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book cover

Part I: Classical chaos

contents    index
1 Overture
2 Flows
3 Maps
4 Local stability
5 Newtonian dynamics
6 Billiards
7 Get straight
8 Cycle stability
9 Transporting densities
10 Averaging
11 Qualitative dynamics, for pedestrians
12 Qualitative dynamics, for cyclists
13 Counting, for pedestrians
14 Trace formulas
15 Spectral determinants
16 Why does it work?
17 Fixed points, and how to get them
18 Cycle expansions
19 Why cycle?
20 Thermodynamic formalism
21 Intermittency
22 Discrete symmetries
23 Deterministic diffusion
24 Turbulence?
25 Irrationally winding

Part II: Quantum chaos

25 Overture
26 Quantum mechanics, briefly
27 WKB quantization
28 Semiclassical evolution
29 Noise
30 Semiclassical quantization
31 Relaxation for cyclists
32 Quantum scattering
33 Chaotic multiscattering
34 Helium atom
35 Diffraction distraction

Part III: Web appendices

A Brief history of chaos
B Stability of Hamiltonian flows
C Implementing evolution
D Symbolic dynamics techniques
E Counting itineraries
F Finding cycles
G Applications
I Discrete symmetries
J Convergence of spectral determinants
K Infinite dimensional operators
L Statistical mechanics recycled
M Noise/quantum trace formulas
N Solutions
O Projects
Cardioid billiard
Ray splitting billiard
list of other projects


Part I: Classical chaos



Chapter 1 - Overture

An overview of the main themes of the book. Recommended reading before you decide to download anything else.
appendix A - You might also want to read about the history of the subject.

Chapter 2 - Flows

A recapitulation of basic notions of dynamics. The reader familiar with the dynamics on the level of an introductory graduate nonlinear dynamics course can safely skip this material, hop to chapter 9: Transporting densities.

Chapter 3 - Maps

Discrete time dynamics arises by considering sections of a continuous flow. There are also many settings in which dynamics is discrete, and naturally described by repeated applications of a map.

Chapter 4 - Local stability

Review of basic concepts of local dynamics: local linear stability for flows and maps.

Chapter 5 - Newtonian dynamics

Review of basic concepts of local dynamics: Hamiltonian flows, stability for flows and their Poincare sections.
appendix C - Stability of Hamiltonian flows (more details, especially for the helium)

Chapter 6 - Billiards

Billiards and their stability.

Chapter 7 - Get straight

We can make some headway on locally straightening out flows.

Chapter 8 - Cycle stability

We can make some headway on locally straightening out flows.

Chapter 9 - Transporting densities

A first attempt to move the whole phase space around - natural measure and fancy operators.

Chapter 10 - Averaging

On the necessity of studying the averages of observables in chaotic dynamics. Formulas for averages are cast in a multiplicative form that motivates the introduction of evolution operators.

Chapter 11 - Qualitative dynamics, for pedestrians

Qualitative dynamics of simple stretching and mixing flows; Smale horseshoes and symbolic dynamics. The topological dynamics is incoded by means of transition matrices/Markov graphs.
appendix E - further, more advanced symbolic dynamics techniques.

Chapter 12 - Qualitative dynamics, for cyclists

Theory of pruning fronts for generic flows.

Chapter 13 - Counting, for pedestrians

You learn here how to count and describe itineraries. While computing the topological entropy from transition matrices/Markov graphs, we encounter our first zeta function.

Chapter 14 - Trace formulas

If there is one idea that one should learn about chaotic dynamics, it happens in this chapter: the (global) spectrum of the evolution is dual to the (local) spectrum of periodic orbits. The duality is made precise by means of trace formulas.

Chapter 15 - Spectral determinants

We derive the spectral determinants, dynamical zeta functions.

Chapter 16 - Why does it work?

This chapter faces the singular kernels, the infinite dimensional vector spaces and all those other subtleties that are needed to put the spectral determinants on more solid mathematical footing, to the extent this can be achieved without proving theorems.

Chapter 17 - Fixed points, and how to get them

Periodic orbits can be determined analytically in only few exceptional cases. In this chapter we describe some of the methods for finding periodic orbits for maps, billiards and flows. There is also a neat way to find Poincare sections.

Chapter 18 - Cycle expansions

Spectral eigenvalues and dynamical averages are computed by expanding spectral determinants into cycle expansions, expansions ordered by the topological lengths of periodic orbits.

Chapter 19 - Why cycle?

In the preceeding chapters we have moved at rather brisk pace and derived a gaggle of formulas. Here we slow down in order to develop some fingertip feeling for the objects derived so far. Just to make sure that the key message - the ``trace formulas'' and their ilk - have sunk in, we rederive them in a rather different, more intuitive way, and extol their virtues. This part is bedtime reading. A few special determinants are worked out by hand.

Chapter 20 _ Thermodynamic formalism

Generalized dimensions, entropies and such.

Chapter 21 - Intermittency

What to do about sticky, marginally stable trajectories? Power-law rather than exponential decorrelations?

