**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.
Proof sketches

**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
proved;

**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

**Nov 30:**

**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)