Phase Transitions in Coupled Nonlinear Oscillators

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Phase Transitions in Coupled Nonlinear
Oscillators

• Tanya Leise
• Amherst College
• tleise_at_amherst.edu
• Materials available at www.amherst.edu/tleise

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Single Finger Oscillation
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Single Finger Oscillation
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Bimanual Oscillations
Left hand Right hand
Right hand
Phase portrait
Left hand
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Bimanual Oscillations
Right hand
Left hand Right hand
Left hand
Right hand
Left hand
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Bimanual Oscillations

• Increasing frequency

In-phase
Out-of-phase
Transition
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Basic Features

• Only two stable states exist in-phase and
  out-of-phase.
• As the frequency passes a critical value,
  out-of-phase oscillation abruptly changes to
  in-phase.
• Beyond this critical frequency, only in-phase
  motion is possible.

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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

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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)

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Differential equation models
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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

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Single Finger Oscillatory Solution
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Single Finger Oscillatory Solution
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Single Finger Oscillatory Solution
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Coupled Nonlinear Oscillators
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Bimanual Oscillatory Solutions
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Bimanual Oscillatory Solutions
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Stability Analysis
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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

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

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An Energy Well Model
Slow twiddling frequency
Fast twiddling frequency
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Basic Twiddling Model
Potential function V for phase difference f
Stable states correspond to energy wells (minima
of V)
Conclusion f0 is always a stable state (minimum
for any a and b), while fp is a stable state
only for parameter values B/Agt1/4.
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Sources

• H. Haken, J.A.S. Kelso, and H. Bunz. A
  theoretical model of phase transitions in human
  hand movements. Biol. Cybern., 51347-356, 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, 3579-87, 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.