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Question 1: (A circular limit cycle) Consider x¨+ ax·(x2 +x·2 -1) + x = 0, where a > 0.
i) Find and classify all the fixed points.ii) Show that the system has a circular limit cycle, and find its amplitude and period.iii) Determine the stability of the limit cycle.iv) Give an argument which shows that the limit cycle is unique, i.e., there are no other periodic trajectories.
Question 2: Sketch the phase portrait for each of the following systems. (As usual, r, θ denote polar coordinates.)
7.1.1 r· = r3 - 4r, θ = 1 7.1.2 r· = r(1 - r2 )(9 - r2), θ·=17.1.3 r· = r(1- r2 )(4 - r2), θ· = 2 - r2 7.1.4 r· = r sin r, θ·= 1
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Explain how probabilistic primality tests are used in practice to produce extremely large numbers that are almost certainly prime. Do such tests have any potential drawbacks?
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The expansion of a 3 X 3 determinant can be remembered by the following device. Write a second copy of the first two columns to the right of the matrix, and compute the determinant by multiplying entries on six diagonals:
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