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BIFURCATION ANALYSIS FOR TIMESTEPPERSLAURETTE

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 eeective preconditioner for solving linear systems involving the full Jacobian via conjugate gradient iteration. The methods presented for steady-state solving, continuation, direct calculation of bifurca-tion 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.