Modules: Discrete Chaos

Chapter 2: Stability of Two-Dimensional Maps

Section 4.3: Fundamental Set of Solutions

Example 4.3.

In this example we will compute the solution of the Linear-2D MAP

for the parameter vlaues a = -2, b = -3, c = 3, d = -2, e = 0, f = 0, satisfying the initial condition x1 = 1, x2 = 2. Although formulae are available for solutions of linear maps, the geometry of particular solutions may not be self-evident. Furthermore, there can be numerical limitations in iterating an initial condition even for moderate number of times.

 


Figure 4.3.1. Twenty five iterates of the initial condition (1, 2) under the linear map for the parameter vlaues a = -2, b = -3, c = 3, d = -2, e = 0, f = 0. Notice the rapid increase in the magnitude of the variables.

 


Figure 4.3.2. The first ten iterates of the initial condition under the linear map from the previous figure are plotted in Xi vs. Time view of Phaser. The bright blue dots are the x1 values and darker blue dots are the x2 values. The graphs of the coordinates of this solution are misleading: to display just the first ten iterates we had to skew the scaling; while the horizontal axis runs from -1 to 10, the vertical axis runs from -7000 to 7000.

 


Figure 4.3.3. Two orbits of Linear-2D MAP with mystery paramater values and initial conditions (marked with larger blue and yellow dots). The plotting (x1, x2)-plane is the Phase Portrait View of Phaser.

 

Activities:

  • Click on the first picture to load it into your local copy of Phaser. Change the Stop Plotting = 600. Go. (PhaserTip: Time) Now the initial condition will be iterated 600 times under the linear map. Scroll down the numbers. What is the largest exponent do you see? NaN stands for "Not a Number."
  • Click on the second picture to load it into your local copy of Phaser. Change Y-Min and Y-Max so that all ten iterates are visible. (PhaserTip: WindowSize) You can determine what values to use from the previous figure. Clear and Go.
  • Click on the third picture to load it into your local copy of Phaser. Find the mystery parameter values and the initial conditions. (PhaserTips: Parameters, Initial Conditions)) Compute the fundamental set of solutions and find the formulas for the two particular solutions.

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