# Arnold's Cat Map — Chaotic Area-Preserving Map

Interactive Arnold Cat Map visualization: (x', y') = ((x+y) mod N, (x+2y) mod N). Iterations scramble the image, but after exactly K steps it restores perfectly — chaos with hidden order.

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

- **Category:** Math

## Overview

Interactive Arnold Cat Map visualization on a discrete N×N grid. The generalized Arnold map (x', y') = ((x+y) mod N, (x+2y) mod N) is an area-preserving Anosov diffeomorphism that scrambles any image into apparent noise, yet after exactly K iterations (the period) the image restores perfectly — demonstrating the coexistence of chaos and periodicity. Three visualization panels: (1) Main iterated image showing the current state after n applications of the cat map; (2) Original image for comparison; (3) Difference heatmap |current − original| using a blue→purple→red→yellow color scale showing pixel-wise deviation. Users can upload any image or use built-in patterns (cat face, color grid, gradient, checkerboard). Generalized map matrix A = [[1,a],[b,1+ab]] with adjustable parameters a and b — when det(A)=1 the map is area-preserving and periodic; when det(A)≠1 it is not. Grid size N (16–256) controls both resolution and the period K. The period K is computed by finding the smallest positive integer such that A^K ≡ I (mod N). Real-time statistics: iteration count, period K, determinant, eigenvalues. Four presets: Classic (1,1), Stretch (1,2), Strong (2,3), Mild (1,0). Step forward/backward, animation with adjustable speed, and jump-to-period button. Educational content covers Arnold's 1960 discovery, Anosov diffeomorphisms and uniform hyperbolicity, Lyapunov exponent λ ≈ 0.962, KAM theory connections, and applications in image encryption, digital watermarking, pseudo-random number generation, Hamiltonian dynamics, and chaotic cryptography. Multi-language support (zh, en, es, fr, de, ru, pt).

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