Hénon figure

Hénon mapping for the parameters a = 1.4 and b = 0.3: horseshoe-shaped structure

The French astronomer Michel Hénon found the following two-dimensional map, the so-called Hénon map , which was derived from a simplification of the Poincaré map belonging to the Lorenz equation . The Lorenz equation originally came from meteorology and was one of the first dynamic systems in which chaotic behavior was found. The Hénon mapping is described by:

${\ displaystyle {\ begin {matrix} x_ {n + 1} = 1 + y_ {n} -ax_ {n} ^ {2} \\ y_ {n + 1} = bx_ {n} \ end {matrix}} }$

with . Here a and b are the control parameters of the system. ${\ displaystyle a> 0.0 \ leq b \ leq 1}$

The Hénon map is made up of a total of three elementary maps with k = 1,2,3: ${\ displaystyle Abb_ {k}}$

non-linear distortion of the y-coordinate: , ${\ displaystyle Abb_ {1} (x, y) = (x, 1 + y + ax ^ {2})}$
Contraction of the x-coordinate: for 0 <b <1, ${\ displaystyle Abb_ {2} (x, y) = (bx, y)}$
Mirroring x on the main diagonal y =: . ${\ displaystyle Fig_ {3} (x, y) = (y, x)}$

It should be mentioned that another important property of this mapping is self-similarity . In simple terms, it means a fractal enlargement of any sub-area that is again similar to its initial object. The attractor of the Hénon map is a strange attractor .

Individual evidence

↑ ^a ^b Kenneth J. Falconer: Fractal Geometry - Mathematical Foundations and Applications . Spektrum Akademischer Verlag, 1993, ISBN 3-86025-075-2 , pp. 207-209 .

[falconer1993-1] Kenneth J. Falconer: Fractal Geometry - Mathematical Foundations and Applications . Spektrum Akademischer Verlag, 1993, ISBN 3-86025-075-2 , pp. 207-209 .