Practical implementation of pseudo-arclength continuation to ensure consistent path direction

Abstract

Pseudo-arclength continuation is a scheme used to generate a set of solutions for nonlinear equations and has been applied in numerous fields. In continuation schemes, solutions are obtained by tracking a parameter associated with the system as it varies, thereby creating a solution path. Pseudo-arclength continuation employs the nullspace vector, which is tangent to the path, as an approximation of the arclength in order to determine the direction of variation at each point on the path. However, the nullspace vector has both positive and negative solutions that are equally valid, leading to uncertainty in the direction of travel along the path at each location. Selecting the wrong nullspace vector can result in returning to a previously located solution instead of continuing the mapping of the solution path. Techniques to ensure consistent path direction have been developed; however, their efficacy does not extend to all scenarios. This paper addresses this gap by introducing a “flipping condition” to track previously determined path directions and ensure continuous following of the same path direction. A clear description of the process used to incorporate the flipping condition into the pseudo-arclength continuation scheme, practical implementation of the scheme, and examples demonstrating its application to the nonlinear circular restricted three-body problem (CR3BP), is provided. The inclusion of the flipping condition ensures a consistent direction of travel along the solution path and allows for families of solutions to be developed.