## Mathematics Illuminated

# The Concepts of Chaos

## Most of us learned at an early age how an apple falling from a tree... inspired Isaac Newton to describe how the universe behaves by certain predictable rules. But what about when the universe doesn't behave so... predictably? Can mathematics explain the often unpredictable behavior of the physical world?

The flapping of a butterfly’s wings over Bermuda causes a rainstorm in Texas. Two sticks start side by side on the surface of a brook, only to follow divergent paths downstream. Both are examples of the phenomenon of chaos, characterized by a widely sensitive dependence of the future on slight changes in a system’s initial conditions. This unit explores the mathematics of chaos, which involves the discovery of structure in what initially appears to be randomness, and imposes limits on predictability.

### Unit Goals

- Chaos is a type of nonlinear behavior characterized by sensitive dependence on initial conditions.
- Chaotic systems can be deterministic, yet unpredictable.
- Linear systems are solvable because of superposition.
- Nonlinear systems are often not solvable in an exact sense.
- Phase space is a way to find the qualitative behavior of nonlinear systems.
- Equilibrium points can be stable or unstable.
- Sensitive dependence is easily seen in certain types of iteration procedures.
- The logistic map provides an example of a system that bifurcates into chaos in a relatively well-understood way.
- Space scientists use sensitive dependence to plan minimal-fuel routes through the solar system.

### Unit Glossary

### ADDITIONAL UNIT RESOURCES: BIBLIOGRAPHY

# Bibliography

## WEBSITES

http://ecommons.library.cornell.edu/handle/1813/97

Belbruno, Edward. *Fly Me to the Moon: An Insider’s Guide to the New Science of Space Travel.*Princeton, NJ: Princeton University Press, 2007.

Chernikov, Aleksander A.; Roald Z. Sagdeev, and Georgii M. Zaslavskii. “Chaos – How Regular Can it Be?” *Physics Today,* vol. 41, (November 1988).

Diacu, Florin and Philip Holmes.* Celestial Encounters: The Origins of Chaos and Stability*. Princeton, NJ: Princeton University Press, 1996.

Glass, Leon, Michael R. Guevara, Alvin Shrier, and Rafael Perez. “Bifurcation and Chaos in a Periodically Stimulated Cardiac Oscillator,” *Physica*, vol. 7, (1983).

Gleick, James. *Chaos: Making a New Science*. New York: Penguin Books, 1988.

Holland, John H. *Emergence: From Chaos to Order*. Reading, MA: Helix Books (Addison-Wesley Press), 1998.

Lo, M.W. “The Interplanetary Superhighway and the Origins Program” IEEE Aerospace Conference Proceedings, vol. 7, (2002).

Nolasco, J.B. and R.W. Dahlen. “A Graphic Method for the Study of Alternation in Cardiac Action Potentials,” *Journal of Applied Physiology,* vol. 25, no. 2 (1968).

Pikovsky, Arkady, Michael Rosenblum, and Jurgen Kurths. *Synchronization : A Universal Concept in Nonlinear Sciences. *Cambridge, New York: Cambridge University Press, 2001.

Smith, Douglas L. “Next Exit 0.5 Million Kilometers,” *Engineering and Science*, vol. 65, no. 4 (2002).

Stewart, Ian. *From Here to Infinity: A Guide to Today’s Mathematics*. Oxford, Great Britain: Oxford University Press, 1996.

Strogatz, Steven. *Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry and Engineering.* Cambridge, MA: Perseus Books Publishing, 2000.

Thornton, Marion. *Classical Dynamics of Particles and Systems*, 4^{th} ed. Orlando, FL: Saunders College Publishing (Harcourt Brace College Publishers), 1995.

Wang, Q.D. “Power Series Solutions and Integral Manifold of the N-Body Problem,” *Regular and Chaotic Dynamics, *vol. 6, no. 4 (2001).

Feynman, Richard. *The Character of Physical Law*. New York: Modern Library: 1994; p 108.

Poincaré, Henri (edited and introduced by Daniel L. Goroff) N*ew Methods of Celestial Mechanics*, vol. 1; Los Angeles, CA: American Institute of Physics: 1993; pp 110-111.