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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