We consider a discrete dynamical system given by a diffeomorphism on . Such a map may be given explicitly, or be obtained as the Poincaré map of a vector field on a four-dimensional state space. Stable and unstable manifolds of invariant manifolds of saddle-type, here fixed points or invariant circles, play important roles in organizing the global dynamics. It is well-known that the stable manifolds form boundaries between different basins of attraction. Furthermore, the transverse intersection of stable and unstable manifolds leads to homoclinic or heteroclinic tangle, associated with chaos. These manifolds are global, often noncompact objects that can have very complicated structure. Only in special situations it is possible to find stable and unstable manifolds analytically. In general they need to be computed with numerical methods.
In this paper we compute the two-dimensional unstable manifold of an invariant manifold of saddle-type of a three-dimensional diffeomorphism. (The stable manifold can be computed by considering the inverse.) There are two different cases: the unstable manifold of a hyperbolic fixed point , and the unstable manifold of a normally hyperbolic invariant circle H. The Unstable Manifold Theorem [Hirsch et al. 1977, Palis and De Melo 1982] guarantees the existence of the local unstable manifold in a neighborhood of the invariant manifold of saddle-type. The two cases are sketched in Figure 1. See also the animations.
The idea is to obtain the global unstable manifold by globalizing the local unstable manifold . In practice, we reliably compute a sufficient piece of , so that conclusions on the global dynamics can be drawn. The unstable manifold is represented by a discrete set of mesh points. The quality of the mesh can be prescribed, and the mesh is adapted whenever necessary. Our algorithm was originally developed for the computation of unstable manifolds of a normally hyperbolic invariant circle of saddle-type. With a slight adaptation, it can also be used to find a two-dimensional unstable manifold of a hyperbolic fixed point.
As starting data for our algorithm we need an approximation of the local unstable manifold . There are different algorithms for computing of a hyperbolic fixed point ; see [Osinga 1996] and biblio therein. The work presented here was motivated by the possibility of obtaining the starting data for the computation of the unstable manifold of an invariant circle H of saddle-type in the form of a linear approximation of with the method in [Osinga 1996, Broer et al. 1996, Broer et al. 1997]. Their method is a variation of the graph transform that allows the computation of a normally hyperbolic invariant circle of saddle-type of a three-dimensional map. The key idea is to start with a known invariant circle H of a map f together with the Df-invariant splitting of the tangent space at H. This splitting induces Df-invariant stable and unstable normal bundles that are embedded in in a neighborhood of H. Because H is normally hyperbolic, these embedded normal bundels form a well-defined coordinate system in a neighborhood of H. The invariant circle of a small perturbation of f is computed as the graph over the known circle H in the coordinate system given by the embedded normal bundles. As a special feature of the method the new -invariant splitting of the invariant circle is computed in a second step, regardless of the dynamics on . Consequently, the method can be used in a continuation setting: a known invariant circle can be followed by increasing in small steps. The embedded unstable normal bundle of [Osinga 1996, Broer et al. 1996, Broer et al. 1997] is the first order approximation of , the local unstable manifold of H.
This allows us to globalize this first order approximation to compute a significant piece of . To be more concrete, the invariant circle H is known in a finite mesh M of points, and at each mesh point we are given the normal direction of the embedded normal bundle. (Since is of saddle-type, is a vector.) We now choose a linear foliation of the state space so that each leaf has a unique intersection with H. From the unstable normal bundle above we compute unit vectors for all , such that is tangent to . Starting from the linear approximation given by M and , we compute the intersection of the unstable manifold with the finitely many leaves of . For this to work we need the following.
In each leaf of there is a unique curve of intersection with the unstable manifold . In other words, the unstable manifold intersects each leaf transversally.
Assuming this foliation condition is satisfied, we can compute the unstable manifold in each of the leaves of as a sequence of points that have a prescribed distance from each other. The set of sequences in a finite number of leaves defines a mesh that represents the unstable manifold. This computation can be done in steps by adding rings or bands, that is, by adding a single new point to the sequence in each leaf. In this way, one can see the unstable manifold grow during the computation.
The linear foliation has no dynamical property, but should be seen as an a priori definition of the mesh. By adding additional leaves during the computation we guarantee the quality of the mesh on the unstable manifold. By construction, our method is independent of the dynamics on the invariant circle. We can use it for the computation of the two-dimensional unstable manifold of a fixed point, if we interpret the fixed point as an invariant circle in polar coordinates. A detailed description of the algorithm can be found in Section 3.
The procedure of adding rings or bands fails when the computed portion of the unstable manifold no longer intersects each leaf of in a unique curve. A piece of the intersection is then missed in the computation, and the algorithm is lacking information about a part of the unstable manifold. Because this information is necessary in the computation, the algorithm stops. This is discussed in more detail in Section 6; see also Figure 6 and Figure 18. Note that by definition the foliation is transverse to so that at least a part of can be computed. Furthermore, we think that many interesting examples satisfy the foliation condition; see Section 5. Possible relaxations of the foliation condition are discussed in Section 6.
In summary, we present an algorithm that computes a growing piece of the two-dimensional unstable manifold of a normally hyperbolic invariant circle of saddle-type or a hyperbolic saddle point, until the unstable manifold becomes tangent to the foliation . The mesh representing the invariant manifold is of a prescribed quality. This allows us to study the global dynamics of many interesting systems, in particular those that satisfy the Foliation Condition.
This paper is organized as follows. In Section 2 we give an overview over the literature on the calculation of global unstable manifolds. In Section 3 we give a detailed description of the algorithm, and Section 4 deals with its correctness. The performance of our method is demonstrated in Section 5 with a number of examples. The applicability of the algorithm and some open problems are discussed in Section 6.
Written by: Bernd Krauskopf
& Hinke Osinga
Created: May 27 1997 --- Last modified: Wed Jul 2 10:50:50 1997