Solved: In this problem we show how small changes in the coefficients of a system of

Chapter 9, Problem 25

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In this problem we show how small changes in the coefficients of a system of linear equations can affect the nature of a critical point when the eigenvalues are equal. Consider the system x _ = _1 1 0 1 _ x. Show that the eigenvalues are r1 = 1, r2 = 1 so that the critical point (0, 0) is an asymptotically stable node. Now consider the system x _ = _1 1 1 _ x, 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 asymptotically stable node. 2