  We propose new methods for the numerical continuation of point-to-cycle
connecting orbits in 3-dimensional autonomous ODE's using projection boundary
conditions. In our approach, the projection boundary conditions near the cycle
are formulated using an eigenfunction of the associated adjoint variational
equation, avoiding costly and numerically unstable computations of the
monodromy matrix. The equations for the eigenfunction are included in the
defining boundary-value problem, allowing a straightforward implementation in
AUTO, in which only the standard features of the software are employed.
Homotopy methods to find connecting orbits are discussed in general and
illustrated with several examples, including the Lorenz equations. Complete
AUTO demos, which can be easily adapted to any autonomous 3-dimensional ODE
system, are freely available.
