Neural Attractors

This project generates and visualizes beautiful chaotic attractors using simple feedforward neural networks with feedback, based on the paper "Artificial Neural Net Attractors" by J.C. Sprott. This is a revamped version of the project developed during the Nonlinear Dynamics course at NTU SPMS; an attractor generated by it won the CoSScienceArt competition.

Overview

The neural network architecture consists of:

• An input layer with D elements
• A hidden layer with N neurons
• A single output, which is fed back to the input

Because of the feedback loop, this NN becomes a nonlinear map capable of chaotic behavior. Input parameters (N, D, scaling factor s) define the time-series evolution of the system, and some sets of parameters lead to the phase space settling into a basin of attraction. These attractors might be point attractors, limit cycles, toroidal and strange (chaotic) attractors.

Installation

1. Clone the repository:

   git clone https://github.com/anyakors/neural_attractors.git
   cd neural_attractors

Usage

Basic Usage

Generate and visualize an attractor with default parameters:

python main.py

Customizing Neural Network Parameters

python main.py --N 4 --D 16 --s 0.75 --tmax 100000 --seed 3160697950

• --N: Number of neurons in the hidden layer (default: 4)
• --D: Dimension of the input vector (default: 16)
• --s: Scaling factor for the output (default: 0.75)
• --tmax: Number of iterations (default: 100000)
• --discard: Number of initial warm up iterations to discard (default: 1000)
• --seed: Random seed for reproducibility

Visualization Options

python main.py --skip-value 8 --figsize 16 16 --cmap viridis --linewidth 0.2 --alpha 0.15

• --skip-value: Number of points to skip for visualization (default: 16)
• --figsize: Figure size in inches (default: 12 12)
• --dpi: DPI for saved figures (default: 300)
• --cmap: Colormap for visualization (default: 'Spectral')
• --linewidth: Line width for visualization (default: 0.1)
• --alpha: Alpha (transparency) for visualization (default: 0.1)
• --interpolate-steps: Interpolation steps for visualization (default: 3)

Plotting Options

python main.py --plot-time-series --plot-scatter --plot-lyapunov --show

• --plot-trajectory: Plot the attractor trajectory (default: True)
• --plot-time-series: Plot the time series
• --plot-scatter: Plot a scatter plot of the attractor
• --plot-lyapunov: Calculate and plot the Lyapunov exponent
• --lyapunov-iterations: Number of iterations for Lyapunov exponent calculation (default: 5000)
• --no-plot: Don't create any plots
• --show: Display plots instead of saving them

Output Options

python main.py --output-dir images --prefix my_attractor --save-data

• --output-dir: Output directory (default: 'output')
• --prefix: Prefix for output files (default: 'attractor')
• --save-data: Save trajectory data to files

Batch Generation

Generate multiple attractors:

python main.py --count 10 --auto-filter --save-data

• --count: Number of attractors to generate (default: 1)
• --auto-filter: Filter out uninteresting attractors

Loading Existing Data

python main.py --load-data output/attractor_x1.npy output/attractor_x2.npy

Examples

Generate a high-quality attractor

python main.py --N 4 --D 32 --s 0.5 --tmax 1000000 --discard 10000 --skip-value 8 --figsize 16 16 --dpi 600

Calculate and plot the Lyapunov exponent

python main.py --plot-lyapunov --lyapunov-iterations 10000

Generate multiple attractors and save interesting ones

python main.py --count 20 --auto-filter --save-data

Create a series of attractors with varying parameters

for s in 0.2 0.3 0.4 0.5 0.6 0.7 0.8; do
    python main.py --s $s --prefix "attractor_s${s}" --save-data
done

Analyze how Lyapunov exponents change with parameter s

lyapunov_multi.sh

Lyapunov Exponent

Lyapunov exponent measures the rate of separation of two trajectories in a dynamical system that start infinitesimally close to each other. It indicates chaotic behavior in the system:

• Positive Lyapunov exponent: Indicates chaos. Nearby trajectories diverge exponentially over time.
• Zero Lyapunov exponent: Indicates stability or a bifurcation point.
• Negative Lyapunov exponent: Indicates stability. Nearby trajectories converge over time.

For neural network attractors, the Lyapunov exponent calculation:

1. Takes an initial state and a slightly perturbed initial state and runs the system forward
2. Measures the divergence between the trajectories
3. Computes the logarithm of this divergence rate

Example usage:

python main.py --N 4 --D 32 --s 0.8 --plot-lyapunov --lyapunov-iterations 8000

License

This project is licensed under the MIT License - see the LICENSE file for details.

Acknowledgments

Inspired by the original work by J.C. Sprott "Artificial Neural Net Attractors" published in Computers & Graphics, Vol. 22, No. 1, pp. 143-149, 1998 (doi.org/10.1016/S0097-8493(97)00089-7).