PHASER: Phaser Modules Discrete Chaos

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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