Deterministic Chaos. caoz. PHYS 306/638 University of Delaware. Chaos vs. Randomness. Do not confuse chaotic with random: Random: irreproducible and unpredictable Chaotic:
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Random:
Chaotic:
Suppose the Universe is made of particles of matter interacting according to Newton laws→ this is just a dynamical system governed by a (very large though) set of differential equations
"It is so complicated that I cannot even draw the figure."
q
q
2
d
d
+
+
q
=
w
+
f
ml
c
mg
sin
A
cos(
t
)
D
2
dt
dt
w
q
d
d
+
+
q
=
w
q
sin
f
cos(
t
)
0
D
dt
dt
New Language for Chaos:
Our Pendulum: 2 < dim < 3
noninteger dimension
N
1 L
1 L
2 L/2
4 L/2
4 L/4
16 L/4
8 L/8
22n L/2n
2n L/2n
What is Dimension?=
e
d
d
N
(
e
)
L
(
1
/
)
=
e
e
d
lim
log
N
(
)
/
log(
1
/
)
c
e
®
0
4 1/9
8 1/27
=
n
n
d
lim
log
2
/
log
3
c
®
¥
n
=
<
d
log
2
/
log
3
1
c
NonTrivial Example: Cantor SetN
1 1
e2t
Lyapunov Exponents: Part IIm

x
x
(
1
x
)


1
1
n
n
n
Logistic Map: Part Im

m
=
d
=

k
k
1
lim
4
.
669201
...
m

m
®
¥
k
+
k
1
k
Shadowing Theorem:Although a numerically computed chaotic trajectory diverges exponentially from the true trajectory with the same initial coordinates, there exists an errorless trajectory with a slightly different initial condition that stays near ("shadows") the numerically computed one. Therefore, the fractal structure of chaotic trajectories seen in computer maps is real.