Chapter 22 - Discrete symmetries

Dynamics often comes equipped with discrete symmetries, such as the reflection and the rotation symmetries. Symmetries simplify and improve the cycle expansions in a rather beautiful way. This chapter explains how symmetries factorize the cycle expansions.
appendix I - further examples of discrete symmetry: rectangles and squares.

Chapter 23 - Deterministic diffusion

We look at transport coefficients and derive exact formulas for diffusion constants when diffusion is normal, and the anomalous diffusion exponents when it is not. All done from first principles without ever invoking any probabilistic notions.

Chapter 24 - Turbulence?

Flows described by PDEs are said to be `infinite dimensional' because if one writes them down as a set of ODEs, one needs infinitely many of them to represent the dynamics of one PDE. The long-time dynamics of many such systems of physical interest is finite-dimensional. Here we cure you of the fear of infinite-dimensional flows.

Chapter 25 - Irrationally winding

Circle maps and their thermodynamics analyzed in detail.


Part II: Quantum chaos

Chapter 25 - Overture

In the Bohr - de Broglie old quantum theory one places a wave instead of a particle on a Keplerian orbit around the hydrogen nucleus. The quantization condition is that only those orbits contribute for which this wave is stationary. Here we shall show that a chaotic system can be quantized by placing a wave on each of the infinity of unstable periodic orbits.

Chapter 26 - Quantum mechanics, briefly

We first recapitulate basic notions of quantum mechanics and define the main quantum objects of interest, the quantum propagator and the Green's function.

Chapter 27 - WKB quantization

A review of the Wentzel-Kramers-Brillouin quantization of 1-dimensional systems.

Chapter 28 - Semiclassical evolution

We relate the quantum propagator to the classical flow of the underlying dynamical system; the semiclassical propagator and Green's function.

Chapter 29 - Noise

About noise: how it affects classical dynamics, and the ways it mimicks quantum dynamics.

Chapter 30 - Semiclassical quantization

This is what could have been done with the old quantum mechanics if physicists of 1910's were as familiar with chaos as you by now are. The Gutzwiller trace formula together with the corresponding spectral determinant, the central results of the semiclassical periodic orbit theory, are derived.

Chapter 31 - Relaxation for cyclists

In Chapter 17 we offered an introductory, hands-on guide to extraction of periodic orbits by means of the Newton-Raphson method. Here we take a very different tack, drawing inspiration from variational principles of classical mechanics, and path integrals of quantum mechanics.

Chapter 32 - Quantum scattering

A brief review of the quantum theory of elastic scattering of a point particle from a repulsive potential, and its connection to the Gutzwiller theory for bound systems.
appendix K - Traces and determinants for infinite-dimensional operators?

Chapter 33 - Chaotic multiscattering

Semiclassics of scattering in open systems with a finite number of non-overlapping scatterers.

Chapter 34 - Helium atom

Helium atom spectrum computed via semiclassical spectral determinants.
appendix C - Stability of Hamiltonian flows, details for the helium

Chapter 35 - Diffraction distraction

Diffraction effects of scattering off wedges, eavesdropping around corners incorporated into periodic orbit theory.


Take-home exam for the third millenium.


Part III: Material which will be kept on the web

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.

Appendix C - Stability of Hamiltonian flows

Symplectic invariance, classical collinear helium stability worked out in detail.

Appendix D - Implementing evolution

To sharpen our intuition, we outline the fluid dynamical vision, have a bout of Koopmania, and show that short-times step definition of the Koopman operator is a prescription for finite time step integration of the equations of motion.

Appendix E:

Deals with further, more advanced symbolic dynamics techniques.

Appendix F Counting itineraries
Further, more advanced cycle counting techniques.
Appendix G Finding cycles
More on Newton-Raphson method.
Appendix H Applications
To compute an average using cycle expansions one has to find the right eigenvalue and maybe a few of its derivatives. Here we explore how to do that for all sorts of averages, some more physical than others.
Appendix I Discrete symmetries
Dynamical zeta functions for systems with symmetries of squares or rectangles worked out in detail.
Appendix J Convergence of spectral determinants
A heuristic estimate of the n-th cummulant.
Appendix K Infinite dimensional operators
What is the meaning of traces and determinants for infinite-dimensional operators?
Appendix L Statistical mechanics recycled
The Ising-like spin systems recycled. The Feigenbaum scaling function and the Fisher droplet model.
Appendix M Noise/quantum trace formulas
The quantum/noise perturbative corrections formulas derived as Bohr and Sommerfeld would have derived them were they cogniscenti of chaos, with some Vattayismo rumminations along the way.
Appendix N Solutions
Solutions to selected problems - often more instructive than the text itself. Recommended.
Appendix O Projects
The essence of this subject is incommunicable in print; the only way to developed intuition about chaotic dynamics is by computing, and you are urged to try to work through the essential steps in a project that combines the techniques learned in the course with some application of interest to you.