Module attractors 

Strange attractor implementations — RK4 integration, Lyapunov exponents, warmup.

All seven classical attractors are implemented with:

• 4th-order Runge-Kutta (RK4) integration for accuracy
• Lyapunov exponent approximation (numerical divergence tracking)
• Warmup / transient discard so initial noise is removed before sampling
• Normalised output bounding boxes (attractor fits ≈ unit cube)
• Bifurcation parameter sweeps for each attractor family

Structs

AttractorPool

A pool of attractor samplers — one per particle stream. Allows spawning particles from multiple independent trajectories.

AttractorSampler

Stateful sampler that integrates an attractor over time. Maintains the current state and advances it on each call to next().

AttractorStats

Statistical metadata about an attractor’s trajectory.

Enums

AttractorPalette

Colour palette for attractor visualisation.

AttractorType

Which strange attractor to use.

Functions

derivatives

Compute continuous-time derivatives at state s for the given attractor. Returns (dx/dt, dy/dt, dz/dt).

initial_state

Canonical initial conditions close to the attractor.

initial_state_warmed

Return a warmed-up initial state on the attractor surface.

kaplan_yorke_dimension

Kaplan–Yorke dimension from Lyapunov spectrum: D_KY = j + Σ(λ_i) / |λ_{j+1}|

largest_lyapunov_exponent

Compute the largest Lyapunov exponent numerically.

lorenz_bifurcation

Lorenz bifurcation diagram: sweep rho from rho_min to rho_max, return last-n states.

lorenz_parametric

Lorenz attractor with configurable parameters.

lyapunov_spectrum

Compute all three Lyapunov exponents via QR decomposition (Gram–Schmidt). Returns [λ1, λ2, λ3] in descending order.

poincare_section

Collect Poincaré section crossings (z = z_level, z going upward). Returns the x,y coordinates at each crossing.

recurrence_plot

Compute a recurrence matrix for a sampled attractor trajectory. matrix[i][j] = 1 if |state_i - state_j| < threshold. Returns a flat n × n bit vector.

rk4_step

Single RK4 step. More accurate than Euler at the cost of 4× function evaluations.

rossler_parametric

Rössler attractor with configurable parameters.

step

Evolve by one step and return (new_state, displacement). Uses RK4 for accuracy.

velocity_color

Map an attractor velocity magnitude to a colour using a gradient. Returns (r, g, b, a) all in [0.0, 1.0].

warmup

Warm up an attractor (discard the transient trajectory). Call before sampling to ensure the state is on the attractor.