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hw3_solution - ME201 Advanced Dynamics(Fall 2007 HW-3...

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ME201 Advanced Dynamics (Fall 2007) HW-3 Solution ( 100 pts ) 1. 50pts Consider the following linear dynamical system on R 2 : ˙ x = ax + by ˙ y = cx + dy In matrix form, this system can be written as ˙ x = A x for a matrix A , where x = ( x, y ). Find the matrix A . Find the eigenvalues and generalized eigenvectors (if they exist) for any value of ( a, b, c, d ). Draw the corresponding phase portraits. Show that the nature of the phase portrait is determined by Det ( A ) and Tr ( A ), the determinant and trace of the matrix A . In Det ( A ) - Tr ( A ) plane, draw the various phase portraits. Do not forget the lines where you have zero eigenvalues. If needed, use help of numerical computation of trajectories. The matrix for this system is: A = a b c d The eigenvalues ( λ i ) and eigenvectors ( v ) are: λ i = 1 2 a + d ± p 4 bc + ( a - d ) 2 · v = ˆ a - d 4 bc +( a - d ) 2 2 c 1 ! where it should be clear that there are two eigenvalues and two eigenvectors. These results are those of Mathematica, they are specific to the case of nonzero entries (ie c 6 = 0). There are probably many different forms and normalized versions if you consider all cases. Along these lines consider the special case of A = a 0 0 d where a = d . The two eigenvalues are clearly equal to a in this case and therefore we need generalized eigenvectors. Using the technique ( A - λ 1 I ) x = 0 we have the conditions: 0 x 1 + 0 x 2 = 0 0 x 1 + 0 x 2 = 0 Because of this ambiguity we define two parameters α = x 1 and β = x 2 . Using this and the equations above we have x 1 x 2 = 1 0 α + 0 1 β for any nonzero α and β . So therefore the generalized eigenvectors for this particular case are 0 1 and 1 0 . To characterize the system with respect to its Trace and Determinant lets see how they are related. By definition for this two by two matrix: Tr ( A ) = λ 1 + λ 2 Det ( A ) = λ 1 λ 2 ME201 Advanced Dynamics 1 HW3 Solution
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Since we have a good concept of phase space characteristics with respect to eigenvalues, lets solve for them using the two equations above: λ 1 , 2 = 1 2 Tr ( A ) ± p Tr ( A ) 2 - 4 Det ( A ) · From here we note that qualitative changes in the properties of the eigenvalues occur when Tr ( A ) = 0, Det ( A ) = 0, Tr ( A ) 2 = 4 Det ( A ). We therefore investigate the regions of the Tr ( A ) vs. Det ( A ) plane which is subdivided by these conditions.
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