Homework 5
AMATH 383, Autumn 2009
Due: Wednesday, December 2, after class
1. (4+6+8+6+8 points)
HopfBifurcation and Limit Cycle:
In class we discussed the Brussela
tor, a chemical reaction that showed an oscillating
limit cycle
after its equilibrium point changed
from stable to unstable. In this problem we study an ODE system similar to the Brusselator
equations. It is given by
x
′
=
αx
−
y
−
x
(
x
2
+
y
2
)
y
′
=
x
+
αy
−
y
(
x
2
+
y
2
)
with solution
(
x
(
t
)
,y
(
t
))
depending on the parameter
α
and some initial conditions.
(a) Find the single equilibrium of this system and compute the eigenvalues of the Jacobian at
that point. Show that the equilibrium is a stable focus (damped oscillations) for
α <
0
and
becomes unstable for
α >
0
.
(b) Using a computer, plot the phase diagram of the system in the range
(
x,y
)
∈
[
−
2
,
2]
2
,
for the parameter choices
α
= 0
.
5
and
α
=
−
0
.
8
. Note the change in behavior of the
equilibrium point, which is called
HopfBifurcation
. For
α
=
−
0
.
8
, compute the solution of
the ODE system numerically with initial conditions
x
(0) = 0
.
01
and
y
(0) = 0
.
01
and plot
it in the time range
t
∈
[0
,
15]
. The solution starts to grow exponentially according to an
unstable equilibrium, but turns into a stationary oscillation for larger times (a limit cycle).
(c) In order to understand this behavior we change the variables
(
x
(
t
)
,y
(
t
))
to polar coordinates
(
r
(
t
)
,ϕ
(
t
))
by
x
(
t
) =
r
(
t
) cos
ϕ
(
t
)
,
y
(
t
) =
r
(
t
) sin
ϕ
(
t
)
with a radius from the origin
r
(
t
)
≥
0
and an angle
ϕ
(
t
)
∈
[0
,
2
π
]
. Show that the derivative
of
(
x
(
t
)
,y
(
t
))
is given by
p
x
′
(
t
)
y
′
(
t
)
P
=
r
′
(
t
)
e
1
+
r
(
t
)
ϕ
′
(
t
)
e
2
with vectors
e
1
= (cos
ϕ,
sin
ϕ
)
and
e
2
= (
−
sin
ϕ,
cos
ϕ
)
, and the right hand side can be
written
p
(
α
−
(
x
2
+
y
2
)
)
x
−
y
(
α
−
(
x
2
+
y
2
)
)
y
+
x
P
= (
αr
(
t
)
−
r
(
t
)
3
)
e
1
+
r
(
t
)
e
2
.
Derive the equations
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 Spring '08
 WAlker
 Math, Calculus, Constant of integration, Boundary value problem, 0 1 J

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