Version 10, the stable version aug 2003 - nov 2004

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	Contents
	Index
	Chapter 1	Overture An overview of the main themes of the book. Recommended reading before you decide to download anything else.	Nov 20 2002 90% finished
	Appendix A:	you might also want to read about the history of the subject.
	Exercises		Jan 30 2002
			Jun 10 2003
	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 chapter.	Jun 18 2003 70% finished
	Exercises		Jan 30 2002
			Aug 21 2002
	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.	Jun 18 2003 70% finished
	Exercises		Jan 30 2002
			Aug 21 2002
	Chapter 4	Local stability Review of basic concepts of local dynamics: local linear stability for flows and maps.	Nov 20 2002 60% finished
	Exercises		Jan 30 2002
			Jan 30 2002
	Chapter 5	Newtonian dynamics Review of basic concepts of local dynamics: Hamiltonian flows, stability for flows, billiards and their stability.	Jun 18 2003 60% finished
	Appendix C:	Stability of Hamiltonian flows: more details, especially for the helium.
	Exercises		Jan 30 2002
			Jan 30 2002
	Chapter 6	Get straight We can make some headway on locally straightening out flows.	Jun 18 2003 60% finished
	Exercises		Aug 30 2003
	Chapter 7	Transporting densities A first attempt to move the whole phase space around - natural measure and fancy operators	Nov 20 2002 60% finished
	Exercises		Jan 30 2002
			10 Feb 2000
	Chapter 8	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.	Nov 20 2002 90% finished
	Exercises		Jan 30 2002
			Aug 10 2002
	Chapter 9	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.	Nov 20 2002 60% finished
	Appendix E:	deals with further, more advanced symbolic dynamics techniques.	8 aug 99 80% finished
	Exercises		Jan 30 2002
			10 Feb 2000
	Chapter 10	Counting, for pedestrians One learns how to count and describe itineraries. While computing the topological entropy from transition matrices/Markov graphs, we encounter our first zeta function.	Aug 30 2003 60% finished
	Exercises		22 aug 98
			10 Feb 2000
	Chapter 11	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.	Nov 20 2002 85% finished
	Exercises		Jan 30 2002
			Aug 10 2002
	Chapter 12	Spectral determinants We derive the spectral determinants, dynamical zeta functions.	Nov 20 2002 85% finished
	Exercises		Jan 30 2002
			Aug 10 2002
	Chapter 13	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.	Nov 20 2002 76% finished
	Exercises		12 aug 2000
			16 May 2001
	Chapter 14	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.	4 Oct 98 70% finished
	Exercises		16 mar 98
			12 aug 2000
	Chapter 15	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.	30 Aug 98 90% finished
	Exercises		Jan 30 2002
			10 Feb 2000
	Chapter 16	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.	Nov 20 2002 50% finished
	Exercises		Jan 30 2002
			Aug 10 2002
	Chapter 17	Thermodynamic formalism Generalized dimensions, entropies and such.	25 aug 2000 50% finished
	Exercises		25 aug 2000
	Chapter 18	Intermittency What to do about sticky, marginally stable trajectories? Power-law rather than exponential decorrelations?	Nov 20 2002 75% finished
	Exercises		7 jun 2000
			7 jun 2000
	Chapter 19	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.	Nov 20 2002
	Appendix I:	deals with further examples of discrete symmetry (rectangles and squares).
	Exercises		10 jan 99
	Chapter 20	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.	Nov 20 2002 85% finished
	Exercises		16 mar 98
			10 feb 2000
	Chapter 21	Irrationally winding Circle maps and their thermodynamics analyzed in detail.	Dec 96 85% finished
	Exercises

Part II: Quantum chaos

	Chapter 22	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.	Jun 15 2003 70% finished
	Chapter 23	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.	Jan 30 2002 85% finished
	Chapter 24	WKB quantization A review of the Wentzel-Kramers-Brillouin quantization of 1-dimensional systems.	Jan 30 2002 85% finished
	Exercises		Jun 15 2003
	Chapter 25	Relaxation for cyclists In Chapter 14 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.	Aug 30 2003 85% finished
	Exercises		Aug 30 2003
			10 Feb 2000
	Chapter 26	Semiclassical evolution We relate the quantum propagator to the classical flow of the underlying dynamical system; the semiclassical propagator and Green's function.	Jan 30 2002 85% finished
	Exercises		Jan 30 2002
			10 Feb 2000
	Chapter 27	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.	Jan 30 2002 80% finished
	Exercises		Jan 30 2002
			Aug 10 2002
	Chapter 28	Chaotic scattering Scattering off N disks, exact and semiclassical.	12 aug 2000 80% finished
	Appendix K:	What is the meaning of traces and determinants for infinite-dimensional operators?
	Exercises		12 aug 2000
			10 Feb 2000
	Chapter 29	Helium atom The helium atom spectrum computed via semiclassical spectral determinants.	17 june 2000 96% finished
	Appendix C:	Stability of Hamiltonian flows: more details, especially for the helium.
	Exercises		12 aug 2000
			Aug 10 2002
	Chapter 30	Diffraction distraction Diffraction effects of scattering off wedges, eavesdropping around corners incorporated into periodic orbit theory.	Jan 30 2002 95% finished
	Exercises		Jan 30 2002
		Epilogue Take-home problem set for the third millenium.	6 Sept 96 10% finished

Part www: Material which will be kept on the web

	Appendix 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.	22 Jul 97 66% finished
	Appendix B	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).	20 nov 2002 30% finished
	Appendix C	Stability of Hamiltonian flows Symplectic invariance, classical collinear helium stability worked out in detail.	12 aug 2000 80% finished
	Appendix D	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.	15 nov 2002 50% finished
	Exercises
	Appendix E	Symbolic dynamics techniques Further, more advanced symbolic dynamics techniques.	9 March 98 60% finished
	Appendix F	Counting itineraries Further, more advanced cycle counting techniques.	30 nov 2001 60% finished
	Exercises
	Appendix G	Finding cycles More on Newton-Raphson method.	9 March 98 60% finished
	Appendix H	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.	Jan 30 2002 60% finished
	Exercises		Jan 30 2002
			10 Feb 2000
	Appendix I	Discrete symmetries Dynamical zeta functions for systems with symmetries of squares or rectangles worked out in detail.	10 Jan 99 80% finished
	Appendix J	Convergence of spectral determinants A heuristic estimate of the n-th cummulant.	12 aug 2000 30% finished
	Appendix K	Infinite dimensional operators What is the meaning of traces and determinants for infinite-dimensional operators?	9 Feb 96 95% finished
	Appendix L	Statistical mechanics recycled The Ising-like spin systems recycled. The Feigenbaum scaling function and the Fisher droplet model.	14 Nov 96 33% finished
	Exercises		9 sep 98
			10 Feb 2000
	Appendix M	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.	5 Jun 1995 50% finished
	Appendix N	What reviewers say Bohr, Feynman and so on turning in their graves. Ignore this.	12 aug 2000 1% finished
	Appendix O	Solutions Solutions to selected problems - often more instructive than the text itself. Recommended.	Jan 30 2002 55% finished
	Appendix P	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. Deterministic diffusion, zig-zag map - solution in appendix Solutions [23 mar 98] Cardioid billiard [23 mar 98] Ray splitting billiard [23 mar 98] list of other projects [23 mar 98]	12 aug 2000 55% finished