J. Rauch
4834 East Hall
Email: rauch@umich.edu
Web page: http://www.math.lsa.umich.edu/~rauch/
Office Hours: Wednesdays, 10-11, 2-3.
| Week | Meeting | Date | In Class | Remarks/Web Postings |
|---|---|---|---|---|
| 1 | Lecture 1 | Tues, Sept. 8 | Causality and ODE, Existence by Fundamental Thm. of Calculus, Power series (Cauchy) and Separation of Variables, Picard's existence and uniqueness theorem |
Course info., hw1., Causality |
| 1 | td>Lecture 2 | Thurs, Sept. 10 | Nonnuniqueness example, Fundamental Thm. of the Phase Line, Differentiable dependence and Perturbation Theory | Fundamental Theorem of the Phase Line |
| 2 | Lecture 3 | Tues, Sept. 15 | Perturbation theory cont. Review Euler's exp(rt) method for scalar const. coeff. linear homogeneous equations. Algorithm for multiple roots. | hw1 due. The Steps of Perturbation Theory. |
| 2 | Lecture 4 | Th., Sept. 17 | Linear homogeneous systems. Linear independence. X(t) to X(t_0) is 1-1 onto linear map from solutions of homog. eqn. to initial data. Fundamental matrices. | section 2.3. |
| 3 | Lecture 5 | Tu, Sept. 22 | Linearization at an equilibrium I (science text approach). Euler's exponential method, exp(rt) v. Distinct eigenvalues. Cauchy's Theorem in the linear case and exp(tA). | pg. 62-64, then, 55-61, hw2 due. hw3 posted. Eigenbasis Theorem. |
| 3 | Lecture 6 | Th., Sept. 24 | Linearization at an equilibrium II (perturbation theory approach). Norms and matrix norms and exp(tA). Introduc generalized eigenspaces X_j. | pg 36, 64-65. Linearization II, |
| 4 | Lecture 7 | Tu., Sept. 29 | Spectral Theorem for general A. | pg. 276-279 and/or Generalized Eigenvectors handout. |
| 4 | Lecture 8 | Th., Oct. 1 | Multiple roots algorithm with examples. | hw3 due. hw4. Additional exs. pg 66-71. |
| 5 | Lecture 9 | Tu., Oct. 6 | Application to phase plane. | Section 2.8. Phase Planes. hw4 due. hw5 |
| 5 | Lecture 10 | Th., Oct. 8 | Phase plane continued. Centers, spirals, ellipses. | Two ellipses handouts. |
| 6 | Lecture 11 | Tu. Oct. 13 | Application to large time asymptotics. Floquet multipliers. Two harmonic oscillators. | Sections 2.7, 2.9. Hirsch et al section 6.2. |
| 6 | Lecture 12 | Th. Oct 15 | Kronecker's Theorem. Ergodicity. Linear nonhomogeneous/Variation of constants. (Will be applied in Secs. 4.3, 4.4.) | Section 2.4 + pg. 72-73. Kronecker's Theorem. |
| 7 | Study Day | Tu., Oct. 20 | No Class | |
| 7 | MIDTERM EXAM | Th. Oct 22 | In class. Closed book. Two sides of a 3x5 cards of notes. No electronics. | Midterm solutions. Revised ellipse axes. |
| 8 | Lecture 13 | Tu. Oct 27 | Logistic w/harvesting, bifurcations. Definitions/examples of stability | hw 6 due. hw 7. Sections 4.1, 4.2, 4.3, 4.4 |
| 8 | Lecture 14 | Th. Oct. 29 | More on Poincare Map and Floquet Theory. Limitations on linearization when evalues are purely imaginary, examples. Proof of Corollary on page 99 (w/o Thm 2.12.) |
2.9. pg. 168-169. |
| 9 | Lecture 15 | Tu, Nov. 3 | Theorem 4.3. (Linearization asympt. stab. implies asymp. stab.) Gronwall Lemma. | 4.4, pg 31. hw 7 due |
| 9 | Lecture 16 | Th, Nov. 5 | Theorem 4.2 (A+B(t) is asymptotically stable when limsup B =small, A asymptotically stable). | 4.3, hw7.prob4.soln in office door binder. |
| 10 | Lecture 17 | Tu, Nov. 10 | Stable/Unstable manifolds. Linear case. Real linear case. Bounded sols of Y'=AY + gamma(t). | hw 8 due, 4.5 |
| 10 | Lecture 18 | Th. Nov. 12 | Stable/Unstable manifolds. Integral eqn for small bounded solutions of Y'=AY + g(Y). Fixed point iteration. Local construction. Globalization. | |
| 11 | Lecture 19 | Tu. Nov. 17 | End Stable/Unstable manifold. Nonlinear pendulum. Nullclines. | hw 9 due, hw 10, 6.3. |
| 11 | Lecture 20 | Th. Nov. 19 | Nonlinear Pendulum continued. Not minimum of V(x) can be stable. Lyapunov's second method, hamiltonian systems. | 5.1, 5.2. Instability assertion on pg 188 false. |
| 12 | Lecture 21 | Tu. Nov. 24 | Hamiltonian systems. Gradient systems. Lyapunov's Stability Theorem. LaSalle's invariance principal. |
hw 10 due. hw 11 (short). Gradient systems. 5.3, 5.4. |
| 12 | Thanksgiving | Th. Nov. 26 | No Class | |
| 13 | Lecture 22 | Tu. Dec. 1 | Lyapunov's asypmtotic stability theorem. Applications to hamiltonian, gradient systems and an alternate proof of asymtotic stability when linearization has e-values in left half plane. |
hw 12. |
| 13 | Lecture 23 | Th. Dec. 3 | hw 11 due. | |
| 14 | Lecture 24 | Tu. Dec. 8 | hw 12 due. | |
| 14 | Lecture 25 | Th. Dec. 10 | Last Class | |
| 15 | Study Day | Tu. Dec. 15 | No Class | |
| Fri. Dec 18 | 10:30-12:30 AM, two sides of TWO 3x5 cards. No electronics. |