Closed Curves on Spheres

U is an open set: if (a, ?) e U, there exists tx 10. Due to the continuous dependence of a solution on the initial value condition, yYs(t\) > 9 provided that (y, 8) is sufficiently close to (a, ?). By (i), the solution yy8 too escapes to +oo and consequently (y, 8) e U. Similar arguments apply to L. Thus U, L are two open, disjoint sets. Take any point (a, ?) not in U U L. By (ii), the graph of the solution ya? cannot enter either U or L. Hence ya? is bounded, ?1 ya?(t\) such that (tx, rj) ? U. Then the graph of the corresponding solution ytiv(t) stays out of U. Since solutions do not intersect, yt[71(t) > ya?(t) for every t. This contradicts the fact that the graph (t, ya?(t)) meets dU at (a, ?). Let us denote the solution whose graph coincides with dU by yu(t). Obviously yu(t + 27r) is a solution, bounded as well. If yu(t) ^ yu(t + 27t) then they never in tersect and, by the previous argument, they must satisfy yu(t + 27i) oo and unstable as t ? ?oo.