]> August 19 Predrag Cvitanović 1. Things fall apart A brief history of motion in time. intro Chapter 1 Overture Read quickly all of it - do not worry if there are stretches that you do not understand yet. The rest is optional reading: appendHist Appendix A Brief history of chaos ChaosBook.org version12.1, Aug 22 2008: iPaper version (let me know if you prefer PDF files) introOverheads lecture overheads August 21 2. Trajectories We start out by a recapitulation of the basic notions of dynamics. Our aim is narrow; keep the exposition focused on prerequsites to the applications to be developed in this text. I assume that you are familiar with the dynamics on the level of introductory texts such as Strogatz, and concentrate here on developing intuition about what a dynamical system can do. flows Chapter 2 Flows flowsOverh lecture overheads homework #1: exercises (1.1), (2.1), (2.7), and (2.8), optional (2.10) - due Tue Aug 26 [solutions to chap. 1 exercises] [solutions to chap. 2 exercises] August 26 3. Flow visualized as an iterated mapping Discrete time dynamical systems arise naturally by either strobing the flow at fixed time intervals (we will not do that here), or recording the coordinates of the flow when a special event happens (the Poincare section method, key insight for much that is to follow). maps Chapter 3 Discrete time dynamics mapsOverh lecture overheads August 28 4. There goes the neighborhood So far we have concentrated on description of the trajectory of a single initial point. Our next task is to define and determine the size of a neighborhood, and describe the local geometry of the neighborhood by studying the linearized flow. What matters are the expanding directions. The repercussion are far-reaching: As long as the number of unstable directions is finite, the same theory applies to finite-dimensional ODEs, Hamiltonian flows, and dissipative, volume contracting infinite-dimensional PDEs. stability Chapter 4 Local stability ChaosBook.org version12.1, Aug 22 2008: skip sect. 4.5.1 stabilityOverh lecture overheads invariants Chapter 5 Cycle stability Read quickly through it, skip sect. 5.3. Skipped in the lectures, but will need some of the definitions in what follows. conjug Chapter 6 Get straight Advanced material, most of it safely skipped. Read at least sect. 6.6, and, if you have had trouble with integrating helium dynamics, sect 6.3. Skipped in the lectures. homework #2: exercises (3.1), (3.5), (4.1), and (4.3), optional (3.6) and (4.4) - due Tue Sep 2 [solutions to chap. 4 exercises] September 1 Labor Day September 2 5. Pinball wizzard The dynamics that we have the best intuitive grasp on is the dynamics of billiards. For billiards, discrete time is altogether natural; a particle moving through a billiard suffers a sequence of instantaneous kicks, and executes simple motion in between, so there is no need to contrive a Poincare section. newton Chapter 7 Hamiltonian dynamics Read cursorily through sects. 7.1 and 7.2. In this course we will focus on far-from-equilibrium dissipative systems (rather than energy-conserving systems typical of quantum-mechanical applications), so this is not material of importance for what follows. billiards Chapter 8 Billiards Read all of it. The 3-disk pinball illustrates some of the key concepts for what follows; invariance under discrete symmetries, symbolic dynamics. Optional: download some simulations from ChaosBook.org/extras, or write your own simulator. billiardsOverh lecture overheads September 4 Evangelos Siminos 6. Group theory - a brief introduction Combinatorics cannot be tought. But we will make a brave attempt. discrete Chapter 9 World in a mirror Study section 9.1. (make sure the date in the page footers is Sep 3 2008 or later, otherwise download from here the current version of the chapter) discreteOverh lecture E. Siminos notes homework #3: exercises 9.2, 9.3, 9.4, 9.5 (a)-(e) - due Tue Sep 9 [solutions to chap. 9 exercises] September 9 Evangelos Siminos 7. Discrete symmetries of dynamics Dynamical systems often come equipped with discrete symmetries, such as the reflection symmetries of various potentials, and they simplify the dynamics in a rather beautiful way: If dynamics is invariant under a set of discrete symmetries, the state space is tiled by a set of symmetry-related tiles, and the dynamics can be reduced to dynamics within one such tile, the fundamental domain. Read sections 9.2 and 9.3. September 11 Evangelos Siminos 8. Continuous symmetries of dynamics If the symmetry is continuous, the interesting dynamics unfolds on a lower-dimensional ``quotiented'' system, with ``ignorable" coordinates eliminated (but not forgotten). The families of symmetry-related full state space cycles are replaced by fewer and often much shorter ``relative" cycles, and the notion of a prime periodic orbit has to be reexamined: it is replaced by the notion of a ``relative'' periodic orbit, the shortest segment that tiles the cycle under the action of the group. Furthermore, the group operations that relate distinct tiles do double duty as letters of an alphabet which assigns