Computing Invariant Manifolds via the Continuation of Orbit Segments
A key feature of packages such as Auto [12, 15], Content  and Matcont  is a collocation solver for two-point boundary value problems (BVPs); see also Chaps. 1 and 2. In conjunction with pseudo-arclength continuation, it is possible to find the solution of a two-point BVP and then continue it in parameters. This basic idea will be known to most readers as the standard technique to compute a one-parameter branch of periodic orbits. However, the continuation of BVPs is a much more versatile tool and the solution need not be a periodic orbit, but may be any specified orbit segment. For example, the continuation of a suitable orbit segment is utilized in the HomCont extension for the computation of connecting orbits; see  and also [13, 21, 43].
In this chapter we focus on the idea of representing an invariant global manifold of a dynamical system as a family of orbit segments, which can then be computed as a solution family of a suitable BVP. Note that the thus computed object lies entirely in the phase space, rather than the product of phase space and parameter space.
KeywordsPeriodic Orbit Invariant Manifold Unstable Manifold Stable Manifold Rotation Number
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