]> January 7 Predrag Cvitanović 1. Trajectories We start out by a recapitulation of the basic notions of dynamics. Our aim is narrow; keep the exposition focused on prerequisites 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 intro Chapter 1 Overture (optional) Read quickly all of it - do not worry if there are stretches that you do not understand yet. appendHist Appendix A Brief history of chaos (optional) A brief history of motion in time. introOverheads overture overheads (optional) Future's So Bright, I Gotta Wear Shades [click right, open in new tab] January 9 2. 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 probably want to print only a chapter at a time on paper - the book is being edited concurrently with the course. You can see when a chapter was edited on the page footer mapsOverh lecture overheads HW1 exercises (2.1) (2 pts), (2.7) (4 pts), and (2.8) (4 pts), bonus (3.5) (2 pts) - due in class Tue Jan 14 January 14 3. 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 stabilityOverh lecture overheads January 16 4. Cycle stability If a flow is smooth, in a sufficiently small neighborhood it is essentially linear. Hence in this lecture, which might seem an embarrassment (what is a lecture on linear flows doing in a book on nonlinear dynamics?), offers a firm stepping stone on the way to understanding nonlinear flows. Linear charts are the key tool of differential geometry, general relativity, etc, so we are in good company. If you know your eigenvalues and eigenvectors, you may prefer to fast forward here. invariants Chapter 5 Cycle stability Skip sect. 5.2.1. Lyapunov Chapter 6 Lyapunov exponents (optional) Awkward stuff, much cited in the literature, most of it safely ignored. Skipped in the lectures. please take this quiz - we are testing the SOOC :) HW2 exercises (3.1) - 6 points, (4.1) - 4 points, and (4.3) - 6 points. Bonus exercises: (3.2) - 6 points, (3.7) - 6 points and (4.6) - 4 points, - due Tue Jan 21 January 21 5. Stability exponents are invariants of dynamics We prove that (1) Floquet multipliers are the same everywhere along a cycle, and (b) that they are invariant under any smooth coordinate transformation. January 23 6. 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. 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 Billiards overheads billiardsOverh (optional) a bit of history While simulating weather patterns 50 years ago, Edward Lorenz overthrew the idea of the clockwork universe with his ground-breaking research on chaos. HW3 exercises (5.1) - 3 points, (B.2) - 3 points, and (8.1) - 10 points. Bonus exercise: (8.6) - 6 points. - due Tue Jan 28 a hint: check out programs ChaosBook.org/extras/ January 28 7. 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 Read all of it. Ask tons of questions in the class. discreteOverh lecture E. Siminos notes (optional) discreteOverh lecture overheads (optional) January 30 8. Discrete symmetry reduction of dynamics to a fundamental domain discrete Chapter 9 World in a mirror Read sects. 9.2 - 9.4 Dynamics for Fundamentalists. Skip sect. 9.5 Invariant polynomials. HW4 ChaosBook ver. 14.5: Exercise 1.1 3-disk symbolic dynamics - 2 points; Exercise 8.3 Stability of billiard cycles - 4 bonus points; Exercise 9.5 Symmetries of an equilateral triangle - 4 points; Exercise 9.6 Reduction of 3-disk symbolic dynamics to binary - 3 points; Exercise 9.7 C2-equivariance of Lorenz system - 3 bonus points; - due Tue February 4 February 4 9. 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. continuous Chapter 10 Relativity for cyclists Read Sects. 10.1 to 10.3. Ask tons of questions in the class. continuousOverh lecture overheads February 6 10. Got a continuous symmetry? Freedom and its challenges HW5 = click here posted here as a pdf file. To get correct cross references, you have to open today's (imperfect) version 14.5.1 of the ChaosBook. - due Tue Feb 11 HW5 = Howey 5 click here exercise (10A.X), mandatory for Howey 5th and 3rd floor folks whose systems have continuous symmetries, optional for everyone else: Is slicing the best thing since invention of sliced bread? It is not clear - it reveals the dynamics hidden behind drifts along symmetry directions, but it gets dicey as one approaches a chart border. This is a problem set: propose a coordinate change (perhaps rescaling of time) that regularizes close passages to the chart border without a need to refine the time steps of your integrator. A suggestion - close to the chart border al trajectories are straight lines, so some kind of simple rescaling should work. February 11 11. 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. continuousOverh lecture overheads February 13 12. 