Phase Transitions in Coupled Nonlinear Oscillators

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

2. Single Finger Oscillation

4. Bimanual Oscillations

5. Bimanual Oscillations

6. Bimanual Oscillations Increasing frequency:

8. 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

10. Differential equation models

11. 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

15. Coupled Nonlinear Oscillators

16. Bimanual Oscillatory Solutions

18. Stability Analysis

19. 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 Loss of Stability Leads to Phase Transition

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