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with
H
2 are said to exhibit anomalous diffusion. This is the case of the long-
memory random walk just discussed. Show by direct calculation that (
4.64
)istrue
and that
=
1
/
X
t
t
2
H
+
2
.
∝
4.4 Logistic maps
What is surprising about the chaotic behavior of the solutions to the logistic equa-
tion is that their successive bifurcation from single-valued saturation to chaos with
increasing values of the control parameter is generic and independent of the partic-
ular one-parameter family of maps. Discuss the general dynamical properties of the
solutions to (
4.75
) in terms of iterated maps
y
n
+
1
=
f
(
y
n
),
where
f
(
y
)
is a nonlinear function, for example, the logistic map on the unit interval is
f
(
y
)
=
λ
y
(
1
−
y
).
One reference you might consider consulting is West [
44
], but there are many others.
Please do not copy what is written, but demonstrate your understanding by putting the
formalism in your own words. In particular, use three successive values of the logistic
map in the chaotic region to empirically determine the value of the control parameter.
4.5 The nonlinear oscillator
Consider a quartic potential
1
4
α
q
4
U
(
q
)
=
whose potential energy looks similar to that shown in Figure 3.4 except that the sides
of the potential are steeper. Use the analysis of the dynamics of the harmonic oscillator
from Section
3.1
and construct the Hamiltonian in terms of the action-angle variables.
Also find the action dependence of the frequency.
References
[1] R. J. Abraham and C. D. Shaw,
Dynamics - The Geometry of Behavior, Parts 1-4
,Santa
Cruz, CA: Aerial Press (1982-1988).
[2] L. A. Adamic and B. A. Huberman, “The nature of markets in the World Wide Web,”
Quart.
J. Electron Commerce
1
, 512 (2000).
[3] V. I. Arnol'd,
Russian Math. Surveys
18
, 85-191 (1963); V. I. Arnol'd and A. Avez,
Ergodic
Problems of Classical Mechanics
, New York: Benjamin (1968).
[4] L. Bachelier,
Annales scientifiques de l'Ecole Normale Supérieure, Suppl. (3) No. 1017
(1900): English translation by A. J. Boness, in
The Random Character of the Stock Market
,
Cambridge, MA: P. Cootners, MIT Press (1954).
[5] A.-L. Barabási,
Linked
, New York: Plume (2003).
