PHASER: Phaser Modules Discrete Chaos

Modules: Discrete Chaos

Chapter 2: The Stability of Two-Dimensional Maps

Section 4.7: Stability of Linear Systems

In Section 4.5 we have explored the geometry of the Phase Portraits, or Phase Space Diagrams, of two-dimensional linear maps of the form

(1)

where a, b, c. d, e, and f are parameters.We take e = f = 0 so that the origin is always a fixed point for all values of the parameters. Here, we will consider the cases when there is at least one eigenvalue of modulus 1 and explore the stability of the fixed point at the origin (non-hyperbolic fixed point).

Example: Nonhyperbolic Linear 2D Maps in Jordan Normal Form

Figure 4.7.1. Linear-2D MAP in Eq.(1) for a = 1, b = 1, c = 0, d = 1. Here, 1 is a double eigenvalue with a single eigenvector. The origin is unstable.

Figure 4.7.2. Linear-2D MAP in Eq.(1) for a = 1.0, b = 0, c = 0, d = 0.95. There is one positive eigenvalue with modulus less than 1 and one eigenvalue 1. The origin is stable. Notice the alternation of the orbits about the vertical axis.

Figure 4.7.3. Linear-2D MAP in Eq.(1) for a = 0.95, b = 0, c = 0, d = 1.0. There is one positive eigenvalue with modulus less than 1 and one eigenvalue 1. The origin is stable.

Figure 4.7.4. Linear-2D MAP in Eq.(1) for a = -1, b = 0, c = 0, d = 0.95. Here, -1 is an eigenvalue and the other eigenvalue is positive with modulus less than 1. The origin is stable. Notice the alternation of the orbits about the vertical axis.

Figure 4.7.5. Linear-2D MAP in Eq.(1) for a = 0.8, b = 0.6, c = -0.6, d = 0.8. The eigenvalues are complex with moduli 1. The origin is stable (center).

Activities:

Click on the first picture to load it into your local copy of Phaser. Set the the parameters to a = 1.0, b = 0.0, c = 0.0, d = 1.0. (PhaserTip: Changing Parameters) Clear and Go. What happens to the number of fixed points? Is the origin stable?
Click on the first picture again to load it into your local copy of Phaser. Set the the parameters to a = -1.0, b = 0, c = 0.0, d = 1.0. (PhaserTip: Changing Parameters) Clear and Go. Is the origin stable?
Click on the first picture again to load it into your local copy of Phaser. Set the the parameters to a = -1.0, b = 1.0, c = 0.0, d = -1.0. (PhaserTip: Changing Parameters) Clear and Go. Is the origin stable?

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