individual chapters in paper printout format 

stable version 12, June 27 2008 
icons explained:

contents 

index 

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.  
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  Get straight  
We can make some headway on locally straightening out flows.  
Chapter 7  Hamiltonian dynamics  
Review of basic concepts of local dynamics: Hamiltonian flows,
stability for flows and their Poincaré sections.
appendix C  Stability of Hamiltonian flows (more details, especially for the helium) 

Chapter 8  Billiards  
Billiards and their stability.  
Chapter 9  World in a mirror  
Symmetries simplify the dynamics in a beautiful way: If dynamics is invariant under a set of discrete symmetries, it can be reduced to dynamics within the fundamental domain. If the symmetry is continuous, the dynamics is reduced to a lowerdimensional desymmetrized system. Families of symmetryrelated cycles are replaced by fewer and often much shorter ``relative" cycles.  
Chapter 10  Qualitative dynamics, for pedestrians  
Qualitative properties of
a flow partition the state space in a topologically invariant way:
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 11  Qualitative dynamics, for cyclists  
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 12  Fixed points, and how to get them  
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 Poincaré sections.  
Chapter 13  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.  
Chapter 14  Transporting densities  
A first attempt to move the whole phase space around  natural measure and fancy operators.  
Chapter 15  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 16  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 17  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 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  Discrete factorization  
Symmetries simplify and improve the cycle expansions in a rather beautiful
way, by factorizing the cycle expansions.
appendix I  Further examples of discrete symmetry: rectangles and squares. 

Chapter 20  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 21  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.  
Chapter 22  Thermodynamic formalism  
Generalized dimensions, entropies and such.  
Chapter 23  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 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.  
Chapter 25  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 26  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.  
Chapter 27  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 28  Irrationally winding  
Circle maps and their thermodynamics analyzed in detail. 
Chapter 29  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 30  Quantum mechanics, briefly  
We recapitulate basic notions of quantum mechanics and define the main quantum objects of interest, the quantum propagator and the Green's function.  
Chapter 31  WKB quantization  
A review of the WentzelKramersBrillouin quantization of 1dimensional systems.  
Chapter 32  Semiclassical evolution  
We relate the quantum propagator to the classical flow of the underlying dynamical system; the semiclassical propagator and Green's function.  
Chapter 33  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 34  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 35  Chaotic multiscattering  
Semiclassics of scattering in open systems with a finite number of nonoverlapping scatterers.  
Chapter 36  Helium atom  
Helium atom spectrum computed via semiclassical spectral determinants.  
Chapter 37  Diffraction distraction  
Diffraction effects of scattering off wedges, eavesdropping around corners incorporated into periodic orbit theory.  
Chapter 38  Epilogue  
Takehome exam for the third millenium. 
Chapter 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 threebody problem, the ergodic hypothesis, and nonlinear oscillators.  
Chapter B  Linear stability  
Symplectic invariance, classical collinear helium stability worked out in detail.  
Chapter C  Implementing evolution  
To sharpen our intuition, we outline the fluid dynamical vision, have a bout of Koopmania, and show that shorttimes step definition of the Koopman operator is a prescription for finite time step integration of the equations of motion.  
Chapter D  Symbolic dynamics techniques  
Deals with further, more advanced symbolic dynamics techniques.  
Chapter E  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.''  
Chapter F  Finding cycles  
More on NewtonRaphson method: the details expunged from the chapter on finding cycles languish here.  
Chapter G  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: multidimensional Lyapunov exponents, dynamo rates of vector fields.  
Chapter H  Discrete symmetries of dynamics  
Dynamical zeta functions for systems with symmetries of squares or rectangles worked out in detail.  
Chapter I  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 nth cummulant.  
Chapter J  Infinite dimensional operators  
What is the meaning of traces and determinants for infinitedimensional operators?  
Chapter K  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.  
Chapter L  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.  
Chapter S  Solutions  
Solutions to selected problems  often more instructive than the text itself. Recommended.  
Chapter T  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. 