11.6. Experiment vs. Model
Bifurcations, low damping
PHY 380.03 Spring 2013
2013 R. Martin
234
Bifurcations: high damping
PHY 380.03 Spring 2013
2013 R. Martin
Bifurcations: high damping
Orbits:
PHY 380.03 Spring 2013
235
2013 R. Martin
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Orbits:
PHY 38

Chapter 13. Self-organized Criticality
I3.1. The Bak, Tang, Weisenfeld 'Sand Pile'
The sand pile CA is defined on a finite grid of Nx by Ny cells, where each cell can have cubical sand grains on it,
to a critical height of Z = Zcr grains before becoming u

5.2 Cos2 q Potential: Strobe Plot
First, to check out the algorithm, then try the method on some sample orbits. Let's set up the equation
of motion as we did in Chapter 4. Here are the potential and the associated forcing function:
In[1]:=
Out[2]=
V = - C

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7. The Current Sheet Magnetic Field Dynamical System
7.1 Review so far
The course is nearly half-way completed and it's a good time to review what we've been through so far.
Throughout the class so far, I've motivated ideas and methods, and used as an

PHY 380.03
Name:
First Exam
Spring 2004
_
1. Which of the following is a characteristic property of all non-dissipative Newtonian
dynamical systems?
a. Complex behavior
c. Randomness
e. The existence of chaos
b. The existence of attractors
d. Conservation

11.2. Analytic results for the Logistic Map
Fixed Points
The logistic map is a great example because of its complex behavior but simple mathematical expression. In fact, you can learn a lot analytically. Lets begin with the fixed points. For a map, the fi

Final Exam Questions 1 through 7:
Predator - Prey with Functional Response (Holling's
Equations)
Consider the following dissipative system. Let
x = PREY population density
y = PREDITOR population density
Then the following equations model predator-prey dy

PHY 380.03
Practice Exam Questions
Spring 2013
1. The damped, driven pendulum exhibits which of the following kinds of behavior? Circle all
correct answers.
a. Stable fixed point
c. KAM tori
e. Limit cycle
b. Unstable fixed point
d. Strange attractor
f. W

NONLINEAR DYNAMICS
FOR UNDERGRADUATES
Richard F. Martin, Jr.
Physics Department
Illinois State University
Draft Chapters 1-12
R. Martin, December 2013
ii
CONTENTS
Foreword
1. Introduction
iii
1
2. The Pendulum, Archetypal Nonlinear System
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3. Fixed

12.3. Complexity in CAs
Class 4
The 1-D CA Rule 110 seems different than weve seen before. If you run it out to longer numbers of
generations youll see some very interesting time-evolution (remember time increases as you go vertically downward), for examp

12.5. CA's and Fundamental Physics
We've discussed a little in class the question of whether rule-based models can ever explain fundamental physics. We have to be a little careful with the word 'fundamental'. If you come from one tradition in
physics, tha

12.4. Mobile CAs
Last time we expanded our view of 1D cellular automata and I'll begin today by expanding even further.
Dr. Clark will talk next class on some of the CA work he and his students have done, which includes
some study of interacting moving ob

196
Chapter 10 : Fractals
Weve seen some hint of what youve heard called fractals: the strange attractor in the forced, damped
pendulum for example, where we saw some hint of self-similarity, or the KAM breakup in conservative
systems which also seemed to

12.3 Complexity (continued)
Searching for complex behavior
We've now seen a range of behavior from various cellular automata. From simple 1D CA's we saw
examples of all 4 of the Wolfram classes, as well as one rule which led to a universal computer. We
sa

180
Chapter 9 : Dissipative systems
So far weve been concentrating more on Newtonian systems without damping (a.k.a Hamiltonian systems).
We have learned a little about damping, for example damping terms in Newtons second law tend to involve
odd-order der

Chapter 11. Bifurcations
11.1 Logistic Map DIY
The logistic map is given by xn+1 = r xn H1 - xn L. The name is historical, although I dont know what it refers to. I do know that this is a simple
population model where xn represents the number of animals o

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8. APPLICATION OF A K-SYSTEM:
THE LORENTZ BILLIARD
The Lorentz billiard system is an idealized model system intended to model atoms (or molecules)
in a gas, liquid, or solid. The system is a set of point particles that scatter off of fixed "hard"

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6.9. The KAM theorem and the route to Hamiltonian chaos
Back in the early days of nonlinear dynamics, the French mathematician and physicist Henri
Poincar was working on the problem of the stability of the solar system: is the sun/planet
system stable

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6.6 Lyapunov Exponent Numerical Computation
Lyapunov exponents by the Benettin, et al. method
We saw in Section 6.5 that the Benettin method of pulling back the test comparator orbit at regular intervals and restarting the exponential divergence calcu

2/21/13
120
6.4 Sensitivity to initial conditions (SIC) example
The equations of motion for the driven pendulum are
q=w
w = -sinq + Acos HwD tL
where the time variable is normalized, as discussed in class.
In Homework #4 we got the following Strobe plot f

124
6.5 Lyapunov exponent calculation: the Benettin, et al. method
Recall that in Section 6.2 we defined the Lyapunov exponent for an orbit as
1 d(t)
L = lim ln
d0 0 t
d0
t
(6.2)
where d(t) is the distance in phase space between the fiducial orbit

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6.7 Predictability in Chaotic Systems
We saw that several definitions of chaos are at least consistent with the idea that chaotic systems
are, in some sense, unpredictable. This could be because of the SIC property or because of
decay of correlations,

117
6.3.
Sensitivity to Initial Conditions
Here's Strogatz' definition of chaos again: Chaos is aperiodic long-term behavior in a
deterministic system that exhibits sensitive dependence on initial conditions. He needs all the
qualifiers (deterministic, lo

2/7/13
79
Section 4.6 Extended phase space example
Motion in the Cos2 q Potential with a driving torque
As an example of the extended phase space lets look at the motion of an angular system under the
influence of the following potential:
In[1]:=
V = - Co

98
5.3 Poincar Surface-of-Section Plots
How do we handle higher dimensional systems that are autonomous, that is if the forces or the
potentials are not time-dependent? Nonlinear dynamics scientists use a variety of tools to study
such systems, including

109
5.4 Dimensionless parameters
OK, we're finally getting into chaos - but there is one more tool you need to learn about before we tackle
a real problem: finding the useful parameters for your system. Let's briefly return to the pendulum
equation o

111
6. CHAOTIC DYNAMICS
OK, its time to begin looking seriously at chaos. Weve examples of what Ive called chaotic
motion, from the double pendulum demonstration to phase portraits to Poincar sections. In
these examples Ive characterized it loosely as bei

85
5. VISUALIZING 3-D AND HIGHER DYNAMICS
5.1. "Strobe Plots": fixed interval phase sampling
For 2-D Newtonian systems weve looked at two main ways of visualizing the dynamics:
plotting the dynamical variables vs. time, and phase space plots (in particula

69
4. DYNAMICAL SYSTEMS AND THE GEOMETRY OF PHASE SPACE
4.1. Dynamical Systems
Weve spent a lot of time exploring two-dimensional Newtonian systems and, in particular, the
plane pendulum. Now its time to start generalizing. Before we do, lets summarize th

49
3.3.
Potential Functions And Fixed Points
I used an analogy in class for the stability of fixed points: a mass sliding on a landscape under
the influence or gravity. Heres a suggestive diagram:
C
A
B
D
For the mass on the landscape, we can see both the