## Mathematics Illuminated

# In Sync

## Many things in the universe behave in a synchronized way — whether manmade, or natural. We see synchronization as an emergence of spontaneous order in systems that most naturally should be disorganized. And when it emerges, there is a beauty and a mystery to it, qualities that often can be understood through the power of mathematics.

Systems of synchronization occur throughout the animate and inanimate world. The regular beating of the human heart, the swaying and near collapse of the Millennium Bridge, the simultaneous flashing of gangs of fireflies in Southeast Asia: these varied phenomena all share the property of spontaneous synchronization. This unit shows how synchronization can be analyzed, studied and modeled via the mathematics of differential equations, an outgrowth of calculus, and the application of these ideas toward understanding the workings of the heart.

### Unit Goals

- The mathematical study of spontaneous synchronization is an emerging field in the study of nonlinear dynamics.
- Spontaneous synchronization occurs in a wide variety of human, biological, and mechanical systems.
- Mathematical descriptions of synchronization require systems of coupled differential equations.
- Differential equations relate quantities and their rates of change.
- Differential calculus makes it possible to deal mathematically with non-constant rates of change.
- Entities that synchronize can be modeled as coupled oscillators.
- The Kuramoto model is a solvable system of coupled differential equations that can represent multiple related oscillators.
- The Millennium Bridge incident represented the interaction of the worlds of both biological and mechanical synchronization.

### Unit Glossary

### ADDITIONAL UNIT RESOURCES: BIBLIOGRAPHY

# Bibliography

## WEBSITES

http://math.nyu.edu/faculty/peskin/heartnotes/index.html

https://www.uaa.alaska.edu/physicsandastronomy/

http://www.arup.com/MillenniumBridge/

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Joseph, George Gheverghese. *Crest of the Peacock : The Non-European Roots of Mathematics*. Princeton, NJ: Princeton University Press, 2000.

Michaels, D.C., E.P. Matyas, and J. Jalife. “Mechanisms of Sinoatrial Pacemaker Synchronization: A New Hypothesis,” *Circulation Research*, vol. 61, no. 55 (November 1987).

Pantaleone, James. “Synchronization of Metronomes,” *American Journal of Physics*, vol. 70, no. 10 (October 2002).

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

Strogatz, Steven. *Sync: The Emerging Science of Spontaneous Order*. New York: Hyperion, 2003.

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.

Watts, Duncan. *Small Worlds: The Dynamics of Networks Between Order and Randomness*. (Princeton Studies in Complexity). Princeton, NJ: Princeton University Press, 1999.

Watts, Duncan. *Six Degrees: The Science of the Connected Age*. New York: W.W. Norton and Co., 2003.

Yeung, M. K. Stephen and Steven H. Strogatz. “Time Delay in the Kuramoto Model of Coupled Oscillators,” *Physical Review Letters*, vol. 82, no. 3 (January 1999).