1. Consider the map
x
n+1
= 2x
n
/(1+x
n
)
. Derive all fixed points and evaluate their stability.
Show your work.
Answer
:
0
)
1
(
1
2
=
−
+
=
x
x
x
x
x
simplifies to…
Therefore,
x = 0 or 1 are the fixed points.
Evaluate the first derivative to determine stability, by the quotient rule the derivative is:
( )
1
,
0
2
1
1
,
0
)
1
(
*
2
)
1
(
*
2
=
−
=
∧
∧
+
−
+
=
x
x
x
x
x
dx
x
df
@
∧
x
= 0 ,
( )
2
=
dx
x
df
@
∧
x
= 1 ,
( )
5
.
0
=
dx
x
df
Therefore, the fixed point at 0 is unstable and the fixed point at 1 is stable.
2
.
For each map below, generate cobweb plots to determine the map’s behavior (ignore
ranges of inputs that give you imaginary numbers. Use these plots in order to fully
describe all fixed points and their stability.
a.
x
n+1
=sqrt(x
n
)
Answer

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