Introduction to
Nonlinear Dynamics and Chaos
Georgia Tech PHYS 4267/6268 - Fall 2023
Tu,Th 9:30 - 10:45am
Room N210, Howey Physics Building
MS Teams:
some lectures might be live-streamed; all are prerecorded
August 22
1. Introduction
Reading: Chapter 1; Chapter 2, sections 2.1-2.3
Cvitanović lecture 0: Personal introduction, optional
Cvitanović lecture 1
Strogatz lecture 1
Optional reading:
ChaosBook.org Brief history of chaos might amuse you.
August 24
2. Flows on the line
Reading: Chapter 2
Cvitanović lecture 2
;
Rant on uniqueness (best skipped)
Strogatz lecture 2 up to minute 34
Homework HW1: 2.1.5, 2.2.10, 2.3.3, 2.4.2, 2.4.9
(click here), due Aug 31
August 29
3. Numerical solution of nonlinear ODEs
Reading: Chapter 2
Optional reading:
Sections
17.0-17.3 Integration of Ordinary Differential Equations
of Numerical Recipes by Press et al.
Cvitanović lecture 3
Rant on symmetry? Don't break it
;
Rant on PDEs for barbarians (best skipped)
Optional viewing:
Computing is like hygiene, personal
John F Gibson solves the Navier-Stokes
(message: that's how you do it - by thinking)
Computing hygiene (obligatory Gibson rant, take #2)
Carl Střrmer
August 31
4. Bifurcations in one-dimensional systems I
Reading: Chapter 3, sections 3.0-3.3
Cvitanović lecture 4
Strogatz lecture 2, starting at minute 34
Due today: HW1
(pdf file upload to Canvas, link above)
HW2: 2.5.3, 2.6.1, 2.7.3, 2.7.4, 2.8.3
(due in a week)
Marked blue: required for grad students, bonus for undergrads
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 5
5. Bifurcations in one-dimensional systems II
Reading: Chapter 3, sections 3.4-3.5
Cvitanović lecture 5
Strogatz lecture 3
Strogatz lecture 4
September 7
6. Flows on a circle
Reading: Chapter 4, sections 4.0-4.5
Cvitanović lecture 6a Flows on a circle
Cvitanović lecture 6b Damped pendulum. Fireflies
Rant on traffic pileups (best skipped)
Fun stuff:
fireflies
and mosquitos synchronization
Due: HW2
(remember, always due next Thursday)
HW3: 3.1.2, 3.1.5, 3.2.4, 3.3.1, 3.4.3, 3.4.8, 3.5.7
September 12
7. Two-dimensional systems
Reading: Chapter 5, sections 5.0-5.2
Cvitanović lecture 7
Strogatz lecture 5
September 14
8. Phase plane analysis
Reading: Chapter 6, sections 6.0-6.4
Optional reading:
Jacobian matrix?
Cvitanović lecture 8
Strogatz lecture 6
HW4: 3.6.7, 3.7.6, 4.1.8, 4.2.2, 4.3.6, 4.4.1, 4.5.1
September 19
9. Conservative systems
Reading: Chapter 6, sections 6.5-6.8
Cvitanović lecture 9
September 21
10. Index theory. Limit Cycles
Reading: Chapter 6, section 6.9
Optional viewing:
Wolfram demo: Topological Singularities
Reading: Chapter 7, section 7.1-7.2
Cvitanović lecture 10
Rant on tangent bundles (best skipped)
;
Rant on professor's job (best skipped)
Strogatz lecture 7
Strogatz lecture 8
HW5: 5.1.1, 5.1.10, 5.2.12, 6.1.8, 6.2.1, 6.3.11, 6.4.7
September 26
11. Limit cycles. Relaxation oscillators
Reading: Chapter 7, section 7.3-7.5
Chris says: use a computer to do plots - in the long run,
it is faster, and it's what you need to learn anyhow.
