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ChaosBook.org version15.9, Jun 24 2017   - continuously updated: you might prefer (and please always cite) the latest stable edition    icons explained:

CHAOS: CLASSICAL AND QUANTUM

book cover

contents    index

Part I: Geometry of chaos

1 Overture
2 Go with the flow
3 Discrete time dynamics
4 Local stability
5 Cycle stability
6 Lyapunov exponents
7 Fixed points
8 Hamiltonian dynamics
9 Billiards
10 Flips, slides and turns
11 World in a mirror
12 Relativity for cyclists
13 Slice & dice
14 Charting the state space
15 Stretch, fold, prune
16 Fixed points, and how to get them

Part II: Chaos rules

17 Walkabout: Transition graphs
18 Counting
19 Transporting densities
20 Averaging
21 Trace formulas
22 Spectral determinants
23 Cycle expansions
24 Deterministic diffusion
25 Discrete symmetry factorization
26 Continuous symmetry factorization

Part III: Chaos: what is it good for?

27 Why cycle?
28 Why does it work?
29 Intermittency
30 Turbulence?
31 Dimension of turbulence
32 Universality in transitions to chaos
33 Complex universality
34 Irrationally winding

Part IV: The rest is noise

31 Noise
32 Relaxation for cyclists

Part V: Quantum chaos

33 Prologue
34 Quantum mechanics - the short version
35 WKB quantization
36 Semiclassical evolution
37 Semiclassical quantization
38 Quantum scattering
39 Chaotic multiscattering
40 Helium atom
41 Diffraction distraction
Epilogue

Part VI: Web appendices

A Brief history of chaos
B Go straight
C Linear stability
D Discrete symmetries of dynamics
E Finding cycles
F Symbolic dynamics techniques
G Counting itineraries
H Implementing evolution
I Transport of vector fields
J Convergence of Fredholm determinants
K Infinite dimensional operators
L Thermodynamic formalism
M Statistical mechanics recycled
N Noise/quantum corrections
O Projects
pinball

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contents, credits, acknowledgments

index

Part I: Geometry of 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 - Go with the flow

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 14: Transporting densities.

Chapter 3 - Discrete time dynamics

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.
appendix B - linear algebra, eigenvectors

Chapter 5 - Cycle stability

Topological features of a dynamical system - singularities, periodic orbits, and the ways in which the orbits intertwine - are invariant under a general continuous change of coordinates. Surprisingly, there exist quantities - such as the eigenvalues of periodic orbits - that depend on the notion of metric distance between points, but nevertheless do not change value under a smooth change of coordinates.

Chapter 6 - Lyapunov exponents

Is a given system `chaotic'? And if so, how chaotic?

Chapter 7 - Fixed points

Sadly, searching for periodic orbits will never become as popular as a week on Côte d´Azur, or publishing yet another log-log plot in Phys. Rev. Letters. This chapter is the first of the series of hands-on guides to extraction of periodic orbits, and can be skipped on first reading - you can return to it whenever the need for finding actual cycles arises.

Chapter 8 - Hamiltonian dynamics

Review of basic concepts of local dynamics: Hamiltonian flows, stability for flows and their Poincaré sections.
appendix B - stability of Hamiltonian flows, details for the helium

Chapter 9 - Billiards

Billiards and their stability.

Chapter 10 - Flips, slides and turns

Symmetries simplify the dynamics in a beautiful way.

Chapter 11 - World in a mirror

If dynamics is invariant under a set of discrete symmetries, it can be reduced to dynamics within the fundamental domain. Families of symmetry-related cycles are replaced by fewer and often much shorter "relative" cycles.
appendix H Discrete symmetries of dynamics

Chapter 12 - Relativity for cyclists

Symmetries relate sets of solutions.

Chapter 13 - Slice & dice

If the symmetry is continuous, the dynamics is reduced to a lower-dimensional desymmetrized system. We describe two methods of symmetry reduction: (a) slice the group orbits (b) rewrite the dynamics in terms of invariant polynomials.

Chapter 14 - Charting the state space

Qualitative properties of a flow partition the state space in a topologically invariant way: symbolic dynamics and kneading theory for 1-dimensional maps. Pruning.

Chapter 15 - Stretch, fold, prune

Does there exist a ``natural,'' intrinsically optimal coordinate system? Yes: The intrinsic coordinates are given by the stable/unstable manifolds, and a return map should be plotted as a map from the unstable manifold back onto the immediate neighborhood of the unstable manifold. The level is distinctly cyclist, in distinction to the pedestrian tempo of the preceding chapter.

Chapter 16 - Fixed points, and how to get them

Some of the methods for finding periodic orbits for maps, billiards and flows. There is also a neat way to find Poincaré sections.
appendix C - Newton-Raphson method details languish here

Part II: Chaos rules

Chapter 17 - Walkabout: Transition graphs

Topological dynamics encoded by means of transition matrices/Markov graphs.

