## Tuesday, 1 March 2016

### How to prepare for the class test of 10 March?

The material to study for the class test is Chapter 3 - Linear autonomous ODEs
Chapter 4 -The flow near an equilibrium and Chapter 5 - Poincare-Bendixson Theorem from the lecture notes, and the problems sheets nr 3, 4 and 5 relating to these chapters.

I now summarize in some detail the main points in this material:

Chapter 3
definition of linearity
solution of autonomous linear ODEs in terms of exponential of matrix (including relevant proofs)
existence and uniqueness of solutions of autonomous linear ODEs
flow map and its computation in elementary examples in the two-and three-dimensional case
geometric interpretation of explicit formulas for the flow map: invariant subspaces, eigenspaces and generalised eigenspaces, projections and the decoupling principle for linear ODEs
Jordan normal form (general result, but not general proof, and ability to determine the jordan normal form in some simple examples); generalised eigenspaces
Jordan Chevalley decomposition; definition, implications for exp(A) and finding the J-C decomposition in some elementary examples.
Lyapunov and asymptotic stability. Application to linear systems; role of eigenvalues and determination of (in)stability based on information about eigenvalues.
Lyapunov functions; proofs and elementary applications

Chapter 4
Linear approximation near an equilibrium point
Lemma 4.1.1, including interpretation (what does the lemma establish?) and proof (with relevant components, like gronwell estimate and variations of constant formula
Theorem 4.1.2 and proof of part (i)
Hartman-Grobman Theorem may be skipped
Hyperbolic equilibria: proposition 4.2.2 and corollary 4.2.3 (inclusive of proof)
prop 4.2.5 & prop 4.2.6 incl proofs
stable and unstable manifolds (only definitions and main results but no proofs - as not given)
simple examples of bifurcations (and use of implicit function theorem in this context)

Chapter 5 (Hirsch, Smale, Devaney chap 10)
limit sets (definitions and identification of limit sets in simple examples) (10.1)
local sections and flow box (10.2)
monotone sequences (10.4)
Poincare-Bendixson Theorem (10.5 and 10.6) + additional note on classification of omega-limit sets for planar ODEs (as discussed in lecture)

Exercise sheets to be studied:
Problem sheet 4 (but nr 6 not important for test).
Problem sheet 5 (but nrs 1 and 6 not important for test).
Problem sheet 6 (but nrs 4, 5 and 7 not important for test)