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 HW6 = click here posted here as a pdf file. To get correct cross references, you have to open today's (imperfect) version 14.5.2 of the ChaosBook. - due Tue Feb 25 Nazmi Burak Budanur's solution set: a Mathematica notebook. February 18 13. The spatial ordering of trajectories from the time ordered itineraries 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 Eye candy: Chaos in Heavens (and a lesson: you will never understand anything by staring at projections of chaotic trajectories onto configuration coordinates - you need to take care of symmetries, look for equilibria, construct Poincare sections, ...) February 20 14. Qualitative dynamics, for cyclists Dynamical partitioning of a plane smale Chapter 12 Stretch, fold, prune Sects 12.1 - 12.3 February 25 15. Finding cycles cycles Chapter 13 Fixed points, and how to get them Read all of it. J. Newman: Mathematica periodic orbits routines A. Basu: Matlab periodic orbits routines Ring of Fire Visualize O(2) equivariance of Kuramoto-Sivashinsky (AKA "Ring of Fire") February 27 16. Finding cycles; long cycles, continuous time cycles HW7 = click here posted here as a pdf file. To get correct cross references, you have to open today's (imperfect) version 14.5.3 of the ChaosBook. - due Tue Mar 4 part II chaos rules Predrag March 4 17. Markov graphs The topological dynamics is encoded by means of transition matrices/Markov graphs. Markov Chapter 14 Walkabout: Transition graphs Read all of it. Predrag March 6 18. Learning hoow to count count Chapter 15 Counting Read sects. 15.1 - 15.4; 15.6 - 15.7, ChaosBook vers. 14.5.3. Please derive yourself the trace formula the determinant and the topological zeta function. If you do not understand how to derive these, you'll be lost for the rest of the semester, and what fun is that? The lecturer seems to flounder while attempting to derive these, totally essential formulas, let him know if you have simpler derivations. HW8 = click here posted here as a pdf file - due Tue March 11 March 11 19. Transporting densities measure Chapter 16 Transporting densities Skip sects. 16.3 and 16.6. March 13 20. Averaging, trace formulas average Chapter 17 Averaging Read sects. 17.1 and 17.2. Skip sect. 17.1.3 "Moments, cumulants", trace Chapter 18 Trace formulas Read all of it. HW9 = click here posted here as a pdf file - due Tue Mar 25 March 17-21 spring break March 25 21. Spectral determinants det Chapter 19 Spectral determinants Skip sects. 17.5 and 17.6. Sleep deprivation may cause brain damage The leading cause of migranes lecturer TBA March 27 22. Cycle expansions recycle Chapter 20 Cycle expansions Skip sects. 20.3.1 "Newton algorithm for determining the evolution operator eigenvalues", and 20.6 "Stability ordering of cycle expansions". (version 14.5.4, Mar 25 2014) HW10 posted here as a pdf file. To get correct cross references, you have to open today's (imperfect) version 14.5.4 of the ChaosBook. - due Tue Apr 1 April 1 23. Deterministic diffusion average Chapter 17 Averaging Read sects. 17.1, including sect. 17.1.3 "Moments, cumulants", diffusion Chapter 24 Deterministic diffusion Read sect. 25.1: Foundations of statistical mechanics illuminated by 2-dimensional Lorentz gas. Overheads for the day's lecture April 3 24. Deterministic diffusion diffusion Chapter 24 Deterministic diffusion Read sect. 25.2: A class of simple 1-dimensional dynamical systems where all transport coefficients can be evaluated analytically. project, step #1: For a project, pick a dynamical systems topic (or a paper to read) related to your research - the idea is that the project in this course can be included into your thesis, perhaps as an appendix. If there are symmetries in the formulation of the problem, do explain them and classify all types of relevant solutions by their symmetry. Follow the literature on the notation and terminology, include detailed bibliography, with proper credits given; this term paper is not meant to be original research (though that would be sweet :). If you are going to write up the project in LaTeX (and not as a part of your CNS subversion blog), download the template from ChaosBook.org/projects/ project, step #2: Append to HW11 pdf pages (built from the template) a brief skeleton of your project: title, your name, names of advisors (professors, other students) who might help you with their advice, an abstract (of any length), perhaps also a paper that you will base your project on. Next homework append the draft of updated project, and so on - it will be refereed by another participant. And so on... If good, the project will be ethernalized on the ChaosBook.org/projects homepage, where you can see descriptions of earlier projects. project refereeing: When you receive a homework to grade, comment on the appended draft project as you would referee a paper submission - annotate, ask questions, suggest improvements, etc. (no numerical