Selected Excerpts  Section 10.0 

[
preface
 section 4.2
 section 10.0
 section 18.0
 section 22.0
]
Section 10.0: Nonlinear Dynamics in Simple
Maps
Chaos is the score upon which reality is written.
Henry Miller
... it may happen that small differences in the initial
conditions produce very great ones in the final
phenomena.
Henri Poincaré
Prediction is difficult, especially of the
future.
Mark Twain (also attributed to Niels Bohr)
In
the previous book part we saw how natural physical
structures could be described in terms of fractal geometry. In this
book part we will be concerned with the related topic of chaos
in nonlinear dynamical systems. A dynamical system can be
loosely defined as anything that has motion, such as swinging
pendulums, bouncing balls, robot arms, reactions in a chemical
process, water flowing in a stream, or an airplane in flight. For
each of these examples there are two important aspects that must be
considered. First, we need to determine what it is about a dynamical
system that changes over time. In the case of the pendulum, both
position and velocity vary over time, so we would be concerned with
the ``motion'' of both of these states of the pendulum. A less
obvious example is found in a chemical reaction, where the ``motion''
can be found in the ratio of reactants to reagents, or perhaps in some
physical aspect of the chemicals, such as temperature or viscosity.
The second aspect of a dynamical system that we must be concerned with
is in the collection of rules that determine how a dynamical system
changes over time. Usually, scientists have a mathematical model of
how a real dynamical system works. The model will typically have
equations that may be parameterized by time and the previous states of
the system. Sometimes these equations can be used to get an estimate
of what the future state of a dynamical system will be. In this
spirit, let's agree that the ``motion'' of a dynamical system is
dependent on how the state of the system changes over time. Moreover,
there must exist a set of rules that governs how a dynamical system in
some state evolves to another state. We may not know what the rules
are, but if a dynamical system is deterministic, a set of rules for
the time evolution of the system exists independently of our knowledge
of it.
There are many different types of motion that can be exhibited by a
dynamical system. The simplest is fixed point behavior, which
can be seen in a pendulum when friction and gravity bring the system
to a halt. Most fixed points can be likened to a ball placed on top
of a hill, which rolls downward until at some point the ball sits on a
flat spot and has no momentum to carry it further.
The next simplest type of motion is known as a limit cycle or
periodic motion, which involves movement that repeats itself
over and over. A lone planet orbiting a star in an elliptical orbit
is an example of a limit cycle. Some limit cycles are more
complicated than others. For example, a child on a swing drives
the motion of the swing by periodically rocking in beat with the
natural frequency of the swing, much like an idealized pendulum. Now,
if the child rhythmically swings one leg at a higher frequency, the
motion of that leg will cause the motion of his or her body to subtly wobble
back and forth. If it is timed accurately, the child may be able to coerce
the motion into behavior that is more complicated than that of the planet,
with his or her body moving back and forth along the main axis of the swing,
and perpendicularly left to right in step with the leg.
A slightly more complicated form of motion is found in
quasiperiodic systems, which are similar to periodic systems
except that they never quite repeat themselves. For example, the moon
orbits Earth, which orbits the sun, which, in turn, orbits the
galactic center, and so on. In order for the combined motion of the
moon and Earth to be truly periodic, they must at some future point
return to some previously occupied state. But in order for that to
happen, all of the individual motions must resonate, which means that
there must exist a length of time that will evenly divide all of the
frequencies. Figure 10.1c shows quasiperiodic motion as
consisting of two independent circular motions that fail to repeat due
to a lack of resonance.




(a) 
(b) 
(c) 

Figure 10.1: Different types of motion: (a)
fixed point, (b) limit cycle (c)
quasiperiodic

Up until the last thirty years or so, almost every scientist believed
that everything in the universe fell into either fixed point,
periodic, or quasiperiodic behavior. The belief in a clockwork
universe, as exemplified by the mathematician PierreSimon de
Laplace [1], held that, in principle, if one had an
accurate measure of the state of the universe and knew all of the laws
that govern the motion of everything, then one would be able to
predict the future with near perfect accuracy. We now know that this
is not true, since science was mistaken in its assumption that
everything is either a fixed point or a limit cycle. Chaotic systems
are not just exceptions to the norm but are, in fact, more prevalent
than anyone could imagine. Chaos is everywhere: in the turbulence of
water and air, in the wobble of planets as they follow complicated
orbits, in global weather patterns, in the human brain's
electrochemical activity, and even in the motion of a child on a
swing. In all of these cases the complicated motion produced by chaos
prohibits predicting the future in the long term. On the other hand,
phenomena that were once thought to be purely random are now known to
be chaotic. The good news in this case is that chaotic systems admit
prediction in the short term.
Chaos is related to the other topics of this book in many ways. The
pathological nature of the incomputable functions from Part I is very
similar to the unpredictability of chaotic systems. The motion of
chaotic systems can be described by fractal geometry. There is
also a hypothesis known as ``computation on the edge of chaos'' that
will be relevant in the next book part when we study complex systems.
In this chapter we will be primarily concerned with getting an
intuitive feel for how chaos works in a simple, discrete time, iterative
system. The nice thing about the examples that we will be looking at
is that the richness, diversity, and beauty of chaos are
exhibited by a deceptively simple dynamical system.
[1] Laplace claimed that ``given for one instant an
intelligence which could comprehend all the forces by which nature is
animated and the respective positions of the beings which compose it
... nothing would be uncertain, and the future as the past would
be present to its eyes.''
[
preface
 section 4.2
 section 10.0
 section 18.0
 section 22.0
]
