# lab11.desc - LAB #11 Linear Systems Goal: Investigate the...

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LAB #11 Linear Systems Goal : Investigate the behavior of a linear system of equations near its equilibrium point. Characterize the behavior in terms of the nature of the eigenvalues. Required tools : Matlab routine pplane , fplot ; solutions of linear homogeneous systems using eigenvalues and eigenvectors. Discussion In this lab you will perform a systematic study of the solutions to the linear homo- geneous system of diﬀerential equations with constant coeﬃcients given by ~ x 0 ( t )= A~ x( t )( * ) near its equilibrium point, where A = ± ab cd ² .I fw ele t T = a + d denote the trace of A and let D = ad - bc denote the determinant of A , then we observe that the characteristic equation of of the matrix A is λ 2 - + D =0 . Assignment (1) Find the equilibrium point(s) for the system ( * ). (2) Show that the eigenvalues of the 2 × 2 matrix A = ± ² are given by λ = T ± T 2 - 4 D 2 , where T = a + d and D = ad - bc . The eigenvalues will be imaginary if T 2 4 <D . Using fplot , plot the function y = x 2 4 for | x |≤ 4and | y 4. Print your plot and label the axes T and D . (You can get Matlab to do the labeling. See the help for the routine “plot.”) (a) Indicate on your picture the set of points ( T,D ) for which the eigenvalues for the corresponding system are not real. Call this region

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## This note was uploaded on 04/23/2011 for the course MA 366 taught by Professor Cho during the Spring '08 term at Purdue University-West Lafayette.

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lab11.desc - LAB #11 Linear Systems Goal: Investigate the...

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