In this problem we show how small changes in the

Chapter 9, Problem 28

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In this problem we show how small changes in the coefficients of a system of linear equationscan affect the nature of a critical point when the eigenvalues are equal. Consider thesystemx=1 10 1x.Show that the eigenvalues are r1 = 1, r2 = 1 so that the critical point (0, 0) is anasymptotically stable node. Now consider the systemx=1 11x,where || is arbitrarily small. Show that if > 0, then the eigenvalues are 1 i,so that the asymptotically stable node becomes an asymptotically stable spiral point.If < 0, then the roots are 1 ||, and the critical point remains an asymptoticallystable node.