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Lorenz Attractor

Lorenz Attractor

Chaos theory classic with butterfly-like structure.

$$\dot{x}=\sigma(y-x),\quad \dot{y}=x(\rho-z)-y,\quad \dot{z}=xy-\beta z$$
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Barnsley Fern

Barnsley Fern

Iconic fractal via iterated function systems.

$$\text{IFS:}\ \begin{cases} f_1(x,y)=(0,0.16y) \\ f_2(x,y)=(0.85x+0.04y, -0.04x+0.85y+1.6) \\ f_3(x,y)=(0.2x-0.26y, 0.23x+0.22y+1.6) \\ f_4(x,y)=(-0.15x+0.28y, 0.26x+0.24y+0.44) \end{cases}$$
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Clifford Attractor

Clifford Attractor

Dense, swirling structures from simple recurrences.

$$x_{n+1}=\sin(a y_n)+c\cos(a x_n),\ y_{n+1}=\sin(b x_n)+d\cos(b y_n)$$
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De Jong Attractor

De Jong Attractor

Expressive non-linear map with rich textures.

$$x_{n+1}=\sin(a y_n)-\cos(b x_n),\ y_{n+1}=\sin(c x_n)-\cos(d y_n)$$
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Hénon Map

Hénon Map

A seminal discrete-time chaotic system.

$$x_{n+1}=1-a x_n^2 + y_n,\quad y_{n+1}=b x_n$$
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Ikeda Map

Ikeda Map

Optical cavity dynamics yielding intricate spirals.

$$t=\kappa-\frac{0.4}{1+x_n^2+y_n^2};\ x_{n+1}=1+u(x_n\cos t - y_n\sin t);\ y_{n+1}=u(x_n\sin t + y_n\cos t)$$
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Lissajous Curve

Lissajous Curve

Harmonic motion interference patterns.

$$x=A\sin(a t+\delta),\ y=B\sin(b t)$$
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Rössler Attractor

Rössler Attractor

Continuous-time chaos with ribbon-like flow.

$$\dot{x}=-(y+z),\ \dot{y}=x+a y,\ \dot{z}=b+z(x-c)$$
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Aizawa Attractor

Aizawa Attractor

Mesmerizing Aizawa attractor with cosmic colors and complex dynamics.

$$\dot{x}=(z-b)x-dy,\ \dot{y}=dx+(z-b)y,\ \dot{z}=c+az-\frac{z^3}{3}$$
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Dadras Attractor

Dadras Attractor

Electric Dadras attractor with neon lightning effect and dynamic patterns.

$$\dot{x}=y-ax+byz,\ \dot{y}=cy-xz+z,\ \dot{z}=dxy-ez$$
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Chen Attractor

Chen Attractor

Chaotic flow similar to Lorenz with distinct twisting structure.

$$\dot{x}=a(y-x),\ \dot{y}=(c-a)x-xz+cy,\ \dot{z}=xy-bz$$
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Four Wing Attractor

Four Wing Attractor

Four-winged continuous system creating dramatic lobes.

$$\dot{x}=ax+yz,\ \dot{y}=bx+cy-xz,\ \dot{z}=-z-xy$$
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Spirograph

Spirograph

Classic hypotrochoid/epitrochoid patterns in precise symmetry.

$$x=(R-r)\cos t + d\cos\!\left(\tfrac{R-r}{r}t\right),\ y=(R-r)\sin t - d\sin\!\left(\tfrac{R-r}{r}t\right)$$
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Julia Set

Julia Set

Complex quadratic iteration yielding infinite detail.

$$z_{n+1}=z_n^2+c$$
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Mandelbrot Set

Mandelbrot Set

The canonical fractal boundary in the complex plane.

$$z_{n+1}=z_n^2+c$$
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Newton Fractal

Newton Fractal

Basins of attraction emerging from Newton's method.

$$z_{n+1}=z_n-\frac{f(z_n)}{f'(z_n)},\ \ f(z)=z^3-1$$
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Tricorn Fractal

Tricorn Fractal

Conjugate-variant of the quadratic map with threefold symmetry.

$$z_{n+1}=\overline{z_n}^2+c$$
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Burning Ship

Burning Ship

Angular, ship-like silhouette with dramatic ridges.

$$z_{n+1}=(\lvert\operatorname{Re}(z_n)\rvert+i\lvert\operatorname{Im}(z_n)\rvert)^2+c$$
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Random Math Art

Random Math Art

Playful generative blend of trigonometric patterns.

$$x=\sin(at)+\sin(bt),\ y=\cos(ct)+\cos(dt)$$
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