Pseudoarclength

Pseudoarclength continuation is a numerical method used to trace solution branches of equations F(u, λ) = 0 as a parameter λ varies. It augments the original system with an arc-length constraint, allowing continuation through turning points (folds) where simple parameter stepping fails and the solution cannot be continued by changing λ alone.

Formally, let F: R^n × R → R^n and suppose (u0, λ0) is a known solution. A unit tangent (tu, tλ) to the solution curve at that point is obtained from the homogeneous linear system F_u(u0, λ0)·tu + F_λ(u0, λ0)·tλ = 0, with the normalization tu^2 + tλ^2 = 1. To advance along the branch by a prescribed arc-length Δs, one solves the augmented system for (u, λ):

F(u, λ) = 0

tu·(u − u0) + tλ·(λ − λ0) = Δs

The solution (u, λ) becomes the new point (u1, λ1). The tangent is recomputed at the new point and the process repeats to march along the curve.

Advantages include robustness in tracking branches past folds where λ cannot be varied monotonically, making pseudoarclength continuation a standard tool in bifurcation analysis and nonlinear boundary-value problems. It is widely used in numerical bifurcation packages and applications such as mechanics, chemistry, and power systems.

Limitations involve the need for careful step-size control and good initial guesses; the method increases the size of the system to solve by one equation and unknown, raising computational cost. It may still encounter difficulties near degenerate bifurcations or higher-codimension points, where more advanced variants or supplemental constraints may be required.