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

QUESTION:

In this problem we show how small changes in the coefficients of a system of linear equations can affect a critical point that is a center. Consider the system x _ = _ 0 1 1 0 _ x. Show that the eigenvalues are i so that (0, 0) is a center. Now consider the system x _ = _ 1 1 _ x, where || is arbitrarily small. Show that the eigenvalues are i. Thus no matter how small || _= 0 is, the center becomes a spiral point. If < 0, the spiral point is asymptotically stable; if > 0, the spiral point is unstable. 2