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CDS 140a: Homework Set 6
Due: Wednesday, November 21, 2007.
Nonlinear Differential Equations and Dynamical Systems
, Ferdinand Ver
hulst,
Second Edition.
•
Consider two systems ˙
x
=
A
x
and ˙
x
=
B
x
with
x
∈
R
n
and
A,B
constant
matrices.
Assume that all the eigenvalues of
A,B
have nonzero real parts.
Find conditions on
A,B
so that there exists a
diffeomorphism
H
:
U
mapsto→
V
from a neighborhood of origin
U
to
V
which takes orbits of one system to
those of other and preserves parametrization by time.
•
Consider ˙
x
=
ax
and ˙
x
=
bx
with
x
∈
R
and
a,b
nonzero real numbers. Find
a
homeomorphism
H
:
R
mapsto→
R
which takes the orbits of one system to those
of the other and preserves parametrization by time.
Show that
H
is not a
diffeomorphism
in general.
•
Consider the vector field
˙
x
=
2
x
+
y
2
(1)
˙
y
=
y
Denote the linearization of (1) at the origin by
J
and the flow generated by
(1) by
ϕ
t
.
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 Fall '09
 Marsden
 Vector field, power series method, Equations and Dynamical Systems, Ferdinand Verhulst

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