13.4 Geometric interpretation of the asymptotics

The analysis of the asymptotics in the various cases leads to the following conclusion [75Jump To The Next Citation Point, Theorem 1.2, p. 661]:

Theorem 6 Consider a solution to Equations (11View Equation)–(12View Equation). Let −P x = (x,y) = (Q, e ) and let dH be the metric induced by the Riemannian metric gH given by Equation (22View Equation). Then there is a K, a T > 0 and a curve Γ such that

dH (x(t,𝜃),Γ ) ≤ Kt −1∕2

for all t ≥ T. The possibilities for Γ are as follows.

Furthermore, it is possible to describe in detail the behavior of the solutions along the circles; see [75Jump To The Next Citation Point, Theorem 1.3–1.5, pp. 662–664] as well as [75Jump To The Next Citation Point, Figure 1.1–1.3, pp. 663–664]. In fact, the solution tends to the boundary along the circle when A2 + 4BC ≥ 0 and A, B and C are not all equal to zero. In the case of A2 + 4BC < 0, the solution oscillates around the circle forever and is asymptotically periodic with respect to a logarithmic time coordinate.


  Go to previous page Go up Go to next page