In this problem we show that the Liapunov function constructed in the preceding problem

Chapter 9, Problem 10

(choose chapter or problem)

In this problem we show that the Liapunov function constructed in the preceding problem is also a Liapunov function for the locally linear system (26). We must show that there is some region containing the origin for which V is negative definite. a. Show that V ( 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 means that, 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|, show by 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.