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Contents
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Index
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Chapter 1 - Overture
An overview of the main themes of the book. Recommended reading before
you decide to download anything else.
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appendix A -
You might also want to read about the history of the subject.
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Chapter 2 - Flows
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 materials,
hop to
chapter 9: Transporting densities.
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Chapter 3 - Maps
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
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Local stability
Review of basic concepts of local dynamics: local linear
stability for flows and maps.
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Chapter 5
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Newtonian dynamics
Review of basic concepts of local dynamics: Hamiltonian flows,
stability for flows and their Poincare sections.
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appendix C -
Stability of Hamiltonian flows
(more details, especially for the helium)
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Chapter 6
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Billiards
Billiards and their stability.
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Chapter 7
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Get straight
We can make some headway on locally straightening out flows.
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Chapter 8
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Cycle stability
We can make some headway on locally straightening out flows.
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Chapter 9 - Transporting densities
A first attempt to move the
whole phase space around - natural measure and fancy operators.
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Chapter 10 - 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.
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Chapter 11 - Qualitative dynamics, for pedestrians
Qualitative dynamics of simple stretching and mixing flows; Smale
horseshoes and symbolic dynamics. The topological dynamics is incoded
by means of transition matrices/Markov graphs.
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appendix E -
further, more advanced symbolic dynamics techniques.
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Chapter 12 - Qualitative dynamics, for cyclists
Theory of pruning fronts for generic flows.
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Chapter 13
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Counting, for pedestrians
You learn here how to count and describe itineraries.
While computing the topological
entropy from transition matrices/Markov graphs, we encounter our first
zeta function.
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Chapter 14
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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 15
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Spectral determinants
We derive the spectral determinants, dynamical zeta functions.
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Chapter 16
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Why does it work?
This chapter faces the singular kernels, the 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.
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Chapter 17
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Fixed points, and how to get them
Periodic orbits can be determined analytically in only few exceptional
cases. 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 Poincare sections.
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Chapter 18
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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 19
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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. A few special determinants are worked out
by hand.
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Chapter 20
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Thermodynamic formalism
Generalized dimensions, entropies and such.
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Chapter 21
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Intermittency
What to do about sticky, marginally stable trajectories? Power-law rather
than exponential decorrelations?
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Chapter 22
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Discrete symmetries
Dynamics often comes equipped with discrete symmetries, such as
the reflection and the rotation symmetries.
Symmetries simplify and improve the cycle expansions in a rather beautiful
way. This chapter explains how symmetries factorize the cycle expansions.
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appendix I -
further examples of discrete symmetry: rectangles and squares.
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Chapter 23
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Deterministic diffusion
We look at transport coefficients and derive exact formulas
for diffusion constants
when diffusion is normal, and the anomalous diffusion exponents when it is
not. All
done from first principles without ever invoking any probabilistic notions.
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Chapter 24
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Irrationally winding
Circle maps and their thermodynamics analyzed in detail.
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Chapter 25
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Overture
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 26
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Quantum mechanics, briefly
We first 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 27
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WKB quantization
A review of the Wentzel-Kramers-Brillouin quantization of 1-dimensional
systems.
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Chapter 28
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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 29 - Noise
About noise: how it affects classical dynamics, and the ways it mimicks
quantum dynamics.
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Chapter 30 - 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 31 - Relaxation for cyclists
In Chapter 17 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 32 - 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.
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appendix K -
Traces and determinants for infinite-dimensional operators?
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Chapter 33 - Chaotic multiscattering
Semiclassics of scattering in open
systems with a finite number of non-overlapping
scatterers.
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Chapter 34 - Helium atom
Helium atom spectrum computed via semiclassical spectral determinants.
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appendix C -
Stability of Hamiltonian flows, details for the helium
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Chapter 35 - 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
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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
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Infinite-dimensional flows
Flows described by
partial differential equations
are infinite dimensional because
if one writes them down as a set of ordinary differential equations (ODEs)
then one needs an infinity of the ordinary
kind to represent the dynamics of one equation of the partial kind
(PDE).
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Appendix C
- Stability of Hamiltonian flows
Symplectic invariance,
classical collinear helium stability worked out in detail.
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Appendix D
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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 E:
Deals with further, more advanced symbolic dynamics techniques.
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Appendix F
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Counting itineraries
Further, more advanced cycle counting techniques.
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Exercises
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Appendix G
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Finding cycles
More on Newton-Raphson method.
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Appendix H
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Applications
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 all sorts of averages, some more physical than others.
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Exercises
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Appendix I
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Discrete symmetries
Dynamical zeta functions for systems with symmetries of squares or rectangles
worked out in detail.
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Appendix J
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Convergence of spectral determinants
A heuristic estimate of the n-th cummulant.
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Appendix K
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Infinite dimensional operators
What is the meaning of traces and determinants for infinite-dimensional
operators?
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Appendix L
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Statistical mechanics recycled
The Ising-like spin systems recycled. The Feigenbaum
scaling function and the Fisher droplet model.
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Exercises
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Appendix M
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Noise/quantum trace formulas
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 N
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Solutions
Solutions to selected problems - often more instructive than the
text itself. Recommended.
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Appendix O
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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.
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