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?

[ Previous Section | Next Section | Main Index | Phaser Tips ]