Capacity Dimension of The Hénon Attractor Eui-Sun Lee Department of Physics Kangwon National University  Hénon Map  Chaotic Attractor with A Positive Lyapunov Exponent (0.367) for b=0.3 and A=2.1.

Geometrical Structure of The Hénon Attractor 1st close-up 2nd close-up Successive Magnification  Self-Similar Structure on All Scales  Hénon Attractor: Strange Attractor with Fractal Structure

Capacity Dimension  Definition To characterize a strange attractor, let’s cover it with boxes of length  . As the length  is decreased, the number of boxes N( ) obeys a power-law scaling. In the limit  0, the exponent characterizing the strangeness of the attractor, is called the Capacity Dimension.

 Plot of log10 N() vs. log10  -1 for the Hénon Attractor The slope of the graph gives the capacity dimension D0(=1.2690.009) of the Hénon attractor.

Summary The Hénon attractor is a strange attractor with the capacity dimension D0(=1.2690.009). Since 1 D0  2, it has a fractal geometry that is intermediate between a line and a plane.