In 10 and 11, we will prove part of Theorem 9.3.2: If the

Chapter 9, Problem 11

(choose chapter or problem)

In 10 and 11, we will prove part of Theorem 9.3.2: If the critical point (0, 0) of thelocally linear systemdx/dt = a11x + a12y + F1(x, y), dy/dt = a21x + a22y + G1(x, y) (i)is an asymptotically stable critical point of the corresponding linear systemdx/dt = a11x + a12y, dy/dt = a21x + a22y, (ii)then it is an asymptotically stable critical point of the locally linear system (i). deals with the corresponding result for instability. In this problem we show that the Liapunov function constructed in the preceding problemis also a Liapunov function for the locally linear system (i). We must show that there issome region containing the origin for which V is negative definite.(a) Show thatV (x, y) = (x2 + y2) + (2Ax + By)F1(x, y) + (Bx + 2Cy)G1(x, y).(b) Recall that F1(x, y)/r 0 and G1(x, y)/r 0 as r = (x2 + y2)1/2 0. This meansthat, given any > 0, there exists a circle r = R about the origin such that for 0 < r < R,|F1(x, y)| < r and |G1(x, y)| < r. Letting M be the maximum of |2A|, |B|, and |2C|, showby introducing polar coordinates that R can be chosen so that V (x, y) < 0 for r < R.Hint: Choose sufficiently small in terms of M.