #6:
exercises
10.8, 12.13, 12.7; optional 12.12
- due Tue
Oct 7
[solutions to chap. 12 exercises]
[solutions from the class]
October 2
14.
Counting
counting
Chapter 13
Counting
Read all of it.
(version 12.3.1, Oct 6 2008)
October 7
15.
Transporting densities
measure
Chapter 14
Transporting densities
Skip sect. 14.6.
(version 12.3.1, Oct 6 2008)
homework #7:
exercises
13.1, 14.1, 14.3, 14.10, 15.4; optional 13.6, 13.14, 14.5, 14.7
- due Thu
Oct 16
[solutions to chap. 13 exercises]
[solutions to chap. 14 exercises]
October 9
16.
Averaging
average
Chapter 15
Averaging
Read all of it.
(version 12.3.3, Nov 10 2008)
October 11-14
fall break
October 16
17.
Lyapunov exponents
First we trash them as stupid, then we nevertheless define them.
October 21
18.
Trace formulas
trace
Chapter 16
Trace formulas
Read all of it.
(version 12.3.3, Nov 10 2008)
projects #1:
If you are going to write up the project in LaTeX
(and not in blog/svn format),
download the template from ChaosBook.org/projects/
October 23
19.
Spectral determinants
det
Chapter 17
Spectral determinants
Skip sects. 17.5 and 17.6.
(version 12.3.3, Nov 10 2008)
homework #8:
exercises
15.1, 15.4; optional 17.10 (use version 12.3.3, Nov 10 2008)
- due Tue
Oct 28
[solutions to chap. 15 exercises]
[solutions to chap. 17 exercises]
projects #2:
Email me a brief description of your project: title,
your name, names of advisors (professors, other students) who
might help with their advice, an abstract (of any length, as *.txt
or *.pdf file), perhaps also a paper that you will base your project on.
This will be ethernalized on the ChaosBook.org/projects homepage, where you can
see descriptions of earlier projects
- due Tue
Oct 28
October 28
20.
Cycle expansions
recycle
Chapter 18
Cycle expansions
Skip sect. 18.6.
(version 12.3.3, Nov 10 2008)
J. Newman: Mathematica periodic orbits routines
A. Basu: Matlab periodic orbits routines
October 29 - November 11
spring registration
October 30
21.
Cycle expansions
homework #8 again:
make sure that
your programs for finding periodic orbits (Henon and/or Rossler) work
- due Thu
Nov 6
November 4
22.
Cycle expansions - heuristscs
getused
Chapter 20
Why cycle?
Skip sects. 20.4 and 20.5.
(version 12.3.3, Nov 10 2008)
November 6
23.
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.
November 11
24.
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 interable, regular motion?
Read sects. 23.1 to 23.2.3, skim the rest.
homework #9:
exercises
18.14, 20.2, 23.3; optional 21.3
- due Tue
Nov 18
[solutions to chap. 18 exercises]
[solutions to chap. 20 exercises]
[solutions to chap. 21 exercises]
November 13
25.
Deterministic diffusion
diffusion
Chapter 24
Deterministic diffusion
Fundation of statistical mechanics illuminated.
Read sects. 24.1 to 24.2, skim the rest.
projects update:
discussion session
November 18
26.
Deterministic diffusion
November 20
27.
Deterministic diffusion
November 25
28.
Projects discussion session
November 27
thanksgiving
December 2
29.
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 #10:
exercises
26.1, 26.2 and 26.3
- not due in this course [work them out anyway, Gaussians will serve you well later on]
December 4
30.
Turbulence
projects update:
discussion session
[notes]
December 5
GT classes end
December 9
10:50 term project due, Predrag's office
to December 13
Course opinion survey
CETL web link
December 15
GT grades due at noon
December 22
have good holidays!
The rest has yet to be worked out.

solutions to the final exam
to be posted:
December 30 - January 9
spring registration