PHASER: Phaser Modules Discrete Chaos

Modules: Discrete Chaos

Chapter 2: The Stability of Two-Dimensional Maps

Section 4.12: Applications

Example: The Henon MAP
In this example we will investigate the dynamics of the famous Henon MAP

(1)

where a and b are two parameters. We will start with the classic parameter values a = 1.4 and b = 0.3 as chosen by Henon.

Figure 4.12.1. One iteration of a rectangle (consisting of 50,000 points) under the Henon MAP in Eq.(1) for a = 1.4, b = 0.3.

Figure 4.12.2. 100,000 iteration of the initial condition (0, 0) under the Henon MAP in Eq.(1) for a = 1.4, b = 0.3. The solution converges to the "strange attractor" of Henon.

Figure 4.12.3. A Phaser Gallery of the details of the Henon attractor. A region near the tip of the "parabola" is enlarged repeatedly. The fine details of the attractor reveal a fractal-like set.

Figure 4.12.4. A bifurcation diagram of the Henon Map while a is increased from 0 to 1.5 and b = 0.3 held fixed. Horizontal axis is the parameter a and the vertical axis is the variable x1.

Activities:

Click on the first picture to load it into your local copy of Phaser. To see the first two iterates of the initial rectangle, set Stop plotting = 2. Clear and Go. (PhaserTip: Time) Notice that the second iterate folds and stretches again. To see the second iterates only set Start plotting = 2 and Stop plotting = 2.
Click on the first picture again to load it into your local copy of Phaser. Click the left mouse button somewhere on the view and while holding the button select a rectangle and release the button; now you have selected a new FlowBox. Clear and Go. You should see your new rectangle and its image drawn. Repeat the process to get a feeling for the action of the Henon MAP. (PhaserTip: Flow)
Click on the second picture to load it into your local copy of Phaser. Set several more initial conditions by clicking the left mouse button. Clear and Go. (PhaserTip: Initial Conditions) Notice that (most likely) all solutions move to the same attactor. Can you find a solution that does not converge to the attractor?
Click on the third Gallery picture to load it into your local copy of Phaser. Play the Slideshow to zoom into attractor. (PhaserTip: Slideshow).
Click on the first picture to load it into Phaser. Set the parameter to a = 0.7. (PhaserTip: Changing Parameters) Clear and Go. Notice the change in the image of the square.
Click on the secondt picture to load it into Phaser. Set the parameter to a = 0.7. (PhaserTip: Changing Parameters). What happens to the attractor? Leave out transients to see the attractor. Try other values of the parameters. Can you locate the value of a for which there is an attractor of period-7 orbit?

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