Continuation of connecting orbits in 3D-ODEs: (II) Cycle-to-cycle connections

TL;DR

This paper extends numerical methods for connecting orbits in 3D autonomous ODEs from point-to-cycle to cycle-to-cycle, using eigenfunction-based boundary conditions to improve stability and implementation simplicity.

Contribution

It introduces a novel approach for cycle-to-cycle orbit continuation using eigenfunctions, avoiding monodromy matrix computations, and provides practical AUTO implementation demos.

Findings

Eigenfunction-based boundary conditions improve numerical stability.

Method successfully applied to a population dynamics example.

AUTO demos are available for easy adaptation.

Abstract

In Part I of this paper we discussed new methods for the numerical continuation of point-to-cycle connecting orbits in 3-dimensional autonomous ODE's using projection boundary conditions. In this second part we extend the method to the numerical continuation of cycle-to-cycle connecting orbits. In our approach, the projection boundary conditions near the cycles are formulated using eigenfunctions of the associated adjoint variational equations, avoiding costly and numerically unstable computations of the monodromy matrices. The equations for the eigenfunctions 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 the connecting orbits are discussed in general and illustrated with an example from population dynamics. Complete AUTO demos, which can be easily adapted to any autonomous 3-dimensional ODE system, are freely available.