Here are my slides (though the animated gifs on a couple of them are stills in this version) of my talk on an introduction to the science of "chaos" at WorldCon 77 in Dublin, Ireland. This is my attempt to give a gentle introduction to the notion of chaos to a science-fiction audience.

1. ”
Mason A. Porter (@masonporter)
Professor of Mathematics
University of California, Los Angeles
The Science of “Chaos”

2. Main Goal of the Talk
Answer the following question:
What is chaos?

3. The Butterfly Effect
“The fluttering of a butterfly’s wing in Rio de
Janeiro, amplified by atmospheric currents, could
cause a tornado inTexas two weeks later.”
— Edward Lorenz (1979), as quoted in: Laura Nader
(1996), Naked Science: Anthropological Inquiry Into
Boundaries, p. 209

4. The Lorenz Attractor
(animated figure by Dan Quinn)

5. Double Pendulum
(animated figure by 100Miezekatzen)

6. “Jurassic Park” (1990 book; 1993 film)
Dr. Ian Malcom: “See?That's exactly what I'm saying. Nobody
could have predicted that Dr. Grant would just jump out of a
moving vehicle. [Ellie opens the car door and follows
Grant]There's another example. Now I'm here by myself, uh,
talking to myself.That, that's ChaosTheory.”
­ [Narrator Mason: No, it’s not.]
­ Note: Earlier in this scene, Dr. Malcom talks about small variations leading
to large change in outcomes.That is a hallmark of chaotic behavior, but
then the scene goes off the rails (at least as far as chaos is concerned)
­ Note 2: I have heard rumors that Dr. Malcom may have been inspired
partly by GeoffWest (Santa Fe Institute). I know Geoff, and this makes me
smile. :P

7. “The Butterfly Effect” (2004)
The so-called ”butterfly effect” — more properly, “sensitive
dependence (on initial conditions)” — is illustrated in this movie
through small changes in decisions or actions having enormous
(and seemingly unpredictable) effects on events in the movie.
Conceptually, this is basically fine, but the mathematical notion
of chaos is that unpredictability is a hallmark of “chaos” in
deterministic systems (so that we know that randomness isn’t a
cause).
­ Including in seemingly very “simple” deterministic systems

8. Double Pendulum
(figure by 100Miezekatzen)

9. Nice Animations of Chaotic Pendula
(by Robert Ghrist, Professor of Mathematics,
University of Pennsylvania)
Rob Ghrist’s animation of chaos in a double pendulum:
https://www.youtube.com/watch?v=bXxCxhZCCUA&list=PL8erL
0pXF3JYiEz1b8eLPq49aDeMBF99C&index=6
Rob Ghrist’s animation of chaos in a triple pendulum:
https://www.youtube.com/watch?v=EuPZ9TVxMHo&list=PL8erL
0pXF3JYiEz1b8eLPq49aDeMBF99C&index=8

10. Chaotic Mixing in Fluids (figure by Peter Mills)

11. OK, so what is Chaos?
Chaos = “sensitive dependence on initial conditions” + “recurrence”
Sensitive dependence (the so-called “butterfly effect”): if you change the starting
situation (such as the starting position and velocity of a double pendulum), no
matter how small the change, you eventually get entirely different behavior
­ Even if you try to do the exact same setup side-by-side (such as with two double pendula),
arbitrarily small differences (e.g., imperfections in setting up the experiment) will be magnified,
eventually leading to totally different behavior of the two objects
­ Therefore: even a “simple” system can be entirely deterministic, yet still be unpredictable after a short period of time
­ Paradigm changing for science!
Recurrence: Fix a region in ”phase space” (such as a region of possible positions
and velocities for a double pendulum); the system will pass through this region
infinitely many times
­ This condition is to discount situations in which a system “escapes”, which allows a system to have
sensitive dependence but without chaos

12. Another way to think of Chaos:
“sensitive dependence” + “mixing”
Sensitive dependence: “butterfly effect”
Mixing: Start with small drop of yellow dye in
a blue liquid (or a large number of infinitely
small rubber ducks, which we start off at the
location of the blob of dye). Measure how
the dye (and the rubber ducks) spreads
around the container of fluid and how many
parts (at ever-smaller scales) of the container
has been reached by the dye, with ever-
smaller filaments of dye over time.
­ Put cream in coffee and watch what happens

13. “Billiard Systems”

14. Chaotic Waterwheel
Many videos online of this famous example
One of them:
https://www.youtube.com/watch?v=7A_rl-DAmUE

15. “Chaos” versus “Nonlinear Dynamics”
“Nonlinear dynamics” (or “nonlinear science”) is the name of the subject that is often
(misleadingly) called “chaos theory”
“Chaos” is a type of complicated behavior — actually, there are several different types of
chaos — that can occur in a “nonlinear” system.
­ Note: most systems in real life are nonlinear
In a nonlinear system, (to put it loosely) adding the results of two inputs is different from
first adding two inputs and obtaining the result of that
­ Abstractly, think of this as F[x1] + F[x2] ≠ F[x1 + x2], where ‘F’ is whatever the nonlinear system does
MyQuora answer on what are nonlinear and complex systems:
­ https://www.quora.com/How-do-I-explain-to-non-mathematical-people-what-non-linear-and-
complex-systems-mean

16. Summary
Chaos is a rich, complicated type of behavior that can occur in
nonlinear systems (and most systems in real life are nonlinear).
­ Seemingly very simple systems, like a double pendulum, can behave in a
very complicated way!
Fundamental features of chaotic behavior
­ Sensitive dependence on initial conditions + recurrent behavior
­ Change the starting conditions slightly and the behavior is completely different
­ Recurrent behavior, combined with sensitive dependence, is what yields complicated (“chaotic”)
behavior, rather than just something that escapes.
­ Alternatively: Sensitive dependence + mixing
­ Notions of mixing, which make sense for systems like fluids, are another way to think about
recurrent behavior
­ Think of what happens to cream when you put it in coffee and it spreads throughout the
container

17. ”
Mason A. Porter (@masonporter)
Professor of Mathematics
University of California, Los Angeles
The Science of “Chaos”