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Phys. 4267/6268
Assignment 11
Problem 1
Do problem 9.3.1 from Strogatz.
I haven’t formally introduceded the Lyapunov exponents in class yet, but you have already seen one of them when
we discussed the exponential sensitivity of chaotic systems to initial conditions. Lyapunov exponents are the general
ization of the Floquet exponents to aperiodic trajectories. The largest exponent
λ
1
determines the average time rate
of divergence of two trajectories, i.e.,
λ
1
= lim
t
→∞
1
t
ln

~η
(
t
)


~η
(0)

,
where
~η
(
t
) =
~x
2
(
t
)

~x
1
(
t
) is an inﬁnitesimal vector.
Problem 2
Do problem 9.3.8 from Strogatz.
Problem 3
To illustrate the ”time horizon” after which the prediction for chaotic dynamics becomes impossible, numerically
integrate the Rossler system
˙
x
=

y

z,
˙
y
=
x
+
ay,
˙
z
=
b
+
z
(
x

c
)
for
a
=
b
= 0
.
2 and
c
= 5.
(a) Starting at two initial conditions separated by a vector
~η
(0) with an arbitrary direction and length 10

6
, compute
and plot the separation
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This note was uploaded on 08/25/2011 for the course PHY 4267 taught by Professor Grigoriev during the Fall '10 term at Georgia Institute of Technology.
 Fall '10
 GRIGORIEV

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