grade). HW11 - due Tue Apr 8. Exercise 26.8 "Diffusion reduced to the fundamental domain" is essential for most CNS PhD projects (a baby version of how to reduce Euclidean symmetry), so please make an attempt at solving this exercise. April 8 25. Discrete symmetry factorization of spectral determinants A triple home run: simpler symbolic dynamics, fewer cycles needed, much better convergence of cycle expansions. Once you master this, going back is unthinkable. symm Chapter 21 Discrete factorization Symmetries simplify and improve the cycle expansions in a rather beautiful, not entirely obvious way, by factorizing cycle expansions. Read sect. 21.1 April 10 26. Discrete symmetry factorization of spectral determinants 3 disk pinball symmetries suffice to illustrate all that is needed to factorized spectral determinants for any system with a discrete symmetry: character. symm2 Chapter 21 Discrete factorization Read sects. 21.2 - 21.6 HW12 - due Tue Apr 15. April 15 27. Projects presentations session mugshots gallery Who's who? projects project TechBurst 2011 Can we do better than TechBurst 2011? projects: discussion - for videos, see ChaosBook.org project homepage 12:05-12:20 Marc Fleury: "Quantum" walkers. 12:20-12:35 Mikel Jon De Viana: Terrible allergies due to pollen. 12:35-12:50 Kimberly Y. Short: The role of time scales in non-linear systems. 12:55-1:10 Benjamin McInroe: Periodic orbit theory of linear response. 1:10-1:25 Michael S. Dimitriyev: A continuum elastic model of thermal fluctuation allosteric regulation. April 17 28. Continuous symmetry factorization of spectral determinants Continuous symmetries simplify and improve the cycle expansions in a rather beautiful, not entirely obvious way, by factorizing cycle expansions. The lecture, however, was not good, and will not be posted as a video. See Apr 22 instead. rpos (To be published) Continuous symmetry reduced trace formulas Why is this paper not published yet? I would like at least one person out there in the universe (a graduate student, per chance?) to understand it before submitting it. I strongly recommend going through this paper and checking it as a term project. HW13 focus on this week's draft of your term project, submit it as the homework - due Tue Apr 22. 1:10-1:25 Jeffrey M. Heninger: Noise is your friend. April 22 29. Symmetry factorization of spectral determinants - attempt 2 Dresselhaus MIT course 6.734 Group Theory: Application to the Physics of Condensed Matter Chapters 1 to 4 of Dresselhaus lecture notes (or the textbook) are perhaps student-friendlier than Tinkham textbook. I recommend learning about the "Wonderful Orthogonality Theorem" on your own - many of you will find that useful in your research. 1:10-1:25 Alexandre Damião: Thermoacoustic Instatilities using Rijke Tube Model. April 24 30. Turbulence The last lecture of the semester: whence from here? tutorialSD project plane Couette movies Can we do better with a "Slice and Dice" tutorial than plane Couette movies? [the project, and what you should do about it] 12:55-1:25 Pavel M. Svetlichnyy and Tingnan Zhang: Cycle averaging formulas applied to a periodic Lorentz gas (for videos, see ChaosBook.org project homepage). April 25 GT classes end May 1 11:30am - 2:20pm term project due Please upload the project, as you would upload a homework. If you have a paper copy, you can stop by Predrag's office, or put it into Predrag's mailbox. to May 5 Course opinion survey We would very much appreciate your input on how to improve the video aspects of the course, and what one could do to make it a viable MOOC course - or whether we should attempt to go online at all. If the format of this survay is not helpful for that, maybe you can discuss that on piazza. CETL web link May 5 GT grades due at noon May 5 have good holidays! some good stuff we did not have time to cover: The rest has yet to be worked out. ?? ?? ??. 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 HW1?: exercises 26.1, 26.2 and 26.3 - not due in this course [work them out anyway, Gaussians will serve you well later on] ?? ?? ??. How good is your Poincare section? Deconstruct exercise (3.7) "Poincare section border". The gang is right - as Roessler 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. ?? ??. Cycle expansions - heuristscs getused Chapter 20 Why cycle? Skip sects. 20.4 and 20.5. (version 12.3.3, Nov 10 2012) ?? ??. 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. ?? ??. 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 integrable, regular motion? Read sects. 23.1 to 23.2.3, skim the rest. homework HW9: exercises 18.14, 20.2, 23.3; optional 21.3 - due Tue ??? ?? ??? ?? ??. Turbulence tutorialSD project plane Couette movies Can we do better with a "Slice and Dice" tutorial than plane Couette movies? projects update: discussion session