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# pset_2 - MIT 18.385j/2.036j Second Problem Set Suggested...

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Second Problem Set ________________________________________________________________________ Suggested Readings (textbook): Chapters 4-5-6. Suggested Problems (textbook): Strongly Recommended: SR Ch. 3: 3.4.11 3.5.6 3.5.7 3.6.6 3.7.5 3.7.6 Ch. 4: 4.3.2 4.3.3 4.6.4 4.6.5 Ch. 5: 5.1.10 5.2.14 ............................... SR: 5.1.10 Ch. 6: 6.1.8 6.1.9 6.1.11 6.1.13 6.2.2 ......... SR: 6.2.2 ________________________________________________________________________ Problems to hand in for grading (textbook): Ch. 3: 3.4.14 3.4.15 3.5.8 3.6.3 Ch. 4: 4.1.1 4.1.8 Ch. 5: 5.2.11 Ch. 6: 6.1.7 6.1.10 6.1.12 Special Problem Below. ________________________________________________________________________ NOTE: you can use the MatLab® scripts: PHPLdemoA, PHPLdemoB, PHPLplot or PHPLplot_v2 with the problems requiring computer plotting. ________________________________________________________________________ SPECIAL PROBLEM Consider a system in the plane: dx/dt = f(x, y), dy/dt = g(x, y) such that the origin P = (x, y) = (0, 0)} is an isolated critical point, with the linearized system there having a stable star. Now consider the following two alternatives for the complete behavior of the system: (a) Linearized: stable star -----> Fully nonlinear: stable spiral. (b) Linearized: stable star -----> Fully nonlinear: stable proper node. Which ones are possible? For each one that is possible, give an example of a system with the desired behavior. Otherwise, explain why you think the particular alternative cannot happen.

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pset_2 - MIT 18.385j/2.036j Second Problem Set Suggested...

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