Continuation of connecting orbits in 3D-ODEs: (I) Point-to-cycle connections

TL;DR

This paper introduces a new numerical method for continuing point-to-cycle connecting orbits in 3D autonomous ODEs, utilizing eigenfunction-based boundary conditions to improve stability and implementation simplicity.

Contribution

The authors develop a novel approach using eigenfunctions for boundary conditions, enabling straightforward continuation of connecting orbits in AUTO without complex monodromy matrix computations.

Findings

Method successfully applied to Lorenz equations.

Implementation in AUTO is straightforward and adaptable.

Provides complete demos for 3D autonomous ODEs.

Abstract

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.