practice3 - these limit cycles? x = x ( r 2-3 r + 2)-y y =...

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Practice Exam 3: MAP 4305 * 1. Given is a fundamental matrix solution X ( t ) of a system ˙ x = Ax where A is a (unknown, at least for now) 2 by 2 matrix: X ( t ) = p e 5 t e 5 t + e - t e 5 t e 5 t - e - t P . What is e tA , and what is A ? 2. Find the matrix exponential of: 2 1 0 0 0 0 2 1 0 0 0 0 2 0 0 0 0 0 1 1 0 0 0 0 1 , 3. Solve the following initial value problem (use variation of constants formula): ˙ x = Ax + b, x (0) = x 0 where A = p 1 2 2 1 P , b = p 1 t P , x 0 = p 0 1 P . 4. Let V : R 2 R be a twice continuously di±erentiable function. Consider the system ˙ x = V x ( x, y ) ˙ y = V y ( x, y ) Let ( x * , y * ) be a critical point of V (ie V x ( x * , y * ) = V y ( x * , y * ) = 0). Then ( x * , y * ) is clearly an equilibrium point of the system. Can it be a spiral (stable or unstable) or a center? 5. Show that there are no non-constant periodic solutions for: ˙ x = 2 x - y + x 3 y 2 ˙ y = x - y 6. Show that the following system has one stable and one unstable (non-trivial) limit cycle. Where are
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Unformatted text preview: these limit cycles? x = x ( r 2-3 r + 2)-y y = y ( r 2-3 r + 2) + x, where r = r x 2 + y 2 . ( Hint : Use polar coordinates) 7. Complete the discussion in class regarding the van der Pol oscillator by verifying that the region containing the origin and bounded by the line segments L 1 : y = x +6 , x [-3 , 0], L 2 : y = 6 , x [0 , 3], L 3 : x = 3 , y [-3 . 6], L 4 : y = x-6 , x [0 , 3], L 5 : y =-6 , x [-3 , 0], and L 6 : x =-3 , y [-6 , 3] is a trapping region. Recall the equations for the Van der Pol oscillator: x = y + x-x 3 / 3 y =-x * Instructor: Patrick De Leenheer....
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