Palle Dahlstedt, Joakim Linde and Mats G. Nordahl (2004), Chaotic Sounds in Coupled Oscillator Networks, Chalmers preprint.
Model
We consider a model consisting of n sinusoidal oscillators, each of which is phase modulated by a weighted sum of the outputs of the other oscillators. The output of oscillator i is denoted y_i(t), its phase theta_i(t), its frequency f_i, and the sampling rate f_0. In discrete time, we consider the following iterated coupled maps:
Connections between oscillators are generated randomly and independently with probability p. The couplings m_ij for the non-zero connections are drawn at random from a Gaussian distribution with zero average and standard deviation s. The diagonal couplings m_ii are all set to zero, so that an oscillator cannot modulate itself.
Visual representations of transitions from order to chaos
A visual representation of the dynamics is obtained through the use of delay embedding. Below are animations of transitions from order to chaos when the magnitude of the connection strength is increased for two different distributions of intrinsic frequencies. Click on the screenshots to view the animations.
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Sound samples from transitions to chaos
The sound samples in a series are all from the same network, but the coupling strength increases between samples. An estimate of the largest non-zero Lyapunov exponent for each sample is also listed in the tables below. In the first example in each series the couplings are quite weak, while the in the last one the output is mostly noise.Size 20 network:
| sound | lambda_1 |
|---|---|
| sample 1 | -1.79 |
| sample 2 | -1.13 |
| sample 3 | -0.74 |
| sample 4 | -0.41 |
| sample 5 | -0.17 |
| sample 6 | 0.02 |
| sample 7 | 0.20 |
| sample 8 | 0.33 |
| sample 9 | 0.44 |
Size 12 network:
| sound | lambda_1 |
|---|---|
| sample 1 | -1.61 |
| sample 2 | -0.90 |
| sample 3 | -0.47 |
| sample 4 | -0.22 |
| sample 5 | 0.03 |
| sample 6 | 0.24 |
| sample 7 | 0.39 |
| sample 8 | 0.52 |
In the examples below the coupling strength increases continuously throughout the sample.
