J. Rauch
4834 East Hall
Email: rauch@umich.edu
Web page: http://www.math.lsa.umich.edu/~rauch/
Office Hours:
Wednesday 10-12, Tuesday 2:30-3:30.
Email "office hours" are encouraged.| Week | Meeting | Date | In Class | Remarks/Web Postings |
|---|---|---|---|---|
| 1 | Lecture 1 | Tues, Sept. 4 | Dynamics in dimension 1. |
Sections 1.1, 1.2, 1.3. Dynamics in Dimension 1 handout. Causality handout (optional). |
| 1 | td>Lecture 2 | Th. Sept. 6 | Structural Stability. Dynamics in dimension 1.5. | Sections 1.4, 1.5. Dynamics in Dimension 1.5 handout. HW1. |
| 2 | Lecture 3 | Tu. Sept. 11 | Periodic harvesting, Fundamental existence theorem and perturbation theory. |
Section 1.5. Dynamics in Dim 1.5. Steps of Perturbation theory. Chapter 7. |
| 2 | Lecture 4 | Th. Sept. 13 | Existence and uniqueness theorem. Example of nonuniqueness. Science text linearization (importance of linear problems). Linearization by perturbation theory. |
Section 7.4. Science text linearization handout. Linearization by Perturbation Theory handout. HW 2. |
| 3 | Lecture 5 | Tu. Sept. 18 | Norms and norms of matrices. Gronwall. No blow up in the linear case. Fundamental theorems for homogeneous linear systems. |
Section 17.3. Chap. 2. Section 7.4. Fundamental Theorems of Homogeneous Linear Systems handout. |
| 3 | Lecture 6 | Th. Sept. 20 | Euler algorithm. Examples of 2x2 systems with distinct eigenvalues. Trace Determinant plane. | Chap. 2, Sections 3.1, 3.2, 4.1. Phase plane for 2x2 linear system handout. HW 3. |
| 4 | Lecture 7 | Tu. Sept. 25 | Genericity of distinct eigenvalues for 2x2 systems. Phase plane analysis of 2x2 systems. |
Chap. 3. Phase plane for 2x2 linear system handout. Section 5.6. |
| 4 | Lecture 8 | Th. Sept. 27 | Phase plane analysis of 2x2 systems. Direction of rotation of centers, spirals. Ellipse axes and eccentricity. | Chap. 3. Phase plane for 2x2 linear system handout. Ellipse axes handout. HW 4. |
| 5 | Lecture 9 | Tu. Oct. 2 | Zoom Zoom Zoom. Taylor series of solutions at t=0. Cauchy's theorem. e^{At}. | Zoom Zoom Zoom handout. Section 6.4. Section 6.2. |
| 5 | Lecture 10 | Th., Oct. 4 | Properties of e^B. 2x2 with repeated eigenvalues. Two oscillators. | Section 6.2. Section 8.3, 8.4. Hw 5. |
| 6 | Lecture 11 | Tu. Oct. 9 | Two oscillators. Kronecker Theorem. Ergodicity. | Section 6.2. Kronecker handout. |
| 6 | Lecture 12 | Th. Oct. 11 | Spectral theorem for NxN matrices. | Chap. 5 SEction 6.3. Spectral Theorem handout. HW 6 assigned. |
| 7 | Study Day | Tu. Oct. 16 | No Class. Study day. | |
| 7 | MIDTERM Exam | Th. Oct. 18 | In class. Two sides of an index card (3in. by 5in.) of notes. No electronics (phones, gps, laptops, tablets, calculators, ... ). | Old exams with solutions on web page. No section 1.6. Only 2x2 repeated eigenvalues on midterm. Variation of parameters (end of Sec. 6.5.) not done yet. No section 7.5. Otherwise, Chapters 1 through 7 plus handouts. |
| 8 | Lecture 13 | Tu. Oct. 23 | Examples of multiple root algorithm. Asymptotically stable equilibria for X'=AX. Variation of Parameters. | HW 6 due. Multiple roots algoritm handout. Midterm Section 5 of Spectral Decomposition handout. solutions posted. |
