Transcript Kuramoto Oscillators

PARADIGMS IN AUDIO-VISUAL
SYNCHRONICITY:
KURAMOTO OSCILLATORS
Nolan Lem
Center for Research in Music and Acoustics (CCRMA)
2
Kuramoto Oscillators and Collective Synchronization
• Models the collective synchronization of interacting systems.
• Developed by
Yoshiki Kuramoto (b. 1940)[1] who solved a basic model
proposed by Winfree [2] in 1967.
• Motivation was to characterize the behavioral dynamics involved in
chemical, biological, ecological systems found in nature.
• fireflies synchronization, circadian rhythms, laser arrays, brain waves,
cricket chirpings, pacemaker cells, superconduction, etc…
[1] Kuramoto, Yoshiki (1975). H. Araki, ed. Lecture Notes in Physics,
International Symposium on Mathematical Problems in Theoretical Physics 39. Springer-Verlag, New York. p. 420.
[2] A.T. Winfree, J. Theoret. Biol. 16 (1967) 15.
Collective synchronicity as an audio-visual medium
• Motivation: explore the range of auditory synchronicity within a
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specific model.
Can this model be used as a paradigm from which to generate selforganizing phenomena for sonic applications?
What types of parameterization are effective at communicating this
type of gesture?
To what extent can an auditor perceive of collective synch as
occurring over time (e.g. via entrainment for instance)?
What is the most interesting/effective way to audibly portray or induce
the this type of behavior?
Kuramoto Model Characterization
• Non-linear system that models system dynamics of phase-coupled oscillators.
• System Behavior is ultimately a function of:
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Limit-cycle oscillators whose natural initial frequencies are drawn from a
Gaussian distribution.
Coupling strength, Kn, determines whether system may change state (e.g.
phase transition, bifurcations).
• Conceptually: a
“swarm of points” running around a circle at different freqs.
[1]
System Transformation => Complex order parameter
• Equation [1] can be written in terms of a complex order parameter, r(t).
• E.g. This can be thought of as the “Collective rhythm” that is produced by
whole oscillator population. R is the vector radius of the circle that moves
at a frequency of Psi.
• [R, Psi] (t) enable a macro-level view of the model:
[2]
R(t) is a good indicator of phase coherence: how well the oscillators are
locking into the same phase. Psi(t) indicates average group velocity.
Model System states
• 3 Ways to Characterize System/Phase States:
1. Complete Incoherence (no phase locking)
2. Partial Synchronicity (partial phase locking)
3. Full Synchronicity – Phase Coherence (full phase locking)
Sonification Program Model Flow
• Processing with java libraries  generates .txt files of instantaneous phase
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per each oscillator.
Python Script to parse .txt data to generate plots of group phase, group
frequency, and r as a function of time. Forwards data to Chuck.
Chuck generates audio from parsed data
Can work in real-time…but examples here are not.
Example of running Processing….
Example 1
• 20 oscillators => 20 sinusoids
• Stft 01.png  01 phase-freq-r plot (mapping)
• 300Hz f_offset + delt_f(-150  150Hz)
• Starting half way through… oscillators are recruited to render a stable
coherence at freq ≈ 400 Hz.
• Choice of frequencies (e.g. mapping) is obviously very important if we are
trying to create the impression of coherence, convergence, divergence,
etc…more work needed here.
• In short, psychoacoustic phenomena are very different from our other sensory
modalities. Simply mapping one onto another does not always create the
same homologous affect.
Ex. 2
• 07.mov  07.pdf
• N = 30 oscillators
• Slower collective group synchronization. Descent in frequency
as group velocity decreases.
Future interests in project
• This is a simple model for conceptualizing group dynamics of coupled
systems.
• Most meaningful approach would be to use this model as a paradigm
for inducing ‘events’ that couldn’t be generated otherwise. Other
parameterization/mappings could be more interesting:
• Ex. Rhythmic convergences (density/timbre compositing), spatial
mapping (circular speaker array), send data to microcontrollers to
control physical systems/hardware, etc…