Midterm The midterm takes place on Wednesday, 11/9 in class.NO BOOKS, NO NOTES, NO CALCULATORS. See below for more detailedinformation about the material and preparation. Office hours for exam week: M2-3, Tu:3-4, no office hour on Wednesday No homework due on Wednesday. Practice problems for midterm (need not be turned in): 5.2.4, 5.2.7. material for midterm

## Nonlinear Dynamics Homework Solutions Week 3

Introduction toNonlinear Dynamics and Chaos Tu,Th 9:35-10:55 Howey S204 ChaosBook.org/predrag/courses/PHYS-4267-16/ August 23 1. IntroductionReading: Chapter 1; Chapter 2, sections 2.1-2.3 Optional reading: ChaosBook.org Brief history of chaos might amuse you. Online: Strogatz lecture 1 August 252. Flows on the lineReading: Chapter 2Online: Strogatz lecture 2 up to minute 34 Fun stuff:snowflakesProblem set #1: 2.1.5, 2.2.10, 2.3.3, 2.4.2, 2.4.9(click here)August 303. Experimental nonlinear dynamicsReading: Chapter 2Presentation: CRAB lab - Will Savoie, Andy Karsai and Dan GoldmanSeptember 14. Numerical solution of nonlinear ODEsDue in class: Homework #1(for homework solutions, go to t-square, links above)Reading: Chapter 2: Reading: Sections16.0-16.3 of Numerical Recipes by Press et al.Problem set #2: 2.5.3, 2.7.6, 2.8.3(for solutions, go to t-square)Note:There is a typo in 2.8.3 part (c): you are to plot ln(E) vs. ln(Delta t).Explain how the slope is related to the order of the method.September 65. Bifurcations in one-dimensional systems IReading: Chapter 3 Online: Strogatz lecture 2, starting at minute 34September 86. Bifurcations in one-dimensional systems IIDue in class: Homework #2(remember, always due next Thursday)Reading: Chapter 3 Online: Strogatz lecture 3Online: Strogatz lecture 4Problem set 3: 3.1.3, 3.2.6, 3.3.1, 3.4.8, 3.5.7September 13, guest lecturer Prof. F. Fentonstein7. Flows on a circleReading: Chapter 4 Fun stuff:firefliesand mosquitos synchronizationSeptember 15, guest lecturer Prof. F. Fentonstein8. Two-dimensional systemsReading: Chapter 5 Online: Strogatz lecture 5Problem set 4: 3.6.2, 3.7.6, 4.1.8, 4.3.7, 4.6.3 September 209. Phase plane analysisReading: Chapter 6 Online: Strogatz lecture 6September 2210. Conservative systemsReading: Chapter 6 Problem set 5: 5.1.10, 5.2.13, 6.1.3, 6.2.1, 6.3.1, 6.4.7Online: Strogatz lecture 7September 2711. Index theory. Limit CyclesReading: Chapter 7 Online: Strogatz lecture 8September 2912. No periodic orbits? Perturbation theoryReading: Chapter 7 Problem set 6: 6.5.10, 6.6.3, 6.7.2, 6.8.5, 6.8.7 Online: Strogatz lecture 9Online: Strogatz lecture 10October 413. Nonlinear oscillators and averagingOptional reading: Bender and Orszag Chapter 7 Online: Strogatz lecture 11October 614. Bifurcations in two dimensionsReading: Chapter 8 Prof. Flavio Fenton: temporal Belousov-Zhabotinsky demonstrationProblem set 7: 7.1.6, 7.2.9, 7.3.9, 7.4.1. Due October 20.Chris says: use a computer to do plots - in the long run, it is faster, and it's what you need to learn anyhow.Online: Strogatz lecture 12October 10-11Fall breakOctober 13Mid-term exam9:35-10:55 Howey S204; see instructionsOctober 1815. Hopf bifurcationReading: Chapter 8Storgatz bores you? Optional ChaosBook.orgreading (none of this will be on the final): Section 4.3 A linear diversion, Example 4.3 Linear stability of 2-dimensional flows,Example 4.4 In-out spirals,Section 5.1 Equilibria,Appendix C.2 Eigenvalues and eigenvectors. Online: Strogatz lecture 13 October 20 16. Global bifurcations of cycles Reading: Chapter 8 Due in class: Homework #7 Prof. Flavio Fenton: Belousov-Zhabotinsky spirals demonstration Problem set 8: 7.5.4, 7.6.3, 7.6.6, 7.6.17 Online: Strogatz lecture 14 October 25 17. Quasiperiodicity and Poincare maps Reading: Chapter 8 October 27 18. Floquet Theory Reading: Chapter 8 Optional reading: Nonlinear Dynamics 1: Geometry of Chaos, week 2 Problem set 9: 8.1.8, 8.1.11, 8.2.1, 8.2.9, 8.3.1 November 1 19. Lorenz equations Reading: Chapter 9, skip Sect 9.1 A Chaotic Waterwheel (Malkus and Howard were my friends, but this is not the why Lorenz work was so great!) Reading: Chapter 9, R. Grigoriev notes Optional reading: Nonlinear Dynamics 1: Geometry of Chaos, week 1 Online: Strogatz lecture 17 Numerical exploration of different dynamical regimes (: Maple! :) November 3 20. Lorenz equations Reading: Chapter 9 Problem set 10: 8.4.2, 8.4.12, 8.5.2, 8.6.1, 8.7.5 Bonus problem set 10+: Geometry of Chaos, homework 1 (email me the grade sent to you by the autograder) Online: Strogatz lecture 18 November 8 21. One-dimensional maps Reading: Chapter 10 Optional reading: Nonlinear Dynamics 1: Geometry of Chaos, week 7 Optional: Matlab simulations of the Rossler system: reduction to 2D and 1D maps and stretching of phase space volumes Online: Strogatz lecture 19 November 10 22. Universality Reading: Chapter 10, R. Grigoriev lecture notes Optional reading, ChaosBook.org bootleg chapters: Universality in transitions to chaos and Complex universality Problem set 11: assignment Online: Strogatz lecture 20 November 15 23. Charting the state space Reading: ChaosBook.org, Chapter 14 and Chapter 15, edited for PG-13 Optional viewing: Nonlinear Dynamics 1: Geometry of Chaos, week 7 November 17 24. Kneading theory Reading: Chapter 11, Sect. 11.2 (rest mostly useless 1980's flounderings) Fractals in nature, biology, and mathematics. Problem set 12: 10.1.11, 10.2.6 (see comments for 10.2.3), 10.3.11, 10.4.1 - due December 1 November 22 25. Learning how to count Reading: ChaosBook.org, Chapter 17 and Chapter 18, edited for PG-13 Optional viewing: Nonlinear Dynamics 1: Geometry of Chaos, week 9 Optional reading: Aligood, Sauer, and Yorke Problem set 13: assignment - due December 1 November 24 Holiday November 29 26. Learning how to measure Reading: ChaosBook.org, Chapter 19 and Chapter 20, edited for PG-13 Optional viewing: Nonlinear Dynamics 1: Geometry of Chaos, week 10 December 1 27. Ah, chaos! December 6 28. Wrapping it all up; overture Reading: ChaosBook.org, Chapters 21 to 23, edited for PG-13 Optional viewing: Nonlinear Dynamics 1: Geometry of Chaos, week 11 December 8 Final exam 2:50-5:40pm Howey S204; see instructions

