Ordinary Differential Equations I Math 645
Fall 2011

Instructor	Brian Hassard
Lectures	M W F 11:00-11:50 Math 150
First-week handout	Syllabus, grading policy, important dates
Homework	See class
Materials
	interchange of limits p. 17
	generalized eigenvector (wiki)
	Jordan Normal Form
	E invariant with respect to flow p58
	test1_examples_645_20111020.pdf
	Test1 question 2 β(x0,y0) for y0 neg. Test1 question 2 show improper integral finite
	section_2.10_examples.html, section_2.10_examples.pdf, examples 1-6, also exer. 5c,5d section 2.9 Maple code/graphs
	645_section_2.11_examples.html, 645_section_2.11_examples.pdf 2d systems: hyperbolic and elliptic sectors, saddle-node (two hyperbolic sectors and on parabolic sector) cusp, two elliptic and two parabolic sectors, also center manifold in 2d system example 1, section 2.12
	645_section_2.12_center_manifold_examples.html, 645_section_2.12_center_manifold_examples.pdf, 3d plots of trajectories and of approximate center manifold; 2d plots of trajectories restricted to appx. center manifold.
	section_2.4_p85_exer3_finite_arclength.pdf
	Background material for section 2.3 exercise 4. Heine_Cantor_for_real_nxn_matrix-valued_function.pdf Includes proof for a real-valued function, and proof of a matrix norm inequality. These results are then used to show that a general real nxn matrix-valued function A(t,y), continuous for (t,y) in K = {-a1≤t≤a1,|y-y0|≤b} is also uniformly continuous. Here "continuous" and "uniformly continuous" are defined using the matrix norm ||.||. See also matrix norm inequalities, Heine-Cantor
	645_section_3.2_exercise4.html Maple workspace with animations showing trajectories Γ such that ω(Γ) consists of one limit orbit one limit orbit and one equilibrium point, two limit orbits and two equilibria. In the following, each trajectory "slows down" as it passes closer and closer to (0,0) and (1,0) which are equilibria. The top half y>0 and the bottom half y<0 of the circle (x-1/2)2+y2 are each limit orbits for each of the three trajectories. 645_section_3.2_exercise4A.html Maple workspace with animations showing trajectories Γ such that ω(Γ) consists of one limit orbit and one equilibrium, two limit orbits and one equilibrium, three limit orbits and two equilibria In the following, trajectories "slow down" as they pass closer and closer to (1/2,0) and ((1/2)(1-sqrt(2)),0) which are equilibria. The outer orbit Γ is such that &omega(Γ) consists of the two equilibria and three limit orbits. The inner left orbit Γ is such that &omega(Γ) consists of the two equilibria and two limit orbits, the top and bottom halves of the left "eye" outline. The inner right orbit &Gamma is such that &omega(Γ) consists of the equilibrium (1/2,0) and a limit orbit, the right "eye" outline.