Instructor: David A. Meyer
Office hours: AP&M 7256, W 1:00pm-2:00pm, or by appointment
Lecture: Center Hall, room 217B, MWF 9:00am-9:50am
Email: dmeyer "at" math "dot" ucsd "dot" edu
Teaching assistant: Maximilian Metti
Office hours: AP&M 5768, F 11:00am-12:00pm, or by appointment
Section: AP&M B412, Tu 9:00am-9:50am
Email: mmetti "at" math "dot" ucsd "dot" edu
This course is the second in the upper division ordinary differential equations sequence. It covers systems of equations in 2 and higher dimensions, emphasizing the qualitative (topological) properties of their solutions, and how those properties differ in 2 versus more dimensions. If time allows we will also discuss discrete dynamical systems or fractals.
The textbook for the Math 130 sequence this year is S. H. Strogatz, Nonlinear Dynamics and Chaos, with Applications to Physics, Biology, Chemistry, and Engineering (Cambridge, MA: Perseus Books 2004). In this course we will cover at least Chapters 7, 8 and 9. There is a copy of the textbook on reserve at the Science and Engineering library.
There will be weekly homework assignments, due in lecture on Wednesdays, or before then section or in the drop box in the basement of AP&M. Students are allowed to discuss the homework assignments among themselves, but are expected to turn in their own work — copying someone else's is not acceptable. Homework scores will constitute 1/6 of the final grade.
There will be two midterms, approximately in the fourth and eighth weeks of the quarter. The final is scheduled for 8:00am Wednesday 13 June 2012. Scores on the two midterms and final will constitute 1/4, 1/4 and 1/3 of the final grade, respectively. There will be no makeup tests.
2 Apr 12 |
overview of course HWK (Extra credit; due any time before F 1 June 12). write up and/or present example of ODE system from outside of class §7.0. Limit cycles definition in/stability §7.1. Examples van der Pol oscillator [Mathematica notebook] |
4 Apr 12 |
§7.2. Ruling out closed orbits gradient descent Lyapunov functions HWK (due Tu 10 Apr 12). Chap. 7: 1.3, 1.9, 2.4, 2.9, 2.12, 2.16 [solutions, pursuit problem] |
6 Apr 12 |
Dulac's criterion §7.3. Poincaré-Bendixson Theorem statement |
9 Apr 12 |
glycolysis example discrete Schrödinger equation as complex and as real system [notes] |
11 Apr 12 |
norm-preserving condition for real system skew-symmetric matrices norm-preserving condition for complex system skew-Hermitian matrices introduction to Lie algebras and Lie groups HWK (due W 18 Apr 12). Chap. 7: 3.3, 3.4, 3.7, 3.9, 3.10, 3.11 [solutions] |
13 Apr 12 |
additional conserved quantities for skew-Hermitian systems reduction of discrete Schrödinger system to two dimensions |
16 Apr 12 |
Hamilton's equations of motion and the conserved energy a discrete nonlinear Schrödinger equation norm-preserving energy-conserving reduction to two dimensions |
18 Apr 12 |
critical point is nonlinear center §7.4. Liénard systems generalization of discrete nonlinear Schrödinger example Liénard's Theorem implies existence of stable limit cycle for van der Pol's equation §7.5. Relaxation oscillations shape of trajectories in van der Pol oscillator for large nonlinearity HWK (due W 25 Apr 12). Assume that A, B, C are skew-symmetric n x n matrices. 1. Is AB skew-symmetric? 2. Is [A,B] = AB - BA skew-symmetric? 3. Show that [A,[B,C]] + [B,[C,A]] + [C,[A,B]] = 0. Chap. 7: 4.1, 5.2, 5.3, 5.6, 5.7 [solutions] |
20 Apr 12 |
two time scales in van der Pol oscillations [Mathematica notebook] estimating the period §7.6. Weakly nonlinear oscillators harmonic oscillator with small nonlinearities example of the failure of regular perturbation theory [Mathematica notebook] |
23 Apr 12 |
example of the success of 2 time perturbation theory [Mathematica notebook] |
25 Apr 12 |
review of Chapter 7 |
27 Apr 12 |
application of 2 time perturbation theory to van der Pol equation HWK (due W 2 May 12). Chap. 7: 6.2, 6.13, 6.16, 6.19 [solutions] |
30 Apr 12 |
Midterm covering Chapter 7 [solutions] |
2 May 12 |
§8.0. Bifurcations revisited review of topology changes in 1 dimension HWK (Extra credit; due any time before W 9 May 12). find a 1 dimensional system with a 2 to 4 fixed points transcritical bifurcation §8.1. Saddle-node, transcritical, and pitchfork bifurcations example of saddle-node bifurcation along unstable manifold in 2 dimensions HWK (due W 9 May 12). Chap. 8: 1.4, 1.5, 1.7, 1.10, 2.1, 2.8, 2.11, 2.12 [solutions] |
5 May 12 |
Hopf bifurcations supercritical, subcritical, degenerate |
7 May 12 |
§8.4. Global bifurcations of cycles blue sky cycle bifurcations infinite period bifurcations homoclinic bifurcations [Mathematica notebook] |
9 May 12 |
§8.6. Coupled oscillators phase space torus uncoupled oscillator trajectories [Mathematica notebook] rational angular velocity ratios give cycles cycles are knotted irrational angular velocity ratios give trajectories dense in the torus measure of trajectory depends on rational approximations which come from continued fraction expansions derivation of continued fraction for square root of 2 HWK (due W 16 May 12). Chap. 8: 4.1, 4.2, 4.12, 6.2, 6.3, 6.7 [solutions] |
11 May 12 |
coupled oscillators blue sky cycle bifurcation §8.7. Poincaré map example in which the Poincaré map can be calculated explicitly stability of fixed point from cobweb diagram |
14 May 12 |
using differential of Poincaré map to prove stability in general example §9.0. Introduction to the Lorenz equations simplified model for atmospheric flow x convection rate y temperature difference between ascending/descending currents z deviation from vertical linear temperature gradient §9.2. Simple properties of Lorenz equations fixed points §9.3. Numerical integration of Lorenz equations σ = 10, b = 8/3, r = 28 [Mathematica notebook] HWK (due W 23 May 12). Chap. 8: 7.2, 7.4, 7.8, 7.9 Chap. 9: 2.1, 2.2, 2.3, 2.4 [solutions] |
17 May 12 |
review session in AP&M 7421, 12:00nn-1:00pm |
21 May 12 |
Midterm covering Chapter 8 (except §8.3 and §8.5) |
23 May 12 |
§9.4. Lorenz map similar to Poincaré map |f'| > 1 so no stable cycles §9.5. Exploring parameter space "transient chaos", periodicity, stable limit cycle HWK (due W 30 May 12). Chap. 9: 3.1, 3.8, 4.2, 5.5abcd [solutions] |
25 May 12 |
§10.0. Introduction to one-dimensional maps discrete time dynamical systems §10.3. Logistic map: analysis fixed points stability of fixed points bifurcation to 2-cycle stability of 2-cycle bifurcation to 4-cycle |
28 May 12 |
Memorial Day; no lecture |
30 May 12 |
§10.2. Logistic map: numerics successive period doublings, closer and closer together limit is attractor with infinite number of points §10.4. Periodic windows 3-cycle followed by period doubling preceded by intermittency HWK (due W 30 Jun 12). Chap. 10: 1.11, 1.12, 3.1, 3.2, 3.5, 3.6, 3.7, 3.13 [solutions] |