Consider the (uncoupled) system x'= sin(2x) , y' = y

(1) Show that the origin is asymptotically stable by showing that V = sin² x + (1/2) y² is a Lyapunov function. (Recall that sin(2x) = 2 sin x cos x.)

(2)Show that the statement of Theorem 9.6.3 in the text would be incorrect if it only required V' to be negative semi-definite, by showing that the region {(x, y)| < x < , 1 < y < 1} would then satisfy all the conditions, but that not all points in this region are in the basin of attraction of the origin. (Note that because the two equations are uncoupled, you can deal with them separately.)

useful information:

Theorem 9.6.3 Let the origin be an isolated critical point of the autonomous system (6). Let the

function V be continuous and have continuous first partial derivatives. If there is a

bounded domain DK containing the origin where V(x, y) < K for some positive K,

V is positive definite, and V is negative definite, then every solution of Eqs. (6) that

starts at a point in DK approaches the origin as t approaches infinity.

In other words, Theorem 9.6.3 says that if x = (t), y = (t) is the solution of

Eqs. (6) for initial data lying in DK, then (x, y) approaches the critical point (0, 0) as

t. Thus DK gives a region of asymptotic stability; of course, it may not be the

entire basin of attraction. This theorem is proved by showing that (i) there are no

periodic solutions of the system (6) in DK, and (ii) there are no other critical points

in DK. It then follows that trajectories starting in DK cannot escape and therefore

must tend to the origin as t tends to infinity.