Here are some basic notions for understanding this program and the above screenshots:

1. The Jacobian matrix

of the dissipative Hénon map

is given by

, where

.

2. Therefore the determinant of the Jacobian matrix is

.

3. For

, the Hénon map is area-contracting; for

, area-preserving; and for

, area-expanding. (See the screenshots 5–7.)

4. For

, something interesting happens because it is the conservative limit of the dissipative Hénon map. By solving fixed point equations of the first and the second iterated Hénon map with

, it is easy to find that there are two types of solutions, two hyperbolic solutions

with period 1 (THS) and two elliptic solutions

with period 2 (TES). For

, the THS is approximately

, while the TES is approximately

, as shown in the fourth screenshot.

5. By carefully locating the points on the initial circle to pass through one of the TES (i.e. either

or

) you can see two beautiful yin-yang-like spirals (one is located at the top-left and the other at the bottom-right), which are shown in the first screenshot [5]. Here the radius

of the initial circle is given by

,

where

. Since this is the conservative limit of the dissipative Hénon map, these yin-yang-like spirals near the TES can exist forever without contracting or expanding.

6. The shape and the color of these spirals become very close to those of the true yin-yang spiral by imagining that the initial circle is filled in with black dots [5].

7. This numerical experiment may give us some hints about why Jupiter's red spot and Saturn's hexagon-shaped hurricane seem to exist forever without contracting or expanding [6–7].

[1] M. Hénon, "A Two-Dimensional Mapping with a Strange Attractor,"

*Communications in Mathematical Physics*,

**50**(1), 1976 pp. 69–77.

[5] In Chinese philosophy, the concept of yin-yang, which is often called "yin and yang", is used to describe how seemingly opposite or contrary forces are interconnected and interdependent in the natural world; and, how they give rise to each other as they interrelate to one another. For more detailed information, see the Wikipedia article for "

Yin and Yang".