PART 2- LINEAR SYSTEMS AND LINEARIZATION

2/5: solution of exam 1,  general facts about lin. systs.

2/7: existence for general linear ystems: Picard iteration

2/9: uniqueness for general linear systems: Gronwall's inequality
HW3: handout, due 2/16 (iteration for 3 linear systems)

2/12: review of linear algebra: change of basis formulas.
The fundamental matrix of a linear system

2/14: discussion of HW3. Solution of diagonalizable systems: fundamental
matrix, solution of IVP by expansion in a basis of eigenvectors

2/16: Examples of 2X2 systems with constant coefficients
HW4 (due 2/23): 7.3 no. 2,4,5,6

2/19: 2 X 2 systems with complex eigenvalues

2/21: 7.5: bifurcation diagram for 2X2 matrices: concepts of bifurcation,
generic properties

2/23:  2X2 systems with repeated eigenvalues: general solution, fundamental matrix
Centers: conserved energy, diagonalization of the quadratic form

2/26: bifurcation from a center: study of a one-parameter family.
x-t graphs of solutions. Concept of stuctural stability.  Structurally stable
2X2 linear systems. Systems whose coefficients are known approximately.

HW5: due 3/2: 7.5 13(a)(b)(c),14(a)(b); 7.6  6,7(a)(b)(c)

2/28: Eigenvalues and global behavior: 3 and higher-dimensional systems
Stable and unstable subspaces (sect. 7.6)

3/2: A one-parameter family in R^3 (example). Systems in 4 and higher dimensions.
Generic systems, structurally stable systems

3/7: Solution of exam 2; introduction to linearization.

3/9: Linearization: examples (damped pendulum, cases with eigenvalues
with zero real part). Statement  of Hartman's theorem.
HW 6, due 3/16: 8.1 1(b)(c), 6(b)(f) 7

3/12 Analysis of global behavior: separatrices, basins of sinks. Examples of
predator-prey systems.

3/14: Nonlinear sinks and Liapunov functions. Def. of stable/asymptotically stable singular
points. Def of Liapunov function (strict/non-strict). Construction for a non-hyperbolic
example in R^3 , for potential motion in R^3 and for Lienard's equation.

3/16: detailed behavior for vector fields in R^2: spiral sinks, node sinks,
saddles and their linearizations.  Stable and unstable "surfaces" in hiigher dimensions.