individual chapters in paper printout format (no hyperlinks!) version17.6.4, May 6 2023   - continuously updated: you might prefer (and please always cite) the latest stable edition    icons explained:


book cover

contents    index    short intro

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 Koopman modes
"32" Universality in transitions to chaos
"33" Complex universality
32 Irrationally winding

Part IV: The rest is noise

33 Noise
34 Relaxation for cyclists

Part V: Quantum chaos

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

Part VI: Web appendices

A1 Brief history of chaos
A2 Go straight
A4 Linear stability
A6 Lyapunov exponents
A8 Hamiltonian dynamics
A10 Flips, slides and turns
A14 Charting the state space
A16 Finding cycles
A18 Counting itineraries
A20 Averaging
A24 Deterministic diffusion
A25 Discrete symmetry factorization
A28 Convergence of spectral determinants
A31 Koopman modes
A36 Thermodynamic formalism
A37 Statistical mechanics recycled
A45 Semiclassical quantization
A46 Infinite dimensional operators
A47 Projects


A very very short introduction

contents, acknowledgments



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 A1 - 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 A4 - 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 A8 - 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 A25 Discrete symmetry factorization

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 A16 - 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 A14 - further, more advanced symbolic dynamics techniques
appendix A18 - 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.
sect 33.1 Deterministic transport - the fluid dynamical vision

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 A6 - 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 - 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 A32 - thermodynamic formalism, generalized dimensions, entropies and such
appendix A33 - statistical mechanics recycled: spin systems, Feigenbaum scaling function, Fisher droplet model

Chapter 25 - Discrete symmetry factorization

Symmetries simplify and improve the cycle expansions in a rather beautiful way, by factorizing the cycle expansions.
appendix A25 - further examples of discrete symmetries of dynamics

Chapter 26 - Continuous symmetry factorization

Higher dimensional dynamics requires inclusion of higher-dimensional compact invariant sets into trace formulas. A trace formula for a partially hyperbolic (N + 1)-dimensional compact manifold invariant under N global continuous symmetries is derived here.
appendix A26 - further examples of continuous symmetries of dynamics

Part III: Chaos - what to do about it?

Chapter 27 - 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 28 - 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 A28 - convergence of Fredholm determinants
appendix A40 - infinite dimensional operators

Chapter 29 - 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 30 - 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 31 - Koopman modes

The nonlinear dynamics of transient states on the way to a stable solution is captured by the eigenfunctions of the linear Koopman operator..

Chapter 32 - Irrationally winding

Circle maps and their thermodynamics analyzed in detail.

Part IV: The rest is noise

Chapter 33 - 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 A39 - derive quantum/noise perturbative corrections formulas as Bohr and Sommerfeld would have

Chapter 34 - 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 35 - 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 36 - 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 37 - WKB quantization

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

Chapter 38 - Semiclassical evolution

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

Chapter 39 - 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 40 - 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 A40 - infinite dimensional operators

Chapter 41 - Chaotic multiscattering

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

Chapter 42 - Helium atom

Helium atom spectrum computed via semiclassical spectral determinants.

Chapter 43 - Diffraction distraction

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


Take-home exam for the third millenium.


Part VI: Web appendices

Appendix A1 - 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 A2 - Go straight

We can make some headway on locally straightening out flows.

Appendix A4 - Linear stability

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

Appendix A6 - Lyapunov exponents

To compute an average using cycle expansions one has to find the leading eigenvalue and maybe a few of its derivatives. Here we explore how to do that for several averages, some more physical than others. We show that multidimensional Lyapunov exponents, and relaxation exponents (dynamo rates) of vector fields, can be expressed in terms of leading eigenvalues of appropriate evolution operators.

Appendix A8 - Hamiltonian dynamics

The symplectic structure of Hamilton’s equations buys us much more than just the incompressibility, or the phase space volume conservation.

Appendix A14 - Charting the state space

Deals with further, more advanced symbolic dynamics techniques: periodic orbits of unimodal mappings and the pruning theory of Bernoulli shifts.

Appendix A16 - Finding cycles

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

Appendix A18 - 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 A20 - Averaging

We review some elementary notions of probability theory that will be useful to you no matter what you do with the rest of your life.

Appendix A24 - Deterministic diffusion

Diffusive properties of a prototype example of chaotic Hamiltonian maps, hyperbolic toral automorphisms.

Appendix A25 - Discrete symmetry factorization

4-disk spectral determinant factorization.

Appendix A28 - Convergence of spectral 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.

Chapter A31 - Koopman modes

The notion of spectrum of a linear operator. Implementing evolution. A symplectic integrator.

Appendix A32 - Thermodynamic formalism

Generalized dimensions, entropies and such.

Appendix A33 - 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 A39 - Semiclassical quantization, with 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 A40 - Infinite dimensional operators

What is the meaning of traces and determinants for infinite-dimensional operators?

Appendix S - Solutions

Solutions to selected problems - often more instructive than the text itself. Available to some, upon request.

Appendix A47 - 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.