Modules: Discrete Chaos

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

### Section 4.9: Liapunov Functions for Nonlinear Maps

Example 4.8:

In this example we will investigate the changes in the stability type of the fixed point at the origin of the nonlinear planar map as the parameters a and b are varied. Using the Liapunov function V(x1, x2) = x12 + x22, Invariance Principle and an Instability Theorem, one can establish the following:

• when |a2| < 1 and |b2| < 1, the origin is (globally) asymptotically stable,
• when |a2| < 1 and |b2| = 1, or when |a2| = 1 and |b2| < 1, the origin is asymptotically stable,
• when |a2| = 1 and |b2| = 1, the origin is stable; indeed, solutions approach 4-cycles. If ab = 1, they approach 2-cycles.
• when |a2| > 1 and |b2| > 1, the origin is unstable, Figure 4.9.1. Four solutions for a = 0.9 and b = 0.85. Note that all solutions are approaching the (globally) asymptotically stable fixed point at the origin. Figure 4.9.2. One thousand solutions for a = 0.9 and b = 0.85. Note that all solutions are approaching the fixed point at the origin --- globally asymptotically stable. Figure 4.9.3. Four solutions for a = 1 and b = 1. Note that all solutions are approaching 2-cycles. Figure 4.9.4. One thousand solutions for a = 1 and b = 1. All solutions are approaching 2-cycles, really.

Activities:

• Click on the first picture to load it into your local copy of Phaser. Click the left mouse button to select half a dozen initial conditions. Go.
• Click on the second picture to load it into your local copy of Phaser. While the solutions are being computed hit the Clear button to to clear the transients. Notice that all solutions eventually approach the origin.
• Click on the second picture to load it into your local copy of Phaser. Use the left mouse button to select another FlowBox (PhaserTip: Flow) near the origin; Clear and Go.
• Click on the third picture to load it into your local copy of Phaser. To see the numerical values of period-2 orbits, hit the Left Arrow button to bring up the Xi Values view; Clear and Go. If the numbers do not quite look like period-2, increse Stop Ploting Time to 10000. (PhaserTip: Time)
• Click on the third picture to load it into your local copy of Phaser. Change the value of the parameter b = -1. Clear and Go. (PhaserTip: Parameters) Do you see 4-cycles?
• Click on the fourth picture to load it into your local copy of Phaser. Set Start Plotting = 8888 and Stop Plotting = 9999. (PhaserTip: Time) Notice the disapperance of the bogus diamond. Why?
• Click on the third picture, again, to load it into your local copy of Phaser. Change the parameter values to a = 2.5 and b = 0.2. Clear and Go. (PhaserTip: Parameters) Are the solutions approaching the origin? Are these parameter values covered by the theoretical considerations above? Change the value of the parameter a = 2.5 and b = 0.5. Clear and Go. Where are the solutions going? Explore other parameter values.

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