Chapter 18 - Counting

You learn here how to count distinct orbits, and in the process touch upon all the main themes of this book, going the whole distance from diagnosing chaotic dynamics to - while computing the topological entropy from transition matrices/Markov graphs - our first zeta function.
appendix D - further, more advanced symbolic dynamics techniques
appendix E - advanced counting: kneading theory (pruning) for unimodal mappings and Bernoulli shifts

Chapter 19 - Transporting densities

A first attempt to move the whole phase space around - natural measure and fancy operators.
appendix F - the fluid dynamical vision, and a bout of Koopmania

Chapter 20 - 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.
appendix G - transport of vector fields, multi-dimensional Lyapunov exponents, dynamo rates

Chapter 21 - 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 22 - Spectral determinants

We derive the spectral determinants, dynamical zeta functions. While traces and determinants are formally equivalent, determinants are the tool of choice when it comes to computing spectra.

Chapter 23 - 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 24 - Discrete factorization

Symmetries simplify and improve the cycle expansions in a rather beautiful way, by factorizing the cycle expansions.
appendix H - further examples of discrete symmetries of dynamics

Part III: Chaos - what to do about it?

Chapter 25 - 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.

Chapter 26 - Why does it work?

We face up to singular kernels, 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.
appendix I - convergence of Fredholm determinants
appendix J - infinite dimensional operators

Chapter 27 - Intermittency

What to do about sticky, marginally stable trajectories? Power-law rather than exponential decorrelations? Problems occur at the borderline between chaos and regular dynamics where marginally stable orbits present still unresolved challenges.

Chapter 28 - Deterministic diffusion

We derive exact formulas for diffusion constants transport coefficients when diffusion is normal, and the anomalous diffusion exponents when it is not. All from first principles, without invoking any Boltzmann-Gibbs probabilistic notions.
appendix K - thermodynamic formalism, generalized dimensions, entropies and such
appendix L - statistical mechanics recycled: spin systems, Feigenbaum scaling function, Fisher droplet model

Chapter 29 - 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 "31" - Dimension of turbulence?

[the chapter not available, but links OK]

Chapter 30 - Irrationally winding

Circle maps and their thermodynamics analyzed in detail.

Part IV: The rest is noise

Chapter 31 - Noise

About noise: how it affects classical dynamics, and the ways it mimicks quantum dynamics. As classical noisy dynamics is more intuitive than quantum dynamics, this exercise helps demystify some of the formal machinery of semiclassical quantization.
appendix M - derive quantum/noise perturbative corrections formulas as Bohr and Sommerfeld would have

Chapter 32 - Relaxation for cyclists

In Chapter 12 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.

Part V: Quantum chaos

Chapter 33 - Prologue

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 34 - Quantum mechanics - the short short version

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

Chapter 35 - WKB quantization

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

Chapter 36 - Semiclassical evolution

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

Chapter 37 - 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 38 - 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 J - infinite dimensional operators

Chapter 39 - Chaotic multiscattering

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

Chapter 40 - Helium atom

Helium atom spectrum computed via semiclassical spectral determinants.

Chapter 41 - Diffraction distraction

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

Epilogue

Take-home exam for the third millenium.

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Part VI: Web appendices


Appendix A - 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 B - Go straight

We can make some headway on locally straightening out flows.

Appendix C - Linear stability

Linear algebra, eigenvalues, eigenvectors, symplectic invariance, stability of Hamiltonian flows, classical collinear helium stability worked out in detail.

Appendix D - Discrete symmetries of dynamics

Dynamical zeta functions for systems with symmetries of squares or rectangles worked out in detail.

Appendix E - Finding cycles

More on Newton-Raphson method: the details expunged from the chapter on finding cycles languish here.

Appendix F - Symbolic dynamics techniques

Deals with further, more advanced symbolic dynamics techniques.

Appendix G - Counting itineraries

Further, more advanced cycle counting techniques: kneading theory (pruning) for unimodal mappings and for Bernoulli shifts. The prime factorization for dynamical itineraries of illustrates the sense in which prime cycles are ``prime.''

Appendix H - 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 I - Transport of vector fields

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 several averages, some more physical than others: multi-dimensional Lyapunov exponents, dynamo rates of vector fields.

Appendix J - Convergence of Fredholm determinants

Why does approximating the dynamics by a finite number of cycles work so well? They approximate smooth flow by a tessalation of a smooth curve by piecewise linear tiles. 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 - Thermodynamic formalism

Generalized dimensions, entropies and such.

Appendix M - Statistical mechanics recycled

Spin systems with long-range interactions (Ising-like spin systems, Feigenbaum scaling function, Fisher droplet model) can be converted into a chaotic dynamical system and recast as a cycle expansion. The convergence to the thermodynamic limit is faster than with the transfer matrix techniques.

Appendix N - Noise/quantum corrections

A formal analogy between the noise and the quantum problem allows us to treat the noise and quantum corrections together. Weak noise is taken into account by corrections to the classical trace formula. 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 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.
Consult the open projects and projects homepages for inspiration. Suggestions welcome.