Class time: Tuesday-Thursday 10:00-11:30.
Room. 3088 East Hall?.
J. Rauch
4834 East Hall
Email: rauch@umich.edu
Web page: http://www.math.lsa.umich.edu/~rauch/
Course materials are on that web page.
Office Hours: Wednesday 2-4, 4834 East Hall
Email "office hours" are encouraged.
| Week | Meeting | Date | In Class | Remarks/Web Postings |
|---|---|---|---|---|
| 1 | Lecture 1 | Tues, Sept. 2 | Causality. Separation of variables; examples and analysis. Dynamics in dimension 1. Structural Stability. |
HSD Sections 1.1, 1.2, 1.3. Causality handout. Dynamics in Dimension 1 handout. |
| 1 | td>Lecture 2 | Th. Sept. 4 | Periodic harvesting. Flow and Poincare map. Dynamics in dimension 1.5. | HSD Sections 1.4, 1.5. Dynamics in Dimension 1.5 handout. HW1. |
| 2 | Lecture 3 | Tu. Sept. 9 | Dynamics in dimension 1.5. End of logistic with periodic harvesting. |
HSD Ch1. Section 1.5. Dynamics in Dim 1.5 handout. |
| 2 | Lecture 4 | Th. Sept. 11 | Example of nonuniqueness. Conversion to a first order system. Fundamental existence theorem and perturbation theory. Science text linearization at equilibria. | HW1 due. HW2 posted. Some of Chapters 7 and 17. Steps of Perturbation theory. Science text linearization handout. Section 7.4. |
| 3 | Lecture 5 | Tu. Sept. 16 | Linearization by perturbation theory. Norms of vectors and matrices. No blow up for linear problems, start. | Linearization by Perturbation Theory handout. Linear Theory handout. HSD Section 2.7, 7.2, 17.4, 17.5. |
| 3 | Lecture 6 | Th. Sept. 18 | No blow up for linear problems, finish. Fundamental theorem for Linear Homogeneous Systems. Euler exponential algorithm, constant coefficient scalar case. Euler algorithm for constant coefficient systems, eigenvalues and eigenvectors. Example with complex eigenvalues from HSD 3.2. | Linear Theory handout. HW2 due. HW3 posted. HSD Ch. 2, Ch. 3., 4.1. 5.6. |
| 4 | Lecture 7 | Tu. Sept. 23 | Genericity of distinct evalues. Phase plane for 2x2 linear systems. Conserved quantities. Transformation by linear change of coordinates. | Phase plane handout. HSD Chap. 2,3,4. |
| 4 | Lecture 8 | Th. Sept. 24 | Ellipse geometry for centers and spirals. Cauchy algorithm. | HW3 due. HW4 posted. HSD 3.3. Ellipse axes handout. |
| 5 | Lecture 9 | Tu. Sept. 30 | Cauchy Algorithm applied to 2x2 systems with mulitple eigenvalues. Convergence of series of matrices. e^B and properties. |
HSD Ch. 5. Section 6.4. |
| 5 | Lecture 10 | Th., Oct. 2 | Generalised eigenspaces. Spectral decomposition for general matrices. | HW4 due. Sections 1 and 2 of spectral decomposition handout. HSD 6.3. Ch. 5,6. |
| 6 | Lecture 11 | Tu. Oct. 7 | Multiple roots algorithm. Examples. Algorithm for exp(tA). Nonhomogeneous linear equations also known as variation of constants. | HSD 6.3, 6.4. 6.5. Section 3 of Spectral theory handout. Multiple roots algorithm handout. |
| 6 | Lecture 12 | Th. Oct. 9 | Stability and asymptotic stability for X'=AX. Transient amplification. Turing Instability. | HW5 due. HW6 posted. Section 5 of Spectral Theory handout. Turing instability handout. |
| 7 | Study Day | Tu. Oct. 14 | Study day. | No office hour Weds. 15. Please email questions. |
| 7 | MIDTERM Exam | Th. Oct. 16 | In class. One 3"x5" index card of notes from home. Two sided. Save for the final. No electronics. | Exam does not cover sections section 4, 6, 7, 8 of the Spectral Decomposition Handout. Does not cover sections 4.3, 5.6, 6.2, 7.5, 7.6. Does cover Section 5 of Spectral Theory Handout and Turing instability. |
| 8 | Lecture 13 | Tu. Oct. 21 | Two oscillators. Kronecker's theorem. Ergodicity and Boltzmann. | Kronecker Theorem handout. HSD 8.4. HW6 due. HW7 posted. |
