Phase Transitions in Coupled Nonlinear Oscillators Tanya Leise

Phase Transitions in Coupled Nonlinear Oscillators Tanya Leise Amherst College tleise@amherst. edu Materials available at www. amherst. edu/~tleise

Single Finger Oscillation

Single Finger Oscillation

Bimanual Oscillations Left hand Right hand Phase portrait: Left hand

Bimanual Oscillations Left hand Right hand Left hand

Bimanual Oscillations • Increasing frequency: Out-of-phase In-phase Transition

Basic Features 1. Only two stable states exist: in-phase and out-of-phase. 2. As the frequency passes a critical value, out -of-phase oscillation abruptly changes to inphase. 3. Beyond this critical frequency, only in-phase motion is possible.

Developing a Model • Goals: – To develop a minimal model that can reproduce these qualitative features – 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

Developing a Model • Control parameter: – Frequency w of oscillation (1 -6 Hz; divide by 2 for radians/sec) • Variables (describing oscillations): – Relative phase f of fingers (0º or 180º) – Amplitude r of finger motion (0 -2 inches)

Differential equation models . . . .

Nonlinear Oscillator • Include nonlinear damping term(s) to yield desired phase shifts as increases • Obtain “self-sustaining” oscillations if use negative linear damping term • Hybrid oscillator (Van der Pol/Rayleigh): . . . • Seek stable oscillatory solution of form

Single Finger Oscillatory Solution. .

Single Finger Oscillatory Solution. . .

Single Finger Oscillatory Solution. . . .

Coupled Nonlinear Oscillators . . .

Bimanual Oscillatory Solutions. . . .

Bimanual Oscillatory Solutions

Stability Analysis. .

Loss of Stability Leads to Phase Transition • Stability of the out-of-phase motion depends on the sign of the eigenvalue. • Increasing frequency beyond a critical value cr leads to change in stability of out-of-phase motion, triggering switch to in-phase motion

Energy Well Analogy • Potential function V( ) defined via . • Minima of V correspond to stable phases • Maxima of V correspond to unstable phases.

An Energy Well Model Slow twiddling frequency Fast twiddling frequency

Basic Twiddling Model Potential function V for phase difference : Stable states correspond to energy wells (minima of V): Conclusion: =0 is always a stable state (minimum for any a and b), while =p is a stable state only for parameter values B/A>1/4.

Sources • H. Haken, J. A. S. Kelso, and H. Bunz. A theoretical model of phase transitions in human hand movements. Biol. Cybern. , 51: 347 -356, 1985. • A. S. Kelso, G. Schöner, J. P. Scholz, and H. Haken. Phase-locked modes, phase transitions and component oscillators in biological motion. Physica Scripta, 35: 79 -87, 1987. • B. A. Kay, J. A. S. Kelso, E. L. Saltzman, and G. Schöner. Space-Time Behavior of Single and Bimanual Rhythmical Movements: Data and Limit Cycle Model. Journal of Experimental Psychology: Human Perception and Performance, 13(2): 178 -192, 1987.