]> January 10 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 introOverheads lecture overheads #1 exercises (1.1), (2.1), (2.7), and (2.8), optional (2.10) - due Tue Jan 17 [solutions to chap. 1 exercises] [solutions to chap. 2 exercises] Future's So Bright, I Gotta Wear Shades [click right, open in new tab] January 12 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 January 17 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 You can now print this chapter on the paper - make sure it has printed "version13.7.1" or later on the page footer mapsOverh lecture overheads #2 exercises (3.1), (3.5), (4.1), and (4.3), optional (3.6) and (4.4) - due Tue Jan 24 [solutions to chap. 4 exercises] January 19 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 skip sect. 4.5.1 stabilityOverh lecture overheads January 24 5. Cycle stability 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 Go straight Advanced material, most of it safely skipped. Try to understand sect. 6.6, though. Skipped in the lectures. #3 exercises 5.1, 6.2, 7.2, optional 5.2, 7.4 - due Tue Jan 31 [solutions to chap. 5 exercises] a hint: check out programs ChaosBook.org/extras/ January 26 6. Newtonian mechanics 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 at least cursorily the whole chapter. udacity web link udacity.com What do you think? Thrun had enrollment of 160,000 students. ChaosBook.org is an attempt to reach any student, anywhere, and it reaches about seven. Could one do better? January 31 7. Pinball wizzard 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 #4 exercises (8.1), (8.3) and (8.5), optional (8.6) - due Tue Feb 7 [solutions to chap. 9 exercises] February 2 8. Discrete symmetries of dynamics 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. discrete Chapter 9 World in a mirror You can print this chapter on the paper now - revision 13.7.3, Feb 11 2012 completed Read all of it. Ask tons of questions in the class. discreteOverh lecture E. Siminos notes discreteOverh lecture overheads optional meet P. Glass (and/or break a leg) [solution] February 7 9. Symmetries of solutions discrete Chapter 9 World in a mirror You can print this chapter on the paper now - revision 13.7.3, Feb 11 2012 completed #5 exercise (3.7) - read Sect 3.4 Charting the state space (version 13.7.1 or later); exercises (9.1), (9.2), (9.3), (9.4); optional exercise: rewrite Example 8.1 3-disk game of pinball following the billiards lecture overheads, eternalize your name as contributor to ChaosBook.org. - due Tue Feb 14 [solutions to chap. 9 exercises] February 9 10. Fundamental domain February 14 11. Continuous symmetries of dynamics NOTE: lecture moved to 5th floor Howey conference room 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. continuous Chapter 10 Relativity for cyclists Read all of it. Ask tons of questions in the class. Best not to print this chapter yet - major revisions under way. Make sure you are reading version 13.7.4 or later. continuousOverh lecture overheads #6 exercise (3.7) do it until you get it right; (10.1) Visualizations of the 5-dimensional complex Lorenz flow; (10.3) SO(2) rotations in a plane; (10.8) Rotational equivariance, infinitesimal angles; [ optional (10.2) An SO(2)-equivariant flow with two Fourier modes; (10.12) in a Hilbert basis ] - due Tue Feb 21 February 16 12. How good is your Poincare section? Keith and the gang deconstruct exercise (3.7) "Poincare section border". The gang is right - as Rossler equatins are quadratic, the borders are conic sections (line, circle, ellipse, parabola, hyperbola). Dr. C. is right - sections not going through equilibria are no good, as they do not intersect all trajectories winding around their real (un)stable eigen-vectors. February 21 13. Slice and dice Actions of a Lie group on a state trace out a manifold of equivalent states, or its group orbit. Symmetry reduction is the identification of a unique point on a group orbit as the representative of this equivalence class. Thus, if the symmetry is continuous, the interesting dynamics unfolds on a lower-dimensional `quotiented', or `reduced' state space M/G, with `ignorable' coordinates eliminated (but not forgotten). In the method of slices the symmetry reduction is achieved by cutting the group orbits with a set of hyperplanes, one for each continuous group parameter, with each group orbit of symmetry-equivalent points represented by a single point, its intersection with the slice. continuous Chapter 10 Relativity for cyclists Read Sect. 10.4 Reduced state space. #7 exercises (10.14) Compute the relative equilibrium TW1 of the 5-dimensional complex Lorenz flow; (10.16) Plot the relative equilibrium TW1 in Cartesian coordinates; (10.