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ChaosBook.org version17.6.1, Jul 7 2022  continuously updated: you might prefer (and please always cite) the latest stable edition 
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A very very short introduction 

contents, acknowledgments 

contributors 

index 
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 loglog plot in Phys. Rev. Letters. This chapter is the first of the series of handson 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 symmetryrelated 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 lowerdimensional 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 1dimensional 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  NewtonRaphson method details languish here 
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, multidimensional 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
BoltzmannGibbs 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 higherdimensional 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 
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? Powerlaw 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 longtime dynamics of many such systems of physical interest is finitedimensional. Here we cure you of the fear of infinitedimensional 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. 
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, handson guide to extraction of periodic orbits by means of the NewtonRaphson method. Here we take a very different tack, drawing inspiration from variational principles of classical mechanics, and path integrals of quantum mechanics. 
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 WentzelKramersBrillouin quantization of 1dimensional 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 nonoverlapping 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.  
Epilogue  
Takehome exam for the third millenium. 
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 threebody 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 HamiltonÃ¢â‚¬â„¢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 NewtonRaphson 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 

4disk 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 nth 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 longrange interactions (Isinglike 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 infinitedimensional 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. 