April 26 2005

CHAOS, AND WHAT TO DO ABOUT IT
  

PROBLEM SETS: Please deliver solutions to problem sets by Tuesday, at the lecture, or place them in Predrag's physics mailbox.

Lecture 1           9:35-10:55 Tue Jan 11 2005 in Howey S104
Lecture 2           9:35-10:55 Thu Jan 13
Overture
That deterministic dynamics leads to chaos is no surprise to anyone who has tried pool, billiards or snooker - that is what the game is about - so we start our course about what is chaos and what to do about it by a game of pinball. This might seem a trifle trivial, but a pinball is to chaotic dynamics what a pendulum is to integrable systems: thinking clearly about what is "chaos" in a pinball will help us tackle more difficult problems, such as computing diffusion constants in deterministic gases, or computing the Helium spectrum. We all have an intuitive feeling for what a pinball does as it bounces between the pinball machine disks, and only high school level Euclidean geometry is needed to describe the trajectory. Turning this intuition into calculation will lead us, in clear physically motivated steps, to almost everything one needs to know about deterministic chaos: from unstable dynamical flows, Poincaré sections, Smale horseshoes, symbolic dynamics, pruning, discrete symmetries, periodic orbits, averaging over chaotic sets, evolution operators, dynamical zeta functions, Fredholm determinants, cycle expansions, quantum trace formulas and zeta functions, and to the semiclassical quantization of helium.

Reading:
 Chapter 1: An overview of the main themes of the course. Recommended reading before you decide to download anything else.
 Appendix - A brief history of chaos: Classical mechanics has not stood still since Newton. The formalism that we use today was developed by Euler and Lagrange. By the end of the 1800's the three problems that would lead to the notion of chaotic dynamics were already known: the three-body problem, the ergodic hypothesis, and nonlinear oscillators.
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 Exercises
problem set 1 (optional)

Lecture 3           9:35-10:55 Tue Jan 18
Lecture 4           9:35-10:55 Thu Jan 20
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. It will be a coarse brush sketch - the full description of all possible behaviors of dynamical systems is anyway beyond human ken.
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Reading:
 Chapter 2:
 Exercises
 Chapter 3:
 Exercises
problem set 2
solutions to problem sets 1 and 2

Lecture 5           9:35-10:55 Tue Jan 25
Lecture 6           9:35-10:55 Thu Jan 27
Local stability
We continue the discussion of local properties of flows and maps: Henon map, linear stability, types of eigenvalues for linear maps, stable/unstable manifolds.

Reading:
 Chapter 4:
 Exercises
problem set 3
pendulum.c code

Lecture 7           9:35-10:55 Tue Feb 1
Lecture 8           9:35-10:55 Thu Feb 3
Newtonian mechanics
Nature is partial to flows invariant with respect to the symplectic structure, that is, flows of Hamiltonian systems. The ability to express mechanical systems in terms of Hamilton's equations provides us with a framework to study general properties and symmetries of this type of flows, and - in particular - study the local stability of equilibria and periodic orbits.

Reading:
  Chapter 5
 Exercises
problem set 4

Billiards
Billiards, Bunimovich-Sinai formula for linear stability in billiards.

Reading:
  Chapter 6
  Chapter 7
  Chapter 8
 Exercises

Lecture 9           9:35-10:55 Tue Feb 8
Lecture 10           9:35-10:55 Thu Feb 10

Transporting densities
So far we learned how to track an individual trajectory, and its small neighborhood. While the trajectory of an individual representative point may be highly convoluted, the density of these points might evolve in a manner that is relatively smooth. The evolution of the density of representative points is for this reason (and other that will emerge in due course) of great interest.

Reading:
 Chapter 9: all, except Sect 7.4.1 - Liouville opertor.
Optional reading:
 Appendix K: Infinite dimensional operators (for students with advanced exposure to quantum mechanics, and mathematically inclined, mostly)
 Exercises
problem set 5
solutions to problem set 4

Dynamical averaging  
In chaotic dynamics detailed prediction is impossible, as any finitely specified initial condition, no matter how precise, will fill out the entire accessible phase space (similarly finitely grained) in finite time. Hence for chaotic dynamics one does not attempt to follow individual trajectories to asymptotic times; what is possible (and sensible) is description of the geometry of the set of possible outcomes, and evaluation of the asymptotic time averages. Examples of such averages are transport coefficients for chaotic dynamical flows, such as the escape rate, mean drift and the diffusion rate; power spectra; and a host of mathematical constructs such as the generalized dimensions, Lyapunov exponents and the Kolmogorov entropy. We shall now set up the formalism for evaluating such averages within the framework of the periodic orbit theory. The key idea is to replace the expectation values of observables by the expectation values of generating functionals. This associates an evolution operator with a given observable, and leads to formulas for its dynamical averages.

Reading:
 Chapter 10:
 Exercises
problem set 5

Rest of the schedule is preliminary

Counting
We finish learning how to count: the traces of powers of the transition matrix count admissible cycles, and the largest eigenvalue of the transition matrix yields the topological entropy. The secular determinant of the transition matrix - the Artin-Mazur zeta function - is expressed in terms of the loops of a Markov diagram.

Dynamics, qualitative I  
We start learning how to count: qualitative dynamics of simple stretching and mixing flows is used to introduce symbolic dynamics.