| 8 | Lecture 14 | Th. Oct. 23 | Zooming at non equilibrium and equilibrium points. Definitions of stability and asymptotic stability. Examples of stability and instability of nonlinear systems. Quadratic forms decreasing on orbits of linear equations, begin. | Zoom zoom zoom handout. Sections 5,6 of Spectral Theory handout. HSD 8.1, 8.2. |
| 9 | Lecture 15 | Tu. Oct. 28 | Quadratic forms decreasing on orbits of linear equations. Proof that asymptotic stability is inherited from the linearization. Statement of instability theorem. | HW7 due. HSD 8.1 (160-161 and, Problem 184/1iii). Decreasing quadratic forms handout. |
| 9 | Lecture 16 | Th. Oct. 30 | Stable unstable manifolds, Linear case. Stable Manifold Theorem. | 8.1 8.3. |
| 10 | Lecture 17 | Tu. Nov. 4 | Examples of Stable Manifold Theorem. Proof of stable curve theorem. |
8.1, 8.3. HW8 due. |
| 10 | Lecture 18 | Th. Nov. 6 | Examples of global stable and unstable manifolds. Conservative mechanics in dimension d=1, level curves of the energy. Computation of orbits and heteroclinic connectors for the undamped pendulum. Stable curves and the phase plane of the damped nonlinear oscillator. | HSD 194-196, 200. V.I. Arnold, Ordinary Differential Equations, MIT press, Chap. 12 Section 2. Available online on at UM Library. Mecahnics examples handout. |
| 11 | Lecture 19 | Tu. Nov. 11 | Linear conjugacy. Differentiable conjugacy at nonequilibrium points. Differentiable conjugacy at an equilibrium implies linear conjugacy of the linearized equations. |
HW9 due. Linear conjugacy is discussed in Section 3.4. Differentiable conjugacy at nonequilibrium for d=2 is in Section 10.2. Topological conjugacy and the inadequacy of differentiable conjugacy is discussed in Sections 4.2 and 8.2. Conjugacy handout. |
| 11 | Lecture 20 | Th. Nov. 13 | Topological conjugacy in dimension d=1. | Topological conjugacy and the inadequacy of differentiable conjugacy is discussed in Sections 4.2 and 8.2. Conjugacy handout. |
| 12 | Lecture 21 | Tu. Nov. 18 | Topological conjugacy of sinks proved and hyperbolic equilibria stated. Bifurcation theory, dimension d=1, begin. | HSD 8.2. 8.5. Bifurcation handout. |
| 12 | Lecture 22 | Th. Nov. 20 | Bifurcation theory d=1 finish. Bifucation in dimension N>1. Hopf Bifurcation. | HW10 due. Bifurcation handout. Lyapunov and LaSalle handout. Youtube videos. |
| 13 | Lecture 23 | Tu. Nov. 25 | Lyapunov functions. Proof of Lyapunov and Lasalle theorems 1,2,3,4. Omega limit sets. | HSD 9.2, Lyapunov and LaSalle handout. |
| 13 | Thanksgiving | Th. Nov. 27 | No Class. | |
| 14 | Lecture 24 | Tu. Dec. 2 | Examples of LaSalle/Lyapunov. Example of stable equililibrium with linearization unstable. Gradient systems. Hamiltonian systems. Dirichlet Theorem and its false converse. | HW11 due. HSD 9.3. 9.4. Gradient handout. Stable with unstable linearization handout. |
| 14 | Lecture 25 | Th. Dec. 4 | Motivation for iteration of maps; dimension 1.5, Floquet Theory, Poincare Bendixon. Divergence of iterates without fixed points. Fixed points and their stability. Bifurcation of fixed points. | |
| 15 | Lecture 26 | Tu. Dec. 9 | Bifurcation of fixed points. Period doubling. Tent map. Dense periodic orbits. | HW 12 due. HSD 15.1. 15.2. 15.3, 15.9 |
| 15 | Lecture 27 | Tu. Dec. 8 | Transitivity. Sensitive dependence. Definition of chaos. Four examples. Tent map. Logistic 4x(1-x). Shift map. Duffing oscillator. Without forcing or friction using 2d mechanics. With friction using Lasalle. Chaotic with friction and driving. | HW13 posted. HSD 15.3. 15.4. 15.6. Duffing oscillator page on scholarpedia.org. |
| Office hours | Dec 10,12 | Dec 12, 10-12. Send me an email warning if possible. | ||
| Dec. 16, 1:30-3:30 | Two 3"x5" cards of notes from home. 30% premidterm, 70% postmidterm. No electronics. |