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

Chapter 9, Problem 10

(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. Consider the linear system (ii).(a) Since (0, 0) is an asymptotically stable critical point, show that a11 + a22 < 0 anda11a22 a12a21 > 0. (See of Section 9.1.)(b) Construct a Liapunov function V(x, y) = Ax2 + Bxy + Cy2 such that V is positivedefinite and V is negative definite. One way to ensure that V is negative definite is tochoose A, B, and C so that V (x, y) = x2 y2. Show that this leads to the resultA = a221 + a222 + (a11a22 a12a21)2, B = a12a22 + a11a21,C = a211 + a212 + (a11a22 a12a21)2,where = (a11 + a22)(a11a22 a12a21)(c) Using the result of part (a), show that A > 0, and then show (several steps of algebraare required) that4AC B2 = (a211 + a212 + a221 + a222)(a11a22 a12a21) + 2(a11a22 a12a21)22 > 0.Thus, by Theorem 9.6.4, V is positive definite.