Reading:
  Chapter 9: Qualitative dynamics for pedestrians
  Exercises
problem set 8
solutions to problem set 8

Dynamics, qualitative II  
We continue learning how to count: qualitative dynamics of Smale horseshoes is used to introduce pruning, finite subshifts, Markov Graphs and transition matrices.
Reading:
  Chapter 10: Qualitative dynamics for cyclists (advanced, optional!    unstable version, hence chapter numbering conflicts with the the stable version)
  ChaosBook excerpts: Unstable manifolds (advanced: to be discusses March 29 at 10am and 1:45pm)
Exercises

Midterm recess: spring break week, Mar 21 to Mar 25 2004

By now we have covered for the first time the whole distance from diagnosing chaotic dynamcs to computing zeta functions. Historically, These topological zeta functions were the inspiration for injecting statistical mechanics into computation of dynamical averages; Ruelle's zeta functions are a weighted generalization of the counting zeta functions.

  Exercises
problem set 9
solutions to problem set 9

last day to drop course

The strategy is this: Global averages such as escape rates can be extracted from the eigenvalues of evolution operators. The eigenvalues are given by the zeros of appropriate determinants. One way to evaluate determinants is to expand them in terms of traces, log det = tr log. The traces are evaluated as integrals over Dirac delta functions, and in this way the spectra of evolution operators become related to periodic orbits.

The rest of the course is making sense out of this objects and learning how to apply them to evaluation of physically measurable properties of chaotic dynamical systems.

Reading:
  Chapter 11: Trace formulas
  Exercises

Lecture 14           9:35-10:55 Thu Feb 17
Spectral determinants
We derive the spectral determinants, dynamical zeta functions.

Reading:
  Chapter 12: Spectral determinants
  Exercises
problem set 10
solutions, chapters 1-15

Lecture 15           9:35-10:55 Tue Feb 22
Why does it work?  I
The heuristic manipulations that led to the trace formulas and spectral determinants are potentially dangerous, as we are dealing with infinite-dimensional vector spaces and singular integral kernels. Intuitively, the theory should converge because long cycles are shadowed by nearby pseudo-cycles. Actually, for clasess of not althogether too idealized smooth flows very strong results exists.

Reading:
  Chapter 13: Why does it work?
  Exercises
problem set 11


Thu Feb 24 2005:    Last day to pick a project

Lecture 16           9:35-10:55 Thu Feb 24
Why does it work? II

For clasess of not althogether too idealized smooth flows very strong results exists. We explain the ideas behind proofs of Ruelle and Rugh which establish that for nice real analytic expanding or hyperbolic flows the spectral (Fredholm) determinants are entire, and that at least in that context the edifice constructed in this course has a mathematical basis.

midterm recess, no lecture           Tue ?? 14 2005

Lecture 17           9:35-10:55 Thu ?? 16
Lecture 18           9:35-10:55 Tue ?? 21
Cycle expansions
So far we have derived a plethora of periodic orbit trace formulas, spectral determinants and zeta functions. Now we learn how to expanded these as cycle expansions, series ordered by increasing topological cycle length, and evaluate average quantites like escape rates. These formulas are exact, and, when the winds are kind, highly convergent. The pleasant surprise is that the terms in such expansions fall off exponentially or even faster, so that a handful of shortest orbits suffices for rather accurate estimates of asymptotic averages.

Reading:
  Chapter 15: Cycle expansions
  Exercises
problem set 12

Lecture 19           9:35-10:55 Thu ?? 23
Lecture 20           9:35-10:55 Tue ?? 28
Fixed points, and how to get them
Periodic orbits can be determined analytically in only very exceptional cases. In order to proceed, we shall need data about unstable periodic orbits, so good numerical methods for their detemination are a necessity. We shall start by determining periodic orbits of a unimodal map, and then proceed to Newton-Raphson method for maps and Poincare maps of flows.

Reading:
  Chapter 14: Fixed points, and how to find them
Extras
  Exercises
  Cristel Chandre's lecture notes
problem set 13

Lecture 21           9:35-10:55 Thu ?? 30
Lecture 22           9:35-10:55 Tue ?? 04
Discrete symmetries
Dynamics often comes equipped with discrete symmetries, such as reflection and discrete rotation symmetries. Symmetries simplify and improve the cycle expansions in a rather beautiful way.

Reading:
  Chapter 19: Discrete symmetries
  Exercises
problem set 14

Lecture 23           9:35-10:55 Thu ?? 06
I (Nonequilibrium) statistical mechanics from deterministic chaos?  
After the excursion into Lyapunov exponents, entropies, dimensions and thermodynamical formalism, we are back to the meat of the subject; can one derive statistical mechanics from deterministic chaos? We apply the periodic orbit theory to evaluation of measurable properties of a very simple model of ideal gas, periodic Lorentz gas, and derive closed-form formulas for the diffusion constant.

Can one one go beyond equilibrium statistical mechanics and derive properties of systems far from equilibrium by the methods discussed in this course? This is currently a very lively research area, and we explain how the periodic orbit theory yields transport properties in models of dissipative driven systems, such as Guassian thermostatted Lorentz gas.

  Exercises
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Friday Apr 29:    classes end

Term papers        due no later than 10:00 Monday May 2 2005 - by email / or Predrag's office?

grades deadline            Mon May 9

Predrag.Cvitanovic at physics.gatech.edu