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,
2/23: 2X2 systems with repeated eigenvalues: general solution,
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
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
3/14: Nonlinear sinks and Liapunov functions. Def. of stable/asymptotically
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
saddles and their linearizations. Stable and unstable "surfaces" in hiigher dimensions.