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


Snapshots


Details

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].

References/Notes

[1] M. Hénon, "A Two-Dimensional Mapping with a Strange Attractor," Communications in Mathematical Physics, 50(1), 1976 pp. 69–77.

[2] S. H. Strogatz, Nonlinear Dynamics and Chaos, New York: Perseus Books Publishing, 1994.

[3] K. T. Alligood, T. D. Sauer, and J. A. Yorke, Chaos: An Introduction to Dynamical Systems, New York: Springer, 1996.

[4] H.-O. Peitgen, H. Jurgens, and D. Saupe, Chaos and Fractals: New Frontiers of Science, 2nd ed., New York: Springer, 2004.

[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".

[6] 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.

[7] See the Wikipedia articles for "Jupiter's Great Red Spot" and "Saturn's Hexagon-Shaped Hurricane".



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