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January 10
Predrag Cvitanović
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
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#1
exercises
(1.1),
(2.1), (2.7), and (2.8), optional (2.10)
- due Tue
Jan 17
[solutions to chap. 1 exercises]
[solutions to chap. 2 exercises]
Future's So Bright, I Gotta Wear Shades
[click right, open in new tab]
January 12
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
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January 17
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
You can now print this chapter on the paper
- make sure it has printed "version13.7.1" or later on the page footer
mapsOverh
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#2
exercises
(3.1), (3.5), (4.1), and (4.3), optional (3.6) and (4.4)
- due Tue
Jan 24
[solutions to chap. 4 exercises]
January 19
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
skip sect. 4.5.1
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January 24
5.
Cycle stability
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
Go straight
Advanced material, most of it safely skipped. Try to understand sect. 6.6,
though. Skipped in the lectures.
#3
exercises
5.1, 6.2, 7.2, optional 5.2, 7.4
- due Tue
Jan 31
[solutions to chap. 5 exercises]
a hint: check out programs
ChaosBook.org/extras/
January 26
6.
Newtonian mechanics
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 at least cursorily the whole chapter.
udacity
web link
udacity.com
What do you think? Thrun had enrollment of 160,000 students.
ChaosBook.org is an attempt to reach any student,
anywhere,
and it reaches about seven. Could one do better?
January 31
7.
Pinball wizzard
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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#4
exercises
(8.1), (8.3) and (8.5), optional (8.6)
- due Tue
Feb 7
[solutions to chap. 9 exercises]
February 2
8.
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
You can print this chapter on the paper now
- revision 13.7.3, Feb 11 2012 completed
Read all of it. Ask tons of questions in the class.
discreteOverh
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E. Siminos notes
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optional
meet P. Glass (and/or break a leg)
[solution]
February 7
9.
Symmetries of solutions
discrete
Chapter 9
World in a mirror
You can print this chapter on the paper now
- revision 13.7.3, Feb 11 2012 completed
#5
exercise
(3.7)
- read Sect 3.4 Charting the state space (version 13.7.1 or later);
exercises (9.1), (9.2), (9.3), (9.4);
optional exercise: rewrite Example 8.1 3-disk game of pinball following the billiards lecture overheads, eternalize your name as contributor to ChaosBook.org.
- due Tue
Feb 14
[solutions to chap. 9 exercises]
February 9
10.
Fundamental domain
February 14
11.
Continuous symmetries of dynamics
NOTE: lecture moved to 5th floor Howey conference room
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 all of it. Ask tons of questions in the class.
Best not to print this chapter yet - major revisions under way. Make sure you are reading version 13.7.4 or later.
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#6
exercise
(3.7) do it until you get it right;
(10.1) Visualizations of the 5-dimensional complex Lorenz
flow;
(10.3) SO(2) rotations in a plane;
(10.8) Rotational equivariance, infinitesimal angles;
[
optional
(10.2) An SO(2)-equivariant flow with two Fourier modes;
(10.12) in a Hilbert basis
]
- due Tue
Feb 21
February 16
12.
How good is your Poincare section?
Keith and the gang deconstruct exercise
(3.7) "Poincare section border". The gang is right - as Rossler 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.
February 21
13.
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.
#7
exercises
(10.14) Compute the relative equilibrium TW1 of the 5-dimensional complex Lorenz
flow;
(10.16) Plot the relative equilibrium TW1 in Cartesian coordinates;
(10.**) Plot the symmetry reduced, 4-dimensional complex Lorenz
flow in a slice of your choice, in several 3-dimensional projections.
#7 optional
exercise (10.23) State space reduction by a slice, ODE formulation
- sorry, this is a bit of a mess - might try to improve it
the formulation by the weekend.
- due Tue
Feb 28
Ring of Fire
Visualize O(2) equivariance of Kuramoto-Sivashinsky (AKA "Ring of Fire")
February 23
14.
Slice and dice
Why gauge-fixing in field theory does not seem smart; and
why experimentalists should slice their raw data.
continuous
Chapter 10
Relativity for cyclists
Read Sect. 10.4 Reduced state space.
Best not to print this chapter yet - major revisions under way. Make sure you are reading version 13.7.4 or later.
exp
a letter
to experimentalists
No need to reconstruct fluid velocities - just need a notion of
distance for your data sets
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February 28
15.
