Modules: Discrete Chaos

## Chapter 2: The Stability of Two-Dimensional Maps

### Section 4.8: The Trace-Determinant Plane

Example 4.7:

In this example we will investigate the dynamics of the Linear-2D MAP (1)

while the parameter b is varied and the the remaining parameters a = -1, c = -2, d = 1, e = 0, f = 0 are held fixed. Since the orbits of a map are a sequence of discrete points, unlike orbits of differential equations which are differentiable curves, interpreting simulations of difference equations can, at times, be subtle. Figure 4.8.1. Two orbits of the Linear-2D MAP in Eq.(1) for b = 0.95, b = 0.99, b = 1, b = 1.01. The initial conditions of the blue and yellow solutions are marked with large dots. In the upper-left frame (b = 0.95) the solutions are approaching to the fized point at the origin. In the upper-right frame (b = 0.99) origin is still asymptotically stable but the rate of approach is slower. In the lower-left frame (b = 1) the solutions are periodic with period four; the origin is a center and the period-4 orbits are on invisible circles. Finally, in the lower-right frame (b = 1.01) the origin becomes unstable, a source. Figure 4.8.2. A Gallery of Phase Portraits of the Linear-2D MAP as the parameter b is increased from b = -0.1 to b = 1.1 with increments of 0.025. A rectangular FlowBox containing 5,000 initial conditions are iterated 22 times in each frame.

Activities:

• Click on the first picture to load it into your local copy of Phaser. Notice the way the orbits seem to trace four lines. Why? You might want to compute the eigenvalues for b = 1 and see if they are fourth roots of unity. Select the first frame by clicking on it; notice that the boundary of the frame becomes red. Now click on Phaserize button to run the frame in Phaser. Investigate the changes in the dynamics as you vary the parameter b about 0 or 0.5 (PhaserTip: Changing Parameters)
• Click on the second picture to load it into your local copy of Phaser. Hit the Slideshow button to play the Gallery as a slide show. (PhaserTip: SlideShow)
• Click on the Animate button on the Gallery window. Now each frame will be recomputed in the Main Phaser window which will be Phase Portrait view. Notice the change in the size and the geometry of the original rectangular FlowBox containing 5,000 initial conditions.

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