Modules: Discrete Chaos

Chapter 2: The Stability of Two-Dimensional Maps

Section 4.11: Stability via Linearization

Example: Pielou Logistic Delay MAP

In this example we will investigate the stability of the fixed points of the map

  (1)

where a and b are two parameters with positive values. Since this is a population model, we will consider the dynamics of this system in the first quadrant only. This system has two equilibria with the following stability types as determined from linearization:

  • (0, 0) is asymptotically stable if a < 1, unstable if a > 1,
  • ((a-1)/b, (a-1)/b)) is asymptotically stable if a > 1. When a < 1, this fixed point is not relevant biologically because it has negative coordinates. When a = 1, the two fixed points coincide.

 


Figure 4.11.1. Phase portrait of the Pielou Logistic Delay MAP in Eq.(1) for a = 2.5, b = 2.0. Note that the positive fixed point is asymptotically stable.

 


Figure 4.11.2. Phase portrait of the Linear-2D MAP corresponding to the linearization at the positive fixed point in the previous figure.

 


Figure 4.11.3. A Gallery of Phase portraits of the Pielou Logistic Delay MAP while a is increased from 0 to 3 and b = 2.0 held fixed.

 

Activities:

  • Click on the first picture to load it into your local copy of Phaser. Move your mouse cursor (without clicking) and determine the coordinates of the positive fixed point. (PhaserTip: Cursor Coordinates)
  • Change the parameter b = 3.0 while keeping a fixed (PhaserTip: Changing Parameters). What changes do you observe in the phase portrait?
  • Click on the second picture to load it into your local copy of Phaser. Change parameters in the Linear-2D MAP so as to get the linearization at the origin. Compare the phase portraits of the linear and the nonlinear maps near the origin.
  • Click on the first picture to load it into Phaser. Set the parameter to a = 1. (PhaserTip: Changing Parameters) Clear and Go. What is the stability type of the fixed point at the origin?
  • Click on the third Gallery picture to load it into Phaser. Play the sequence of phase portraits as the parameter a is varied from 0 to 3 while b is fixed at b = 2.0 (PhaserTip: Slideshow)

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