Cvitanović lecture 11
Strogatz lecture 9
Strogatz lecture 10
September 28
12. Perturbation theory. Averaging
Reading: Chapter 7, section 7.6
Reading: Bender and Orszag chapter 7, sect. 7.1, example 1
Cvitanović lecture 12
Strogatz lecture 11
HW6: 6.5.6, 6.6.3, 6.6.7, 6.7.1
October 3
13. Bifurcations in two dimensions
Reading: Chapter 8, section 8.1
Strogatz lecture 12, more than section 8.1, but good
Cvitanović lecture 13, the last 20 min only (apologies)
October 5
14. Hopf bifurcations, chemical oscillations
Reading: Chapter 8, sections 8.2, 8.3
Optional viewing: A stroll through 61,506 dimensions
Flavio Fenton: spiral wave demonstration
Cvitanović lecture 14
Strogatz lecture 13
HW7: 7.1.6, 7.2.9, 7.3.1 due October 19.
October 9-10
Fall break
October 12
Midterm exam
9:30 - 10:45pm Howey N210; see instructions
October 17
15. Global bifurcations of cycles
Reading: Chapter 8, sections 8.4, 8.5
Cvitanović lecture 15
Strogatz lecture 14
October 19
16. Quasiperiodicity; Poincaré maps
Reading: Chapter 8, sections 8.6, 8.7
Cvitanović lecture 16
Strogatz lecture 14 start at 44:00
Due: HW7
HW8: 7.5.4, 7.6.3, 7.6.6, 7.6.17
October 24
17. Lorenz. Finite time stability
Reading: Chapter 9, Sect 9.0 (skip Sect 9.1 A Chaotic Waterwheel !)
Reading: ChaosBook Sect 4.1
Cvitanović lecture 17
October 26
18. Lorenz equations
Reading: Chapter 9, Sect 9.2
Reading: ChaosBook Sect 4.7
Cvitanović lecture 18
Strogatz lecture 17
HW9: 8.1.8, 8.1.11, 8.2.1, 8.2.9, 8.3.1
October 31
19. Strange attractors. Chaos
Reading: Chapter 9, Sects 9.2, 9.3, 9.4
Cvitanović lecture 19
Strogatz lecture 18
November 2
20. One-dimensional maps
Reading: Chapter 10, Sects 10.1 to 10.5
Reading: ChaosBook slides -
"discrete time dynamics"
Reading: ChaosBook slides -
"what is ‘chaos’?"
Optional: Matlab simulations of the Rossler system:
reduction to 2D and 1D maps and
stretching of phase space volumes
Cvitanović lecture 20
Strogatz lecture 19
HW10: 8.4.2, 8.4.12, 8.5.2, 8.6.1, 8.7.5
November 7
21. Experimental nonlinear dynamics
A 1 Fentonsecond lecture: Chaos Lab
Fenton lecture 21
November 9
22. Universality
Reading: Chapter 10, Sects 10.1 to 10.3, 10.5, 10.6
Optional reading,
ChaosBook.org bootleg: Universality in transitions to chaos
HW11: 9.3.1, 9.3.8, 9.3.9, 9.3.4
Cvitanović lecture 22
Strogatz lecture 20
November 14
23. Renormalization theory
Reading: Chapter 10, Sects 10.7
Cvitanović lecture 23
Strogatz lecture 21
Strogatz lecture 22
November 16
24. Fractals
Reading: Chapter 11 (mostly skim over)
Optional reading:
ChaosBook.org Fractals / schmactals
ChaosBook figure "fractal lens"
Cvitanović lecture 24
Strogatz lecture 23
Fractals in
nature,
biology, and
mathematics.
HW12: 10.1.11, 10.2.6 (see comments for 10.2.3),
10.3.11, 10.4.1 - due November 28
November 21
25. Strange attractors
Reading: Chapter 12, Sects 12.0 to 10.3, Sect 12.5
Smale says -
Smale does
Cvitanović lecture 25
Strogatz lecture 24
HW13:
10.5.2,
10.5.7,
11.3.3,
11.3.5,
11.4.6 (3 bonus points),
12.2.18.
- due December 5
November 24
Thanksgiving
November 28
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
Cvitanović lecture 26
Due: HW12
November 30
27. Ah, chaos!
Lai-Sang Young and ChaosBook slides
Cvitanović lecture 27
December 5
28. Wrapping it all up; project presentations
Mobille and Sharaf: "Binocular Rivalry"
Max Comstock: "Now you know"
Cvitanović lecture 28
Due: HW13
December 11
Final exam
8:00-10:50 am Howey N210; see instructions