Aug 19: Introduction: Definitions, Examples, Modelling;
(cf pp 1-14 of Chicone) Hwk 1-8
Aug 24: Discussion; Introduction cont'd; Hwk 1-8 still on the
table; see how much progress you can make, and ask questions where needed;
a first glance at the existence and uniqueness theorem.
Aug 26: Existence and uniqueness theorem; Peano's existence thm.
Aug 31: extensibility; sufficient cond's for global existence; flow;
continuity/differentiability of flow.
Sep 02: sketch of proof for continuity/differentiability of flow;
first sketch toward functional analytic language. Formal proof of existence
and uniqueness started. Hwk 1-5,10,11 due Tue; Hwk 12 due Thu;
other handed out hwk due TBA
Sep 07: Formal proof of existence
and uniqueness finished. Banach spaces.
Sep 09: Example: Brief comparison C^0 with `its own' norm vs the
foreign L^1 norm. Banach's fixed point theorem; abstractly proved.
Outline one future uses of the FPT. Discussion of homework.
Homework due Tue (soft): 5-9 as far as not done yet; 13+14; 18-20
Sep 14: Discussion of Hwk; phase portraits and energy conservation;
hints for use of weighted Banach spaces in #18,20.
Sep 16: Review of Ban'FPT. Continuous and Lipschitz dependence of
FP on parameter
Sep 21: Lemma on perturbed contraction proved; extensibility thm
Sep 23: Review on norms and Banach spaces, and their purpose,
in light of difficulties with pblms 18-20. Just begun with pf of
Peano's existence thm.
Sep 28: proof of Peano's existence theorem and Arzela-Ascoli
Sep 30: proof of Arzela-Ascoli finished and continuous dependence
started; some help for hwk given
Oct 05: continuous dependence finished; review over theorems so far.
Difficulties with global-existence proofs discussed. Review and notation:
derivatives. Hwk: let's get 14c-21 done soon
Oct 07: Invariant sets; stability; Lyapunov function
Oct 12: Some questions on Hwk discussed.
Proof of stability theorem; Defn and examples of omega-limit set
Hwk: 14c hard deadline tomorrow. Hard deadline 18-20 Wed after break.
15-17 also due after the break (if not done yet). Think about 22,
start with 23, 24, 25.
Oct 14: FALL BREAK
Oct 19: Hwk 14c-16 fully discussed. 2nd proof for asymptotic stability.
Hwk 27 soft deadline Thu, hard deadline Tue (omit `linearization' part)
Oct 21: lemma on invariant hypersurfaces; the implicit function theorem
seen in action.
Oct 26: Linearized stability theorem for equilibria stated and
explained. Shortcuts in actual usage, via det and trace.
Hartmann--Grobmann briefly stated.
Oct 28: Discussion of homework
Nov 02: Discussion of homework
Nov 04: **** EXAM ****
Nov 09: Discussion exam; Hwk 21 and 27: using the omega-limit set in
an asymptotic stability proof. Hwk: 22, 28 Thu; 29,30 Thu or Tue
Nov 11: amendments and hwk. Useful theorems for locating eigenvalues.
Nov 16: Hwk discussions; linear systems of ODEs; matrix norms.
Hwk: the olde ones 23, 32 due Thu; or ask questions Thu
Nov 18: Matrix Exponentials
Nov 23: Jordan normal form and its relevance for matrix exponentials
Nov 25: THANKSGIVING
Dec 02: STUDY PERIOD. regular class time will become extra office hour
Dec 06: (Wed) 5-7pm FINAL EXAM
Dec 07: no class (general exam period)
Dec 09: no class (general exam period)