# Hénon Map - Chaotic Visualization

Interactive exploration of chaotic dynamics, fractal structure, and strange attractors in the Hénon map system

> Canonical page: https://elysiatools.com/en/visualizations/henon-map

- **Category:** Math

## Overview

Interactive exploration of the Hénon map, a classic 2D discrete dynamical system exhibiting chaos and fractal structure. Discovered by Michel Hénon in 1976, this simple mapping demonstrates how deterministic nonlinear systems produce complex chaotic behavior. Features: Strange Attractor visualization with density-based coloring showing the iconic banana-shaped attractor, adjustable parameters a (0.8-1.5) and b (0.1-0.4) with classic chaotic values a=1.4, b=0.3, up to 500,000 iterations with real-time rendering, animation mode with parameter oscillation. Fractal Structure Explorer with interactive zoom (drag to pan, scroll to zoom) and auto-zoom animation, multiple color modes (velocity, iteration count, distance), self-similarity demonstration showing Cantor-set-like structure, box-counting dimension ≈1.26. Lyapunov Exponent calculation displaying chaos measure in real-time (λ>0 chaotic, λ≈0 periodic, λ<0 stable), bifurcation diagram visualization across parameter range showing period-doubling route to chaos. System Comparison tab contrasting Hénon map (discrete 2D, difference equations) vs Lorenz attractor (continuous 3D, differential equations) with side-by-side visualizations and feature comparison table. Mathematical Principles section covering iteration equations x_{n+1}=1-ax_n²+y_n, y_{n+1}=bx_n, Jacobian matrix analysis, fixed point calculation, bifurcation sequence (period-2 emerges at a≈1.06, chaos at a≈1.4), applications in chaos theory, cryptography, and signal processing. Educational content on chaos theory, strange attractors, sensitivity to initial conditions (butterfly effect), and fractal geometry. Multi-language support (zh, en, es, fr, de, ru, pt).

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