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
#8
exercise
(10.**),
we do it until we get it right:
Plot the symmetry reduced, 4-dimensional complex Lorenz
flow in an atlas of your making, consisting of two slices,
such that the strange attractor avoids the slice borders and the associated
jumps.
Exercises
11.6 and 11.8.
#8 optional
exercise
(3.7),
we do it until we get it right:
* put a section through each equilibrium, try to optimize
their orientation so that the ridge (their intersection) is
the shortest distance from both equilibria.
** For Roessler flow the section border is a conic section. Derive
analytic formula for this elipse - line - parabola - hyperbola,
replot the section borders using the formula rather than the orientation
of v(x).
- due Tue
Mar 6
March 1
16.
Symbolic dynamics
^M
The two faces 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
March 6
17.
Qualitative dynamics, for cyclists
^M
First we trash them as stupid, then we nevertheless define them.
smale
Chapter 12
Stretch, fold, prune
Prune danish (if we get that far)
#9
exercise
12.3
- due Tue
Mar 13
#9 optional
exercise
12.7
- due Tue
Mar 13
March 8
18.
Finding cycles
cycles
Chapter 13
Fixed points, and how to get them
Read all of it.
tutorial
project
TechBurst 2011
Can we do better with a "Slice and Dice" tutorial than TechBurst 2011?
tutorialSD
project
plane Couette movies
Can we do better with a "Slice and Dice" tutorial than plane Couette movies?
March 13
19.
Finding cycles
March 15
20.
Walkabout: Transition graphs
The topological dynamics is encoded
by means of transition matrices/Markov graphs.
Markov
Chapter 14
Walkabout: Transition graphs
Read all of it.
March 19-23
spring break
March 27
21.
Counting
counting
Chapter 15
Counting
Read sects. 15.1 - 15.4.
#10
exercise
(15.1) Transition matrix for 3-disk pinball
#10 optional
exercise
(15.4) loop expansions;
(15.14) 3-disk pinball topological zeta function.
- due Thu
Apr 5
March 29
22.
Counting
April 3
23.
Transporting densities
measure
Chapter 16
Transporting densities
Skip sects. 16.3 and 16.6.
#11
exercise
(16.1) Dirac delta function;
#11 optional
exercise
(16.5) Invariant measure
- due Tue
Apr 10
April 5
24.
Averaging
average
Chapter 17
Averaging
Read sects. 17.1 and 17.2.
[solutions to chaps. 1 to 13 exercises are in DropBox]
April 10
25.
Trace formulas
trace
Chapter 18
Trace formulas
Read all of it.
#12
exercise
(17.1) How unstable is the Henon attractor? parts (d) and (e) optional
#12 optional
exercise
(16.10) generator of translations.
- due Tue
Apr 17
Apr 12
26.
Spectral determinants
^M
det
Chapter 19
Spectral determinants
Skip sects. 19.3.1, 19.3.2, 19.5 and 19.6.
recycle
Chapter 20
Cycle expansions
Read 20.1,
20.2.1,
20.3,
20.3.1,
and
20.6
[
J. Newman: Mathematica periodic orbits routines
]
[
A. Basu: Matlab periodic orbits routines
]
April 17
32.
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.
April 19
28.
How well can one resolve the state space of a chaotic flow?
LipCvi08
Lippolis
P R Lett
Noise smooths out all the kinky determinstic stuff
LipCvi07
Lippolis
the devil is in the details
Ask Gable and Daniel to guide you through all this fancy
stochastic stuff
April 24
29.
Much noise about nothing
intractVid
video
Physicist's life is intractable
Pretty clear discussin of the interplay of noise and determinism - recommended
intract
overheads
Physicist's life is intractable
Overheads for the above lecture
#13:
exercises
26.1, 26.2 and 26.3
- not due in this course [you might want to work them out anyway, Gaussians will serve you well later on]
April 26
30.
The rest is noise
Wherein the Master Slicer Prize will be presented
relax
Chapter 29
Relaxation for cyclists
Optional: might be useful if you need to find some cycles
crete02
Y. Lan
Turbulent fields and their recurrences
A variational principle for robust invariant solutions searches
[prize ceremony]
[projects update]
April 27
GT classes end
May 1
11:30am - 2:20pm term project due, Predrag's office
to May 5
Course opinion survey
CETL web link
May 7
GT grades due at noon
May 7
have a good summer!