Bifurcation Analysis for Timesteppers

A collection of methods is presented to adapt a pre-existing time-stepping code to perform various bifurcation-theoretic tasks. It is shown that the implicit linear step of a time-stepping code can serve as a highly effective preconditioner for solving linear systems involving the full Jacobian via conjugate gradient iteration. The methods presented for steady-state solving, continuation, direct calculation of bifurcation points (all via Newton’s method), and linear stability analysis (via the inverse power method) rely on this preconditioning. Another set of methods can have as their basis any time-stepping method. These perform various types of stability analyses: linear stability analysis via the exponential power method, Floquet stability analysis of a limit cycle, and nonlinear stability analysis for determining the character of a bifurcation. All of the methods presented require minimal changes to the time-stepping code.