## Chapter 2: The Stability of Two-Dimensional Maps

## Section 4.6: Stability Notions

Example :A Globally attracting but unstable pointConsider the following planar map

We demonstrate in the Phaser simulations below that the origin is globally attracting yet unstable. Notice that the map is not defined at the origin.

Figure 4.6.1.Four solutions of the map above. The yellow and dark blue solutions starting near the origin are moving away from the origin; thus, the origin is unstable. However, all four solutions evetually approach the origin.

Figure 4.6.2.Five hundred randomly chosen initial conditions iterated under the map. All solutions eventually approach the origin which signals the global attractivity of the origin.

Activities:

- Click on the first picture to load it into your local copy of Phaser.
- Set several more initial conditions by clicking the left mouse button near the origin. Clear and Go. (PhaserTip: Initial Conditions)
- Click on the second picture to load it into your local copy of Phaser. Select various FlowBox rectangles, especially near the origin to explore the dynamics of the map near the origin. Clear and Go. (PhaserTip: Flow)

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