Chaos Engine

Simulate chaotic systems, attractors, and bifurcation.

Parameters

σ (sigma)

ρ (rho)

β (beta)

Steps

dx/dt = σ(y−x)
dy/dt = x(ρ−z)−y
dz/dt = xy−βz

r (Growth Rate) 3.700

xₙ₊₁ = r · xₙ · (1 − xₙ)

The Bifurcation diagram shows the route to chaos.

L₁ (m)

L₂ (m)

m₁ (kg)

m₂ (kg)

θ₁ (°)

θ₂ (°)

Chaos Theory & Nonlinear Dynamics Handbook

Mathematical Principles & Theorems

Nonlinear dynamical systems exhibit sensitive dependence on initial conditions. The Logistic Map \(x_{n+1} = r x_n(1 - x_n)\) models population growth with resource constraints; as \(r\) increases from 1 to 4, it undergoes period-doubling bifurcations at ratios governed by the universal Feigenbaum constant \(\delta \approx 4.6692016\). The 3D Lorenz attractor (\(\dot{x} = \sigma(y-x), \dot{y} = x(\rho-z)-y, \dot{z} = xy-\beta z\)) generates a strange attractor with non-integer fractal Hausdorff dimension.

Operating Instructions

Select a system: Logistic Map, Bifurcation Diagram, or Lorenz Attractor.
Set the parameter \(r\) (for Logistic map) or \((\sigma, \rho, \beta)\) (for Lorenz system) using precise value boxes.
Inspect the bifurcation tree showing period doubling cascades and chaotic bands.
Rotate the 3D Lorenz phase portrait to examine trajectory divergence and strange attractor geometry.