In problem 4 we will require the following fact: Given any 2 2 matrix A of real numbers, there exists an invertible matrix P so that A = P M P 1 and M is one of the real canonical forms
0
0
0
1
where , > 0.
1. (From Strogatz Section 5.2) For each of the following linear systems, sketch the phase portrait, classify the fixed point (x ; y ) = (0; 0), and indicate the directions of the eigenvectors.
(a)
( y =
2x 3y
x = y
(b)
( y =
2x + 3y
x =
3x + 4y
(c)
( y = 8x
6y
x = 4x
3y
(d)
( y = 2x + 3y
x = 6x
y
2. Consider the system
◦ x = x + y2 + 32 y y = x + y
(a) Find all fixed points of this system and compute the corresponding linearizations.
(b) For each linearized system, classify the (unique) fixed point.
3. Solve problem 5.2.11 from Strogatz.
4. Given a 2 2 matrix A, let u, v be the solutions of
8
u = Au
1
and
>
u(0) =
0
<
>
:
Define the (time-dependent) matrix (t) =
u(t) v(t) .
(a) Show that the solution of
(
x = Ax
x(0) = x0
is given by
8
v = Av
>
<
• v(0) = 0
:1
x(t) = (t)x0:
(b)
Find (t) for each of the three real canonical forms.
(c)
Suppose that B = P M P 1 for an invertible 2 2 matrix P . Show that the solution of
( x(0) = x0
x = Bx
is given by
x(t) = P (t)P 1x0:
(d)
For each of the following matrices A, find a matrix M so that A = P M P 1, where M is one of
the real canonical forms above, and P = 1
1 . Then apply your answers to parts (b) and (c) to
1
2
find the corresponding (t).
(i)
A =
52
41
(ii)
A =
12
55
(iii)
A =
2
4
1
6
5. For the nonlinear ODEs in (a)-(c), show that the origin is the only fixed point. What type of phase portrait does the linearization predict near the fixed point?
Use a computer program to draw the actual phase portrait. Does it look like the prediction of the linear system?
a) x = x2;
y = y
b) x = y;
y = x2
c) x = x2 + xy; y =
1
y2
+ xy
2
