Stable Limit Cycles as Tunable Signal Sources

Abstract: This paper presents a method for synthesizing audio signals from nonlinear dynamical systems exhibiting stable limit cycles, with control over frequency and amplitude independent of changes to the system's internal parameters. Using the van der Pol oscillator and the Brusselator as case studies, it is demonstrated how parameters are decoupled from frequency and amplitude by rescaling the angular frequency and normalizing amplitude extrema. Practical implementation considerations are discussed, as are the limits and challenges of this approach. The method's validity is confirmed experimentally and synthesis examples show the application of tunable nonlinear oscillator in sound design, including the generation of transients in FM synthesis by means of a van der Pol oscillator and a Supersaw oscillator bank based on the Brusselator.

Read the full paper here.

Supplementary Materials

Find the source code and Pure Data obects in the repo here.

FM Synthesis

• Sinusoidal (classical):
  200 Hz:
  400 Hz:
  500 Hz:
• Van der Pol oscillators as carriers with percussive envelope, enveloped µ:
  200 Hz:
  300 Hz:
  400 Hz:
  500 Hz:
  600 Hz:
• Van der Pol oscillators as carriers and modulators with percussive envelope, enveloped µ:
  200 Hz:
  300 Hz:
  400 Hz:
  500 Hz:
  600 Hz:

Supersaw

• Sawtooth (classical):
  50 Hz to 440 Hz and back, Detune from 0 to 0.8:
• Brusselator:
  50 Hz to 440 Hz and back, Detune from 0 to 0.8, A = 1, B = 6:
  50 Hz, Detune = 0.8, A = 1, B from 6 to 8 and back:
  220 Hz, Detune = 0.8, A = 2.2, B from 8 to 6 and back:
  110 Hz, Detune = 0.8, A = 2.2, B from 8 to 6 and back:
  440 Hz, Detune = 0.8, B = 6, A from 2.2 to 1 and back: