Zusammenfassung
Arclength continuation and branch switching are enormously successful
algorithms for the computation of bifurcation diagrams. Nevertheless, their
combination suffers from three significant disadvantages. The first is that
they attempt to compute only the part of the diagram that is continuously
connected to the initial data; disconnected branches are overlooked. The second
is that the subproblems required (typically determinant calculation and
nullspace construction) are expensive and hard to scale to very large
discretizations. The third is that they can miss connected branches associated
with nonsimple bifurcations, such as when an eigenvalue of even multiplicity
crosses the origin. Without expert knowledge or lucky guesses, these techniques
alone can paint an incomplete picture of the dynamics of a system.
In this paper we propose a new algorithm for computing bifurcation diagrams,
called deflated continuation, that is capable of overcoming all three of these
disadvantages. The algorithm combines classical continuation with a deflation
technique that elegantly eliminates known branches from consideration, allowing
the discovery of disconnected branches with Newton's method. Deflated
continuation does not rely on any device for detecting bifurcations and does
not involve computing eigendecompositions; all subproblems required in deflated
continuation can be solved efficiently if a good preconditioner is available
for the underlying nonlinear problem. We prove sufficient conditions for the
convergence of Newton's method to multiple solutions from the same initial
guess, providing insight into which unknown branches will be discovered. We
illustrate the success of the method on several examples where standard
techniques fail.
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