1. Coupled Oscillators: Joggers, Fireflies, and Finger Coordination
Tanya Leise
Amherst College
Friday Dec 2 at noon
Abstract:  I'll describe a simple oscillator model that can describe the position of a jogger going around a circular track, a firefly blinking on and off, or the motion of a finger waving back and forth.  We can add a second oscillator to model two joggers, two fireflies, or two fingers trying to coordinate their motion.  I'll demonstrate how to use some basic analysis involving derivatives to analyze the various scenarios, emphasizing the role of derivative as rate of change.

2. Jogger on a Circular Track
Looking down on track from above, watching jogger’s position on track as the blue dot

3. Jogger on a Circular Track
We assume the jogger runs on a circular track and represent the jogger’s position as an angle ϕ and the jogger’s speed as an angular rate of change dϕ/dt.

4. Two Joggers on a Track

5. Two Joggers Trying to Jog Together
Look at the problem from the point of view of the “red” jogger:
“If the blue jogger is behind me, I should slow down.”
“If the blue jogger is ahead of me, I should speed up.”
To model this, we want to adjust the red jogger’s speed using an appropriate function of their relative positions:
C is the coupling strength: how much effort put into matching positions

6. Two Joggers Trying to Jog Together
“If the blue jogger is behind me, I should slow down.”
Increase speed to catch up to jogger in front
Decrease speed to let jogger behind catch up
“If the blue jogger is ahead of me, I should speed up.”

7. Two Joggers Matching Speeds on a Track
Now the joggers want to jog together, need to match their speeds; red is faster than blue
Can we make the coupling strong enough to make the angle difference zero? (Small C may be too weak to overcome difference in normal running speeds)

8. Two Joggers Matching Speeds
Find equilibrium where relative position is maintained over time

9. Fireflies
Certain species of fireflies try to flash simultaneously: synchronized flashing.
Model synchronization of a population of N identical oscillators, n=1,2,…,N:
Angle phi keeps track of position in flashing cycle (say, flash when phi=0 on circle, otherwise waiting and “counting down” to next flash
(Kuramoto model with mean field coupling)

10. Synchronization of Fireflies

11. Synchronized Fireflies
Orange=flashing in movie
Want to measure how clustered fireflies’ phases are

12. Synchronization Index R
Picture represents a particular time t

13. Synchronized Fireflies

14. Single Finger Oscillation
Move index (pointer) finger up and down

15. Single Finger Oscillation
Now go back and forth more and more rapidly

16. Coupled Oscillations Left hand Right hand
Explain basic idea, then have people increase frequency, starting with out-of-phase motion

17. Bimanual Oscillations
Increasing frequency of motion:
Out-of-phase
In-phase
Transition

18. Developing a Model Goals:
To develop a minimal model that can reproduce the qualitative features of this experiment
To gain insight into underlying neuromuscular system (how both flexibility and stability can be achieved)
“Nature uses only the longest threads to weave her pattern, so each small piece of the fabric reveals the organization of the entire tapestry.”
R.P. Feynman

19. A Minimal Model
Relate to previous coupling function: -sin(difference in angle). Same here, but adding additional term. Also, coupling strength A varies with frequency omega (control parameter)
See my paper in the Monthly for the derivation and analysis of the complete Haken-Kelso-Bunz model

20. Analysis of Minimal Model
Phi-stars are critical points of V(phi)

21. Analysis of Minimal Modelp
Stable=local minimum of V
Unstable=local maximum of V

22. An Energy Well Model Slow twiddling frequency Fast twiddling frequency
Use potential function approach for graphical interpretation
Slow twiddling
frequency
Fast twiddling
frequency

23. Explanation for Dynamics
Increasing frequency w of motion:
Out-of-phase
In-phase
As w increases, A(w) decreases.
Once A(w) falls below ¼, out-of-phase motion becomes unstable so system switches to in-phase motion.

24. Further Reading
H. Haken, J.A.S. Kelso, and H. Bunz. A theoretical model of phase transitions in human hand movements. Biol. Cybern. 51: , 1985.
T. Leise and A. Cohen. Nonlinear oscillators at our fingertips. American Mathematical Monthly 114(1):14-28, 2007.
S. Strogatz. Human sleep and circadian rhythms: a simple model based on two coupled oscillators. J. Math. Biol. 25: , 1987.
S. Strogatz. Nonlinear dynamics and chaos, 1994.
S. Strogatz. From Kuramoto to Crawford: exploring the onset of synchronization in populations of coupled oscillators. Physica D 143:1-20, 2000.
S. Strogatz. SYNC: The emerging science of spontaneous order, 2003.
Synchronous fireflies,