Prerequisites: Undergraduate courses in engineering thermodynamics, fluid dynamics and heat transfer (MEC ENG 40, MEC ENG 106, and MEC ENG 109). Each student must have access to a PC, Macintosh or workstation machine with scientific programming capabilities for use in homework and projects

Student Learning Outcomes: Acquired necessary knowledge and scientific maturity to apply methods of nonlinear and uncertainty analysis in engineering design and optimization.An ability to apply knowledge of mathematics, science, and engineering. An ability to identify, formulate, and solve engineering problems. The broad education necessary to understand the impact of engineering solutions in a global and societal context. A knowledge of contemporary issues. An ability to use the techniques, skills, and modern engineering tools necessary for engineering practice.This course provides valuable training in the modeling and analysis of mechanical engineering systems using nonlinear and uncertainty analysis. It also serves to reinforce and emphasize the connection between fundamental engineering science and practical problem solving.

Prerequisites: Undergraduate courses in engineering thermodynamics, fluid dynamics and heat transfer (Mechanical Engineering 40, Mechanical Engineering 106 and Mechanical Engineering 109 or equivalent). Each student must have access to a PC, Macintosh or workstation machine with scientific programming capabilities for use in homework and projects

Course Description: Numerical analysis is a fundamental topic in applied mathematics. Many practical problems that scientists try to solve are based on mathematical models, but few can be solved analytically, mainly due to their large size. Therefore computational algorithms are needed for approximating these solutions. It is critically important to maintain the important mathematical properties of the underlying system when developing numerical algorithms, and moreover to ensure accuracy, efficiency and convegence. Numerical analysis is about developing good computational techniques for broad based problems and demonstrating that these properties hold both theoretically and computationally. Numerical analysts make sure that computational algorithms are trustworthy so that domain scientists can be confident in the results of their experiments. Numerical analysts also answer the question, ``what assumptions of the underlying problem are necessary for this computational method to succeed?'' In this course we will focus on numerical linear algebra, interpolation and approximation, which are all essential when solving problems in data science, signal and image processing, and evolutionary dynamics. We will use MA