Modules: Discrete Chaos

## Chapter 1: The Stability of One-Dimensional Maps

### Section 1.6: Graphical Iteration and Stability

Example 1.7: A Quadratic MAP

Consider the quadratic map of the form (1)

where c is a parameter in the range [-2, 0]. In the figure below, for c = -0.5, the graph of the map, 45-degree line and two solutions are plotted. Figure 1.6.1. Quadratic MAP in Eq.(1) for c = -0.5. Notice that there are two fixed points, the points of intersections of the graph and the 45-degree line. The fixed point on the left is asymptotically stable, while the fixed point on the right is unstable.

Activities:

• Click on the picture to load it into your local copy of Phaser.
• Move your mouse cursor (without clicking) and determine the coordinates of the two fixed points. (PhaserTip: Cursor Coordinates)
• Set several more initial conditions by clicking the left mouse button near the fixed points. Clear and Go. (PhaserTip: Initial Conditions)
• Set the parameter to c = -2. (PhaserTip: Changing Parameters) Clear and Go. What happens to the stability types of the fixed points?

Example 1.8: The Tent MAP

In this example, we will examine the piece-wise linear Tent MAP of the form where a is a parameter. In the figure below, the graph of the Tent MAP for the parameter value a = 2.0, the 45-degree line, and two solutions are plotted. Figure 1.6.2. Two fixed points of the Tent MAP (2) for a = 2.0. Notice that both fixed points are unstable.

Activities:

• Click on the picture to load it into your local copy of Phaser.
• Move your mouse cursor (without clicking) and determine the coordinates of the two fixed points. (PhaserTip: Cursor Coordinates)
• Set several more initial conditions by clicking the left mouse button near the fixed points. Clear and Go. (PhaserTip: Initial Conditions)
• Set the parameter to a = 1. (PhaserTip: Changing Parameters) Clear and Go. What happens to the number and the stability types of the fixed points?
• Set the parameter to a = 0.5. (PhaserTip: Changing Parameters) Clear and Go. What happens to the number and the stability types of the fixed points?

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