| 8 | Lecture 14 | Th. Oct. 25 | Transient amplification. Definitions of stability, asymptotic stability. Necessary and sufficient conditions in linear case. Nonlinear examples. | Sections 8.1, 8.4 of HSD. |
| 9 | Lecture 15 | Tu. Oct. 30 | Decreasing positive definite quadratic form for linear asymptotically stable system. Asymptotic stability when linearization is asymptotically stable. | HW 7 due. Section 8.2. Decreasing quadratic form handout. |
| 9 | Lecture 16 | Th. Nov.1 | Turing Instability. Stable and Unstable manifolds in linear case. Stable manifold theorem, statement. Begin proof. | Turing handout. Section 8.3. Stable manifold handout. |
| 10 | Lecture 17 | Tu. Nov. 6 | End proof of stable manifold theorem. Examples of stable manifolds. Nonlinear pendulum. |
Section 8.3 Linearization unstable and nonlinear equilibrium stable handout. Page 195-196. Stable manifold handout. |
| 10 | Lecture 18 | Th. Nov. 8 | Linearization unstable and nonlinear stable example. Need stability in definition of asymptotic stability, 186/12. Linear conjugacy. |
Conjugacy in dimension 1 handout. |
| 11 | Lecture 19 | Tu. Nov. 13 | Differentiable, and topological conjugacy. Nonequilibria and equilibria. Topological conjugacy in 1-d. |
Section 4.2. Section 8.2. Topological conjugacy in dimension 1 handout. |
| 11 | Lecture 20 | Th. Nov. 15 | Topological conjugacy of sinks/sources.. Begin Bifurcations | Section 4.2 and 8.2. Section 8.5. Conjugacy handout. Bifurcation handout. |
| 12 | Lecture 21 | Tu. Nov. 20 | End bifurcations. | Section 8.5. Bifurcation handout. Hw. 10 due. |
| 12 | Thanksgiving | Th. Nov. 22 | No Class. | |
| 13 | Lecture 22 | Tu. Nov. 27 | Conservative mechanical systems d=1. Lyapunov's method. Lasalle's Invariance Prinicpals. | Section 9.2, 9.3. V. Arnold, Ordinary Differential Equations, Ch. 2 Sec. 12. Available online at UM Library. |
| 13 | Lecture 23 | Th. Nov. 29 | End LaSalle. Gradient systems.Hamiltonian Systems. Dirichlet Theorem. Dissipative mechanics (damped nonlinear pendulum). |
Sections 9.2, 9.3, 9.4. V. Arnold, Ordinary Differential Equations, Ch. 2 Sec. 12. Gradient system handout. Hw.11 due. |
| 14 | Lecture 24 | Tu. Dec. 4 | Maps of an interval. Importance of fixed points. Web diagrams. Bifurcations. | Ch 15.1, 15.2. Bifurcation handout. |
| 14 | Lecture 25 | Th. Dec. 6 | Period doubling bifurcation. n-cycles. Tent map. Sensitive dependence, transitivity, chaos. Tent map is chaotic. Symbolic dynamics | 15.4. Conjuation of logistic handout. Hw. 12 due. |
| 15 | Lecture 26 | Tu. Dec. 11 | Shift map is chaotic. Discrete logistic map. Chaos for lambda=4 logistic. Lambda>4 and symbolic dynamics. Chaos on seeds with orbit in [0,1]. Chaos on attractor of the forced Duffing oscillator. |
15.5. 15.6. Conjugation of tent map handout (not on exam). www.scholarpedia.org/article/Duffing_oscillator |
| Dec. 18, 4-6 PM | Two 3"x5" cards of notes from home. 30% premidterm, 70% postmidterm. |