Abstract and Contents

3.3 The case of a hyperbolic fixed point

Because we insist that all mesh points lie in fixed leaves, there is no accumulation of mesh points on one-dimensional submanifolds of . In this sense, our algorithm is independent of the dynamics on the circle. For the same reason it can be used to compute the two-dimensional unstable manifold of a hyperbolic fixed point .

Let be the two-dimensional (generalized) unstable eigenspace of embedded in . We define the linear foliation as follows. The leaf is the half-plane that contains the ray with angle in and is perpendicular to . In fact, any linear foliation that is transverse to would do, but is `locally most transverse', in the sense that each leaf is perpendicular to . Let us assume that satisfies the Foliation Condition.

Take a very small circle around in and let M be its intersection with a prescribed finite number of leaves of , which we take to be equally spaced in . The points in M now parametrize the foliation, as it was the case in the previous section. The linear approximation is then immediate by defining as the unit vector in pointing away from ; see Figure 8.

With these definitions we can use the procedure GLOBALIZE to compute a prescribed number of rings of with prescribed tolerances , and . It is clear from the radial nature of the finite set of leaves that it is necessary to add leaves of during the computation to ensure that the maximal distance between neighboring mesh points does not become too big. (This effect is less pronounced for unstable manifolds of invariant circles of saddle-type; see Section 5.)

Abstract and Contents

Written by: Bernd Krauskopf & Hinke Osinga
Created: May 27 1997 --- Last modified: Fri May 30 19:51:59 1997