MATH 130B (Spring Quarter 2012). Ordinary Differential Equations II

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

Course description

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.

Syllabus (subject to modification)

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]

Last modified: 11 June 12.