Math 6324
Ordinary Dierential Equations I
Dr. Vaughn Climenhaga, PGH 651A
Fall 2012
PROBLEM SET 2
Due 5pm Thursday Oct. 18. You must give complete justication
for all answers in order to receive full credit.
1. Let A, B Mnn commute, and let V Cn be a gene
Notes on the stable manifold theorem
University of Houston, Math 6324, Fall 2012, Vaughn Climenhaga
The stable manifold theorem is one of the most important in the theory
of non-linear ODEs and dynamical systems. Unfortunately, some of the
standard introd
Math 6324
Ordinary Dierential Equations I
Dr. Vaughn Climenhaga, PGH 651A
Fall 2012
PROBLEM SET 1 selected solutions
1. Let be the operator norm on L(Rn ) induced by some norm on Rn .
(a) Let T L(Rn ) be such that T I < 1. Show that T is invertible
and th
Math 6324, Differential Equations - Final exam review sheet
Linear theory.
Basic denitions: vector space, norm, metric, topology
All norms are equivalent in nite dimensions
Operator norm on linear transformations/matrices
Matrix exponential, basic pro
Lorenz system
x = (y x )
y = x ( z ) y
z = xy z
Fix = 10, = 8 , let vary.
3
0 < < 1: only xed point is 0, globally attracting
1 < < 0 := 13.926 . . . : now 0 has an unstable direction, two other
attracting xed points
()
December 4, 2012
1/5
At = 0 there i
Math 6324
Ordinary Dierential Equations I
Dr. Vaughn Climenhaga, PGH 651A
Fall 2012
PROBLEM SET 2
Due 5pm Thursday Oct. 18. You must give complete justication
for all answers in order to receive full credit.
1. Let A, B Mnn commute, and let V Cn be a gene
Math 6324
Ordinary Dierential Equations I
Dr. Vaughn Climenhaga, PGH 651A
Fall 2012
PROBLEM SET 3 solutions
Due 5pm Tuesday Nov. 20. You must give complete justication
for all answers in order to receive full credit.
1. Consider the non-linear system of O
Math 6324
Ordinary Dierential Equations I
Dr. Vaughn Climenhaga, PGH 651A
Fall 2012
PROBLEM SET 4 solutions to selected problems
Due 5pm Thursday Dec. 6. You must give complete justication
for all answers in order to receive full credit.
1. (Perko, exerci
Math 6324
Ordinary Dierential Equations I
Dr. Vaughn Climenhaga, PGH 651A
Fall 2012
PROBLEM SET 4
Due 5pm Thursday Dec. 6. You must give complete justication
for all answers in order to receive full credit.
1. (Perko, exercise 3.5.2, page 213) Consider th
Math 6324
Ordinary Dierential Equations I
Dr. Vaughn Climenhaga, PGH 651A
Fall 2012
PROBLEM SET 4 solutions to selected problems
Due 5pm Thursday Dec. 6. You must give complete justication
for all answers in order to receive full credit.
1. (Perko, exerci
Math 6324
Ordinary Dierential Equations I
Dr. Vaughn Climenhaga, PGH 651A
Fall 2012
PROBLEM SET 3
Due 5pm Tuesday Nov. 20. You must give complete justication
for all answers in order to receive full credit.
1. Consider the non-linear system of ODEs
x1 = x
Math 6324
Ordinary Dierential Equations I
Dr. Vaughn Climenhaga, PGH 651A
Fall 2012
PROBLEM SET 3 solutions
Due 5pm Tuesday Nov. 20. You must give complete justication
for all answers in order to receive full credit.
1. Consider the non-linear system of O
Math 6324
Ordinary Dierential Equations I
Dr. Vaughn Climenhaga, PGH 651A
Fall 2012
PROBLEM SET 2
Due 5pm Thursday Oct. 18. You must give complete justication
for all answers in order to receive full credit.
1. Let A, B Mnn commute, and let V Cn be a gene