Chaos Benchmark

A rigorous dataset generator for nonlinear dynamical systems.

I. Generator Configuration

Target Systems

Note: Rössler's Stable regime never appears in the output because the fixed point decays too slowly to register within the simulation window.

Parameters

Target number of successfully labeled trajectories for each system and regime (Stable, Periodic, Chaotic).

Failsafe limit to prevent infinite loops if a specific behavior is very rare.

Number of parallel background processes. Lower this if your computer is lagging.

II. Dynamical Systems

This tool simulates trajectories from five canonical chaotic systems. Labels (Stable, Periodic, Chaotic) are derived objectively via real-time calculation of the maximal Lyapunov exponent.

Logistic Map

$$x_{n+1} = r x_n (1 - x_n)$$

A classic demographic model of bounded population growth. Chaos arises via a period-doubling cascade as the parameter \(r\) increases, leading to dense orbits.

Hénon Map

\begin{align*} x_{n+1} &= 1 - a x_n^2 + y_n \\ y_{n+1} &= b x_n \end{align*}

A two-dimensional discrete-time dynamical system that maps a plane into itself. Famous for exhibiting a strange attractor with a beautiful fractal structure.

Lorenz System

\begin{align*} \frac{dx}{dt} &= \sigma(y-x) \\ \frac{dy}{dt} &= x(\rho-z)-y \\ \frac{dz}{dt} &= xy - \beta z \end{align*}

Originally formulated to model atmospheric convection rolls. The system is chaotic due to the stretching and folding of phase space around two repelling fixed points (the "butterfly").

Rössler System

\begin{align*} \frac{dx}{dt} &= -y-z \\ \frac{dy}{dt} &= x+ay \\ \frac{dz}{dt} &= b+z(x-c) \end{align*}

Designed by Otto Rössler to be the simplest possible continuous-time chaotic system, featuring only a single nonlinear term (\(zx\)).

Mackey-Glass

$$ \frac{dx}{dt} = \frac{\beta x(t-\tau)}{1+x(t-\tau)^n} - \gamma x $$

A time-delay differential equation modeling the physiological control of blood cell production. The time delay \(\tau\) technically creates an infinite-dimensional phase space.

III. Known Limitations

  • Rössler Stable Regime: The Stable bucket (\(c \in [1.0, 2.0]\)) will never produce successful rows. The mathematical fixed point is real, but it decays too slowly to register as fully settled within our finite simulation window.
  • Lorenz Periodic Regime: The Periodic hit rate is exceedingly low (~85% failure rate). Periodic windows in the Lorenz system are incredibly narrow in parameter space. High max_attempts is recommended if this class is required.

IV. Feature Extraction Glossary

For every generated trajectory, 13 scalar features are extracted to form the dataset:

mean, variance: Standard central tendency and dispersion.
skewness, kurtosis: Symmetry and tail-heaviness of the trajectory distribution.
lyapunov: Maximal Lyapunov exponent. Positive values strictly indicate chaos.
entropy: Shannon entropy of the binned trajectory values.
hurst: Hurst exponent, measuring long-term memory/autocorrelation.
corr_dim: Correlation dimension, a measure of fractal dimensionality.
lz_complexity: Lempel-Ziv complexity of the binarized signal.
num_peaks: Count of local maxima in the trajectory.
dom_freq: The dominant frequency extracted via FFT.
spectral_entropy: Entropy of the power spectrum (flatness).
bandpower: Total power in the signal spectrum.