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)

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)

Looking for the neet resultsCheck NEET Exam Results 2016 here.

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