individual chapters in paper printout format |
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stable
version13,
Dec 31 2009
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you might prefer
the entire version 13
webbook in the Acrobat PDF format, with hyperlinks
(please always cite the latest stable version) |
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contents |
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index |
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| Chapter 1 - Overture |
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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 |
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| 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. | ||||
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| 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. | ||||
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| Chapter 4 - Local stability |
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Review of basic concepts of local dynamics: local linear stability for flows and maps.
appendix B - linear algebra, eigenvectors |
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| 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. | ||||
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| Chapter 6 - Get straight |
| We can make some headway on locally straightening out flows. | ||||
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| Chapter 7 - Hamiltonian dynamics |
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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 |
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| Chapter 8 - Billiards |
| Billiards and their stability. | ||||
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| 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. Families of symmetry-related cycles are replaced by fewer and often much shorter "relative" cycles. | ||||
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| Chapter 10 - Relativity for cyclists |
| 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. | ||||
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| Chapter 11 - 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. | ||||
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| Chapter 12 - 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. | ||||
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| Chapter 13 - Fixed points, and how to get them |
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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 |
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| Chapter 14 - Walkabout: Transition graphs |
| Topological dynamics encoded by means of transition matrices/Markov graphs. | ||||
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Chapter 15 - Counting |
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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 |
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Chapter 16 - Transporting densities |
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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 |
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Chapter 17 - Averaging |
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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 |
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Chapter 18 - 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. | ||||
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Chapter 19 - 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. | ||||
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Chapter 20 - 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. | ||||
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Chapter 21 - Discrete factorization |
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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 |
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Chapter 22 - 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. | ||||
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Chapter 23 - Why does it work? |
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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 |
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Chapter 24 - 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. | ||||
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Chapter 25 - Deterministic diffusion |
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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 |
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Chapter 26 - 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. | ||||
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Chapter 27 - Irrationally winding |
| Circle maps and their thermodynamics analyzed in detail. |
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Chapter 28 - Noise |
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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 |
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Chapter 29 - 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. |
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Chapter 30 - 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. | ||||
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Chapter 31 - 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. | ||||
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Chapter 32 - WKB quantization |
| A review of the Wentzel-Kramers-Brillouin quantization of 1-dimensional systems. | ||||
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Chapter 33 - Semiclassical evolution |
| We relate the quantum propagator to the classical flow of the underlying dynamical system; the semiclassical propagator and Green's function. | ||||
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Chapter 34 - 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. | ||||
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Chapter 35 - Quantum scattering |
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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 |
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Chapter 36 - Chaotic multiscattering |
| Semiclassics of scattering in open systems with a finite number of non-overlapping scatterers. | ||||
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Chapter 37 - Helium atom |
| Helium atom spectrum computed via semiclassical spectral determinants. | ||||
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Chapter 38 - Diffraction distraction |
| Diffraction effects of scattering off wedges, eavesdropping around corners incorporated into periodic orbit theory. | ||||
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Epilogue |
| Take-home exam for the third millenium. |
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| 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. | ||||
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| Appendix B - Linear stability |
| Linear algebra, eigenvalues, eigenvectors, symplectic invariance, stability of Hamiltonian flows, classical collinear helium stability worked out in detail. | ||||
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| Appendix C - Finding cycles |
| More on Newton-Raphson method: the details expunged from the chapter on finding cycles languish here. | ||||
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| Appendix D - Symbolic dynamics techniques |
| Deals with further, more advanced symbolic dynamics techniques. | ||||
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| Appendix 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.'' | ||||
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| Appendix F - 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. | ||||
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| Appendix 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: multi-dimensional Lyapunov exponents, dynamo rates of vector fields. | ||||
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| Appendix H - Discrete symmetries of dynamics |
| Dynamical zeta functions for systems with symmetries of squares or rectangles worked out in detail. | ||||
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| Appendix 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 n-th cummulant. | ||||
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| Appendix J - Infinite dimensional operators |
| What is the meaning of traces and determinants for infinite-dimensional operators? | ||||
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Appendix K - Thermodynamic formalism |
| Generalized dimensions, entropies and such. | ||||
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| Appendix L - 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. | ||||
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| Appendix M - 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. | ||||
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| Appendix S - Solutions |
| Solutions to selected problems - often more instructive than the text itself. Recommended. | ||||
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| Appendix T - Projects |
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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. |
