Playing with the Hénon Map Starting with a Circle or a Square
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This Demonstration shows iterates of the dissipative Hénon map [1–4], , where and are real, acting on 500 equally spaced initial points on a circle or a square. You can drag the crosshairs-shaped locator to change the position of .
Contributed by: Ki-Jung Moon (January 2014)
Open content licensed under CC BY-NC-SA
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 . 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 .
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].
 M. Hénon, "A Two-Dimensional Mapping with a Strange Attractor," Communications in Mathematical Physics, 50(1), 1976 pp. 69–77.
 S. H. Strogatz, Nonlinear Dynamics and Chaos, New York: Perseus Books Publishing, 1994.
 K. T. Alligood, T. D. Sauer, and J. A. Yorke, Chaos: An Introduction to Dynamical Systems, New York: Springer, 1996.
 H.-O. Peitgen, H. Jurgens, and D. Saupe, Chaos and Fractals: New Frontiers of Science, 2nd ed., New York: Springer, 2004.
 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".
 M. Michelitsch and O. E. Rössler, "A New Feature in Hénon's Map," Computers & Graphics, 13(2), 1989 pp. 263-275. Reprinted in Chaos and Fractals, A Computer Graphical Journey: Ten Year Compilation of Advanced Research (C. A. Pickover, ed.), Amsterdam, Netherlands: Elsvier, 1998 pp. 69-71.
 See the Wikipedia articles for "Jupiter's Great Red Spot" and "Saturn's Hexagon-Shaped Hurricane".