# 010 - 326 CHAPTER 5 NONLINEAR SYSTEMS(b y y 4 52 4 2 2 48 X...

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Unformatted text preview: 326 CHAPTER 5 NONLINEAR SYSTEMS (b) y y )' 4 52 4 2 2 48 X X X 2 4 2 4 98 102 (e) We could compute the eigenvalues and eigenvectors for each of the linear systems, how- ever. we can determine much of the qualitative information ab0ut behavior of the solutions from the ﬁgures in part (b). It is interesting to note the size of the eigenvalues (which are quite large in absolute value). so solutions move toward or away from the equilibria very quickly. 11. (a) The equilibrium points in the ﬁrst quadrant are (0. 0), (0. 50) and (40, 0). To classify these equilibrium points we compute the Jacobian matrix which is —2x — y + 40 -x —-2xy —x2 - 3y2 + 2500 ‘ At (0, 0) the Jacobian matrix is 40 0 0 2500 which has eigenvalues 40 and 2500, so (0. 0) is a source. At (0. 50) the Jacobian matrix is -10 0 0 -5000 which has eigenvalues — 10 and —5000. so (0. 50) is a sink. At (40, 0) the Jaeobian matrix IS -40 —40 o 900 which has eigenvalues —40 and 900, so (40. 0) is a saddle. (b) y Jv‘ y 4 52 4 2 2 48 x x x 2 4 2 4 38 42 (c) We could compute the eigenvalues and eigenvectors for each of the linear systems. how- ever, we can determine much of the qualitative information about behavior of the solutions from the ﬁgures in part (b). It is interesting to note the size of the eigenvalues (which are quite large in absolute value), so solutions move toward or away from the equilibria very quickly. ...
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