lab_12_expectations

# lab_12_expectations - b Use Matlab to find the eigenvalues...

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Lab 12 Expectations Submit Plots: 1a, 1b, 2a, 2b, 3b, 3c, 4a, 5d 1. a. Use pplane to plot several orbits for (*). What kind of solution does the origin seem to be? b. Plot these solutions on the square |x,y|<0.1 c. Let A be the 2x2 matrix for (*). Find the eigenvalues and eigenvectors for A and use these to prove what you found in a. is correct. 2. a. Repeat parts a. and b. from part 1 for this system. 3. a. Determine the approximating linear system for each of the systems listed. Find the eigenvalues for these systems and use them to prove what kind of point the origin is. b. Plot some orbits for the linear system to support your claim in part a. c. Plot some orbits for the corresponding non-linear system to show that these predictions still hold. 4. a. Use pplane to plot several orbits for the system (B). Find the approximating linear system for (B*) about (0,0). Plot several orbits for this approximate system. Show that when a small enough scale is used, the uv plot around the origin looks like the xy plot around (.5, .5).

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Unformatted text preview: b. Use Matlab to find the eigenvalues and eigenvectors for the linear approximation to (B*) about (0,0). Then, using the general solution to the system, prove your origin is a sink. 5. The seed is 2 a. Plot a few orbits of the corresponding linear system. They should appear to be circular about the origin. Prove this is indeed true. b. Use pplane to plot the system (C) in the interval |x,y|<2. c. Choose the smaller scale so that you can really see what’s happening around the origin. Use (C) to prove (***). How does it follow that the orbits move constantly toward the origin? How does the sign of the derivative function relate to the behavior of the function? How can you explain the fact that the computer shows you that the orbits are loops when in fact they are spirals? d. Use pplane to plot the specific orbit which starts at x = 0, y = -0.2, use the intervals |x,y|<3 and the zoom-in feature several times near the origin to approximate this radius....
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lab_12_expectations - b Use Matlab to find the eigenvalues...

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