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The Quadratic Map
This is a general form of a two-dimensional iterated map where the next position is determined by a quadratic function of the current position. Finding parameters that result in chaotic behavior can be difficult, so a gallery of interesting pre-sets is provided.
Equations
- x_new = a + bx + cx² + dxy + ey + fy²
- y_new = g + hx + ix² + jxy + ky + ly²
The Pickover System
Named after Clifford Pickover, this system uses trigonometric functions to create intricate, biomorphic, and often surprising patterns from very simple equations. The resulting images can resemble everything from ghostly apparitions to complex organic structures.
Equations
- x_new = sin(b*y) + c*cos(b*x)
- y_new = sin(a*x) + d*cos(a*y)
The Hénon Map
Introduced by Michel Hénon, this is one of the most studied discrete-time dynamical systems that exhibits chaotic behavior. Despite its simplicity, it generates a structure of surprising complexity, showing how points stretch and fold back onto themselves.
Equations
- x_new = 1 - a*x² + y
- y_new = b*x
The De Jong Attractor
This attractor, defined by Peter de Jong, uses a set of trigonometric equations to produce flowing, organic, and often ethereal shapes. By changing the four parameters, a huge variety of visually stunning patterns can be explored.
Equations
- x_new = sin(a*y) - cos(b*x)
- y_new = sin(c*x) - cos(d*y)