Continuation of connecting orbits in 3D-ODEs: (I) Point-to-cycle connections
This work provides a more numerically stable and accessible approach for researchers studying connecting orbits in 3D ODEs, though it is incremental as it builds on existing projection boundary condition techniques.
The authors propose new methods for numerically continuing point-to-cycle connecting orbits in 3D ODEs using projection boundary conditions that avoid computing the monodromy matrix, enabling straightforward implementation in AUTO. The methods are demonstrated on examples including the Lorenz equations.
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.