Lecture Schedule / News and Updates
- Sat 12/15: Final Exam.
2pm-5pm in Barus and Holley 158.
- Wed 12/12: Symbolic dynamics. Ch 15.5.
Extra office hours this week: Friday 12/14, 11am-1pm.
- Mon 12/10: Chaos. Ch 15.4.
- Fri 12/7: Discrete dynamics. Ch 15.1, 15.3.
- Wed 12/5: Hamiltonian systems and classical mechanics. Intro to the Lorentz system. Ch 9.4, 13.1, 13.2, 14.1-3.
- Mon 12/3: Predator-prey systems. Ch 11.2.
- Fri 11/30: Consequences of Poincare-Bendixon, the Poincare map. Ch 10.6, 10.3.
- Wed 11/28: The Poincare-Bendixon Theorem and its consequences. Ch 10.5, 10.6.
Errata: When writing down the P-B theorem, I forgot an important hypothesis: the theorem is that a nonempty limit set in the plane which is closed and bounded either contains an equilibrium or is a closed orbit.
HW6 has been posted, due next Wednesday, 12/6.
- Mon 11/26: Local sections, flow boxes and monotonicity. Ch 10.2, 10.4.
- Mon 11/19: Midterm Exam 2.
Concept list: nonlinear first order autonomous ODE, equilibria, heteroclinic solutions, linearization and the Hartman-Grobman theorem for hyperbolic equilibria, stability and asymptotic stability, nullclines, Liapunov functions, limit sets, invariant sets, bifurcations (especially saddle-node and Hopf). Be able to describe the qualitative changes of a system as it undergoes a bifurcation.
- Fri 11/16: Infectious disese models. Ch 11.1.
I will hold extra office hours today from 2-3:45 to accommodate questions in advance of the midterm.
Check out the paper When Zombies Attack of Munz, et. al. for some SIR type models of the zombie apocalypse.
- Wed 11/14: Models of competitive species. Ch 11.3.
Since we will not be able to cover the Poincare-Bendixon Theorem by Friday, I have decided to change course a bit and cover some applications in the lectures prior to Exam 2.
- Mon 11/12: Limit sets
and flow boxes. Ch 10.1, 10.2.
Some midterm practice problems: Ch. 9: #3, #6, #7.c, Ch 10: #1, #2, #5.
Update 11/13: (Any four of) these may be turned in as an optional HW5.
- Fri 11/9: Liapunov stability continued (we will skip Lasalle's invariance principle), gradient systems. Ch 9.2, 9.3
- Wed 11/7: Nullclines continued, Lyapunov stability. Ch 9.1, 9.2.
- Mon 11/5: The Hopf bifurcation. Introducing nullclines. Ch 8.5, 9.1.
HW 4 due.
- Fri 11/2: Bifurcations continued. Ch 8.5.
- Wed 10/31: Saddles and the stable curve theorem. Stability Ch 8.3, 8.4.
- Mon 10/29: No class - Hurricane Sandy.
Saddles and the stable curve theorem. Ch 8.3.
- Fri 10/26: Linearization for nonlinear systems continued. Ch 8.1, 8.2.
- Wed 10/24: Linearization for nonlinear systems. Ch 8.1.
- Mon 10/22: Guest lecturer Prof. Holmer on local existence and uniqueness for nonlinear systems. Ch 7.1-7.3.
- Fri 10/19: Midterm exam 1.
Includes topics from chapters 1.1-1.3, 2, 3, 4, 5, 6.1, 6.3, 6.4.
- Wed 10/17: Forcing and resonance (nonautonomous systems). Ch 6.5
- Mon 10/15: Systems of harmonic oscillators. Ch 6.2
- Fri 10/12: The matrix exponential. Ch 6.4.
Next week's midterm will cover material up through today's lecture.
- Wed 10/10: Higher dimensional linear systems. Ch 6.1, 6.3.
- Mon 10/8: No class - Brown university holiday.
- Fri 10/5: Repeated eigenvalues and genericity. Ch 5.4, 5.5.
According to the in class vote, we will have Midterm 1 on Friday Oct. 19.
- Wed 10/3: Higher dimensional linear algebra continued. Ch 5.3, 5.4.
- Mon 10/1: Conjugacy of hyperbolic 2D systems continued. Higher dimensional linear algebra. Ch 4.2, 5.1-2.
- Fri 9/28: Dynamical classification of systems continued. Ch 4.2.
- Wed 9/26: Spring mass systems revisited, Dynamical classification
of systems. Ch 4.2.
The complete list of problems for HW 2 has been set.
- Mon 9/24: Canonical form continued: repeated
eigenvalues. Trace-determinant classification of 2D
linear systems. Ch 3.4, 4.1.
Some problems have been posted for HW 2, to be due a week from today.
- Fri 9/21: Linear changes of coordinates. Ch 3.4.
- Wed 9/19: Phase portraits for planar systems: spiral sinks/sources and repeated eigenvalues. Ch 3.2-3.3.
- Mon 9/17: Phase portraits for planar systems: saddles, sources, sinks and centers. Ch 3.1-3.2.
- Fri 9/14: Eigenvalues, eigenvectors and
solutions to 2D linear systems. Ch 2.5-2.7.
Problems from Ch. 2 have been added to HW 1.
- Wed 9/12: 2D linear algebra, determinants,
equilibrium solutions of planar linear systems. Ch 2.3-2.4.
New office hours schedule: Mon 3-4pm, Wed 2-3pm.
Here is a brief guide to the terminology of differential equations that we have been developing.
- Mon 9/10: Intro to planar systems. Ch 2.1-2.2.
- Fri 9/7: The logistic equation with harvesting. Ch 1.2-1.3
- Wed 9/5: Intro to ODE, examples of
equilibria, bifurcations. Ch 1.1
At the end of class, I mislabled the stable and unstable equilibria on the phase lines for x' = ax. The equilibrium is of course stable for a < 0 and unstable for a > 0. Sorry for any confusion!