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Unformatted text preview: 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 ( i ) and eigenvectors ( v ) are: i = 1 2 a + d p 4 bc + ( a- d ) 2 v = 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 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- 1 I ) x = we have the conditions: x 1 + 0 x 2 = 0 x 1 + 0 x 2 = 0 Because of this ambiguity we define two parameters = x 1 and = x 2 . Using this and the equations above we have x 1 x 2 = 1 + 1 for any nonzero and . So therefore the generalized eigenvectors for this particular case are 1 and 1 . 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 ) = 1 + 2 Det ( A ) = 1 2 ME201 Advanced Dynamics 1 HW3 Solution Since we have a good concept of phase space characteristics with respect to eigenvalues, lets solve for them using the two equations above: 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. The following two figures illustrate both this subdividedwhich is subdivided by these conditions....
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