symbolic itineraries to trajectories. Read sections 9.4 and 9.5 if Vaggelis covers the continuous symmetries, skip if the lecture stops at the fundamental domain discussion. discreteOverh lecture overheads homework #4: exercises (9.7), (9.8) and (9.12), optional (9.13) - due Tue Sep 16 [solutions to chap. 9 exercises] September 16 9. Symbolic dynamics knead Chapter 10 Qualitative dynamics, for pedestrians (version 12.3, Sep 30 2008) Qualitative properties of a flow partition the state space in a topologically invariant way. The topological dynamics is incoded by means of transition matrices/Markov graphs. September 18 10. Kneading theory smale Chapter 11 Qualitative dynamics, for cyclists Skip section 11.5. homework #5: exercises 10.4, 10.6, 3.2; optional 10.5, 11.7 - due Tue Sep 23 [solutions to chap. 10 exercises] [solutions to chap. 11 exercises] September 23 11. Finding cycles cycles Chapter 12 Fixed points, and how to get them (version 12.3.1, Oct 6 2008; includes a sketch of chapter `12a' on transition graphs) September 25 12. Finding cycles September 30 13. Finding cycles homework #6: exercises 10.8, 12.13, 12.7; optional 12.12 - due Tue Oct 7 [solutions to chap. 12 exercises] [solutions from the class] October 2 14. Counting counting Chapter 13 Counting Read all of it. (version 12.3.1, Oct 6 2008) October 7 15. Transporting densities measure Chapter 14 Transporting densities Skip sect. 14.6. (version 12.3.1, Oct 6 2008) homework #7: exercises 13.1, 14.1, 14.3, 14.10, 15.4; optional 13.6, 13.14, 14.5, 14.7 - due Thu Oct 16 [solutions to chap. 13 exercises] [solutions to chap. 14 exercises] October 9 16. Averaging average Chapter 15 Averaging Read all of it. (version 12.3.3, Nov 10 2008) October 11-14 fall break October 16 17. Lyapunov exponents First we trash them as stupid, then we nevertheless define them. October 21 18. Trace formulas trace Chapter 16 Trace formulas Read all of it. (version 12.3.3, Nov 10 2008) projects #1: If you are going to write up the project in LaTeX (and not in blog/svn format), download the template from ChaosBook.org/projects/ October 23 19. Spectral determinants det Chapter 17 Spectral determinants Skip sects. 17.5 and 17.6. (version 12.3.3, Nov 10 2008) homework #8: exercises 15.1, 15.4; optional 17.10 (use version 12.3.3, Nov 10 2008) - due Tue Oct 28 [solutions to chap. 15 exercises] [solutions to chap. 17 exercises] projects #2: Email me a brief description of your project: title, your name, names of advisors (professors, other students) who might help with their advice, an abstract (of any length, as *.txt or *.pdf file), perhaps also a paper that you will base your project on. This will be ethernalized on the ChaosBook.org/projects homepage, where you can see descriptions of earlier projects - due Tue Oct 28 October 28 20. Cycle expansions recycle Chapter 18 Cycle expansions Skip sect. 18.6. (version 12.3.3, Nov 10 2008) J. Newman: Mathematica periodic orbits routines A. Basu: Matlab periodic orbits routines October 29 - November 11 spring registration October 30 21. Cycle expansions homework #8 again: make sure that your programs for finding periodic orbits (Henon and/or Rossler) work - due Thu Nov 6 November 4 22. Cycle expansions - heuristscs getused Chapter 20 Why cycle? Skip sects. 20.4 and 20.5. (version 12.3.3, Nov 10 2008) November 6 23. Why does it work? converg Chapter 21 Why does it work? Some of the mathematical ideas that underpin trace formulas. Read only sect. 21.1, skim the rest. November 11 24. Why doesn't it work? inter Chapter 23 Intermittency Everything that we have done so far hinges on exponential separation of nearby trajectories. What happens if we get stuck close to the border of interable, regular motion? Read sects. 23.1 to 23.2.3, skim the rest. homework #9: exercises 18.14, 20.2, 23.3; optional 21.3 - due Tue Nov 18 [solutions to chap. 18 exercises] [solutions to chap. 20 exercises] [solutions to chap. 21 exercises] November 13 25. Deterministic diffusion diffusion Chapter 24 Deterministic diffusion Fundation of statistical mechanics illuminated. Read sects. 24.1 to 24.2, skim the rest. projects update: discussion session November 18 26. Deterministic diffusion November 20 27. Deterministic diffusion November 25 28. Projects discussion session November 27 thanksgiving December 2 29. Much noise about nothing noise Chapter 26 Noise We derive the continuity equation for purely deterministic, noiseless flow, and then incorporate noise in stages: diffusion equation, Langevin equation, Fokker-Planck equation, Hamilton-Jacobi formulation, stochastic path integrals. homework #10: exercises 26.1, 26.2 and 26.3 - not due in this course [work them out anyway, Gaussians will serve you well later on] December 4 30. Turbulence projects update: discussion session [notes] December 5 GT classes end December 9 10:50 term project due, Predrag's office to December 13 Course opinion survey CETL web link December 15 GT grades due at noon December 22 have good holidays! The rest has yet to be worked out. solutions to the final exam to be posted: December 30 - January 9 spring registration