**) Plot the symmetry reduced, 4-dimensional complex Lorenz flow in a slice of your choice, in several 3-dimensional projections. #7 optional exercise (10.23) State space reduction by a slice, ODE formulation - sorry, this is a bit of a mess - might try to improve it the formulation by the weekend. - due Tue Feb 28 Ring of Fire Visualize O(2) equivariance of Kuramoto-Sivashinsky (AKA "Ring of Fire") February 23 14. Slice and dice Why gauge-fixing in field theory does not seem smart; and why experimentalists should slice their raw data. continuous Chapter 10 Relativity for cyclists Read Sect. 10.4 Reduced state space. Best not to print this chapter yet - major revisions under way. Make sure you are reading version 13.7.4 or later. exp a letter to experimentalists No need to reconstruct fluid velocities - just need a notion of distance for your data sets continuousOverh lecture overheads February 28 15. Qualitative dynamics, for pedestrians Qualitative properties of a flow partition the state space in a topologically invariant way. knead Chapter 11 Charting the state space Sects 11.1 and 11.2 #8 exercise (10.**), we do it until we get it right: Plot the symmetry reduced, 4-dimensional complex Lorenz flow in an atlas of your making, consisting of two slices, such that the strange attractor avoids the slice borders and the associated jumps. Exercises 11.6 and 11.8. #8 optional exercise (3.7), we do it until we get it right: * put a section through each equilibrium, try to optimize their orientation so that the ridge (their intersection) is the shortest distance from both equilibria. ** For Roessler flow the section border is a conic section. Derive analytic formula for this elipse - line - parabola - hyperbola, replot the section borders using the formula rather than the orientation of v(x). - due Tue Mar 6 March 1 16. Symbolic dynamics ^M The two faces of qualitative dynamics: (1) temporal ordering, or itinerary with which a trajectory visits state space regions and (2) the spatial ordering between trajectory points, the key to determining the admissibility of an orbit with a prescribed itinerary. Kneading theory. knead Chapter 11 Charting the state space Sects 11.3 - 11.6 March 6 17. Qualitative dynamics, for cyclists ^M First we trash them as stupid, then we nevertheless define them. smale Chapter 12 Stretch, fold, prune Prune danish (if we get that far) #9 exercise 12.3 - due Tue Mar 13 #9 optional exercise 12.7 - due Tue Mar 13 March 8 18. Finding cycles cycles Chapter 13 Fixed points, and how to get them Read all of it. tutorial project TechBurst 2011 Can we do better with a "Slice and Dice" tutorial than TechBurst 2011? tutorialSD project plane Couette movies Can we do better with a "Slice and Dice" tutorial than plane Couette movies? March 13 19. Finding cycles March 15 20. Walkabout: Transition graphs The topological dynamics is encoded by means of transition matrices/Markov graphs. Markov Chapter 14 Walkabout: Transition graphs Read all of it. March 19-23 spring break March 27 21. Counting counting Chapter 15 Counting Read sects. 15.1 - 15.4. #10 exercise (15.1) Transition matrix for 3-disk pinball #10 optional exercise (15.4) loop expansions; (15.14) 3-disk pinball topological zeta function. - due Thu Apr 5 March 29 22. Counting April 3 23. Transporting densities measure Chapter 16 Transporting densities Skip sects. 16.3 and 16.6. #11 exercise (16.1) Dirac delta function; #11 optional exercise (16.5) Invariant measure - due Tue Apr 10 April 5 24. Averaging average Chapter 17 Averaging Read sects. 17.1 and 17.2. [solutions to chaps. 1 to 13 exercises are in DropBox] April 10 25. Trace formulas trace Chapter 18 Trace formulas Read all of it. #12 exercise (17.1) How unstable is the Henon attractor? parts (d) and (e) optional #12 optional exercise (16.10) generator of translations. - due Tue Apr 17 Apr 12 26. Spectral determinants ^M det Chapter 19 Spectral determinants Skip sects. 19.3.1, 19.3.2, 19.5 and 19.6. recycle Chapter 20 Cycle expansions Read 20.1, 20.2.1, 20.3, 20.3.1, and 20.6 [ J. Newman: Mathematica periodic orbits routines ] [ A. Basu: Matlab periodic orbits routines ] April 17 32. 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. April 19 28. How well can one resolve the state space of a chaotic flow? LipCvi08 Lippolis P R Lett Noise smooths out all the kinky determinstic stuff LipCvi07 Lippolis the devil is in the details Ask Gable and Daniel to guide you through all this fancy stochastic stuff April 24 29. Much noise about nothing intractVid video Physicist's life is intractable Pretty clear discussin of the interplay of noise and determinism - recommended intract overheads Physicist's life is intractable Overheads for the above lecture #13: exercises 26.1, 26.2 and 26.3 - not due in this course [you might want to work them out anyway, Gaussians will serve you well later on] April 26 30. The rest is noise Wherein the Master Slicer Prize will be presented relax Chapter 29 Relaxation for cyclists Optional: might be useful if you need to find some cycles crete02 Y. Lan Turbulent fields and their recurrences A variational principle for robust invariant solutions searches [prize ceremony] [projects update] April 27 GT classes end May 1 11:30am - 2:20pm term project due, Predrag's office to May 5 Course opinion survey CETL web link May 7 GT grades due at noon May 7 have a good summer!