tem. For example, the number zero is an attractor for

the system given by f(x) = x: no matter the starting

value, all iterates converge to the value zero.

The iterates of very simple functions fcan exhibit

extremely surprising behavior. Take, for instance, the

iterates of the function f(x) = 1 – cx2, with initial value

x= 0. (Here cis a constant. Each value of cdetermines

its own dynamical system.)

Set c= 0.1. The first five iterates of the system xn+1

= 1 – 0.1xn2are:

0, 1.000, 0.900, 0.919, 0.916, 0.916

The system seems to stabilize to the value 0.916. One

checks that the same type of behavior occurs if we

repeat this exercise for cset to any value between 0 and

0.75. There is a marked change in behavior for c=

0.75, however—the system no longer converges to a

single value but rather oscillates between two values:

0.60 and 0.72. We say that the value c= 0.75 is a bifur-

cation point and that the system has undergone

“period doubling.”

At the value c= 1.25, the system bifurcates again

to yield systems that oscillate between four separate

values. For higher values of c, the system continues to

bifurcate, until finally a so-called

CHAOS

is reached,

where the results jump around in a seemingly haphaz-

ard manner. This phenomenon is typical of many

dynamical systems: the behavior they exhibit is highly

dependent on the value of some parameter c. (Such

dynamical systems are said to be “sensitive” to the

parameter set.)

Researchers have shown that many natural pro-

cesses that appear chaotic, such as the turbulent flow

of gases and the rapid eye movements of humans, can

be successfully modeled as dynamical systems, usually

with very simple underlying functions defining them.

Meteorologists model weather as a dynamical system,

which helps them make forecasts. However, extreme

sensitivity to parameters can easily lead to erroneous

predictions: one small change in the value of just one

parameter may produce very different outcomes. The

so-called butterfly effect, for instance, claims that the

minute changes in air pressures caused by a butterfly

flapping its wings might be all that is needed to tip a

meteorological dynamical system into chaos.

Iteration of functions with

COMPLEX NUMBERS

leads to a study of

FRACTAL

s.

1

–

2

dynamical system 151