Chaos Engine

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.