Is Chaos Predictable?
By: Aga Freund
What Is Chaos Theory?
 “Chaos theory attempts to explain the fact that complex and
unpredictable results can and will occur in systems that are sensitive to their
initial conditions. A common example of this is known as the Butterfly
Effect. It states that, in theory, the flutter of a butterfly's wings in China
could, in fact, actually effect weather patterns in New York City, thousands
of miles away. In other words, it is possible that a very small occurance can
produce unpredictable and sometimes drastic results by triggering a series of
increasingly significant events.” – from http://library.thinkquest.org/3120/
History of Chaos
 Meteorologist, Edward Lorenz, was working on weather prediction problem
in 1960, using a computer set up with a set of twelve equations modeling the
weather. It predicted theoretically what might be the weather and not what it
was. In 1961, he went back to a specific sequence, starting in the middle and let
it run. After one hour, the sequence developed otherwise. Ended up totally
different than the original. Difference: computer saved to six decimal places, he
did to three on the printout to save the paper.
History of Chaos Cont.
 Lorenz proved that digits after the third decimal place can have a big effect on
the outcome of the experiment, which became to be known as the butterfly effect.
Meaning, that small events can have huge effect on changing the prediction of longterm behavior. This phenomenon is also known as sensitive dependence on initial
conditions. So, he indicated that predicting the weather accurately is impossible,
which led him to eventually develop chaos theory. He came down from twelve to
three equations in the system (Lorenz system), of which the outcome always stayed
on a curve, double spiral. He called the image he got the Lorenz attractor.
Lorenz System:
𝑑𝑥
𝑑𝑡
=𝜎 𝑦−𝑥
𝑑𝑦
𝑑𝑡
=𝑥 𝜌−𝑧 −𝑦
𝑑𝑧
𝑑𝑡
= 𝑥𝑦 − 𝛽𝑧
x, y, z − the system state
t − time
σ, ρ, β − the system parameter
Mathematics of Chaos
 Scientists and mathematicians started to play with plotting and exploring
equations. It produced nature-like looking pictures (ferns, clouds, mountains,
and bacteria). They acted the same as stock change, populations, and
chemical reactions simultaneously. The theory had to do with lots of
different intellectual domains, so they began plotting fractals. Chaotic systems
have defining features and are not random.
 Fractal geometry, described in algorithms, explains chaotic systems found
in nature.
Defining Features of Chaotic Systems
1. Deterministic – something determines their behavior.
2. Very sensitive to initial conditions, making the system fairly
unpredictable.
3. Appear disorderly or random, but are not.
Animated Fractals: http://library.thinkquest.org/3120/library.html
Real World & Chaos
 Analysis of chaos indicated that the market prices, while highly random,
have a trend, the amount of which differs from market to market and from
time frame to time frame.
 http://library.thinkquest.org/3120/realife.html
 The coastline of Great Britain is infinite
 http://library.thinkquest.org/3120/realife.html
 Long range weather forecasting is not possible to be completely right due
to effects of chaos.
Example
 George T. Yurkon tried to
use eight significant digits on
initial conditions, however, the
butterfly turned out lopsided.
𝑑𝑥
𝑑𝑡
=𝜎 𝑦−𝑥
𝑑𝑦
𝑑𝑡
=𝑥 𝜌−𝑧 −𝑦
𝑑𝑧
𝑑𝑡
= 𝑥𝑦 − 𝛽𝑧
𝜎 = 10, 𝜌 = 28, 𝛽 =
2
3
Example
𝑑𝑥
𝑑𝑡
=𝜎 𝑦−𝑥
𝑑𝑦
𝑑𝑡
=𝑥 𝜌−𝑧 −𝑦
𝑑𝑧
𝑑𝑡
= 𝑥𝑦 − 𝛽𝑧
Example
2
𝜎 = 10, 𝜌 = 28, 𝛽 = 3
http://www.csuohio.edu/sciences/dept/phys
ics/physicsweb/kaufman/yurkon/chaos.html
Some Examples of Chaos
 The Cantor Set
 The Sierpenski Triangle and its Area
 The Mandelbrot Set
 The Julia Set
 http://library.thinkquest.org/2647/chaos/chaos.htm
Conclusion
 I think that chaos can be predicted to happen in some cases.
However, its outcome cannot be calculated precisely. The range for
the outcome to occur in, though, is possible to be calculated using
the Lorenz System’s three differential equations.
Questions
Resources -Websites Used
 http://library.thinkquest.org/3120/
 http://library.thinkquest.org/3120/history.html
 http://en.wikipedia.org/wiki/Lorenz_system
 http://library.thinkquest.org/3120/history.html
 http://library.thinkquest.org/3120/library.html
 http://library.thinkquest.org/3120/realife.html
 http://www.csuohio.edu/sciences/dept/physics/physicsweb/kaufman/yurkon/chaos.html
 http://library.thinkquest.org/2647/chaos/chaos.htm