Dynamical systems occur in many areas of science, especially fluid dynamics. One is often
interested in examining the structural changes in dynamical systems, and these are often
related to the appearance or disappearance of solution trajectories connecting one or more
stationary points. Homoclinic orbits are trajectories that connect a hyperbolic saddle-type
stationary point to itself, and often arise as the limiting case of periodic solutions. In global
bifurcation analysis, the occurrence of homoclinic orbits is closely tied to chaotic behaviour
in dynamical systems.
Numerical computation of homoclinic orbits requires solving a boundary value problem
(BVP) over a doubly infinite interval. There are three ways that this can be done. The
first is to truncate the problem to a large finite interval and create appropriate projected
boundary conditions to ensure that the solution over the infinite interval is approximated
well. The second is to transform the problem to one over a finite interval via an exponential
change of independent variable. The third is to use a method appropriate for an infinite
interval by discretising the problem using a set of approximating functions appropriate to
an infinite interval. In this thesis we will compare algorithms of all three types, and test our
implementations of them on several test problems.
• The first algorithm we will examine will be the original time method of Beyn. This
method truncates the infinite interval to a finite one, and produces projected boundary
conditions that force the endpoints of the trajectory to lie in the appropriate invariant
subspaces of the stationary point. This algorithm also includes a method for determining
whether or not the truncated interval is large enough to obtain a solution
accurate to a required tolerance.
• The second algorithm is called time + subspace, and is a variation on the original
time method that fixes the distance between the stationary point and the endpoint
of the solution, but allows the interval to vary – the reverse of which was true for
original time.
• The third algorithm is of the second type and is based upon the arclength parameterisation
of the orbits. This method uses the arclength of the trajectory as the independent
variable to transform the problem to a finite interval. However, the BVP
formulated in this way can have a singularity at the end of the domain, and thus a
special collocation method is required to handle this.
• As this method does not perform well on orbits exhibiting ˇ Silnikov behaviour, a
variation of this method called the partial arclength method which uses a different
exponential transformation of the independent variable near the stationary point is
presented to address the deficiency.
• The final algorithm presented in this thesis uses Laguerre polynomials to compute
points on the stable and unstable manifolds of the stationary point by solving a BVP
over a semi-infinite interval. These are then used to produce boundary conditions for
the truncated BVP that ensure that the endpoints are on the appropriate manifold and
not just the appropriate subspace.