Computation of maximal turning points by a variational approach
For researchers studying bifurcations in nonlinear PDEs and parametric problems, this offers a new geometric perspective, though the results are preliminary and incremental.
The paper introduces a new variational approach for detecting bifurcations in nonlinear problems, proving conditions for maximal turning points and presenting an algorithm. Simulation experiments demonstrate the method's behavior but provide no quantitative performance comparisons.
We develop a new paradigm for finding bifurcations of solutions of nonlinear problems, which is based on the detection of extreme values of new type of variational functional associated with the considering problem. The variational functional is obtained constructively by the extended functional method which can be applied to a wide class of parametric problems including nonlinear partial differential equations and problems in non-variational form. The main feature of the approach is that it allows to consider the problem of finding bifurcations from another geometric point of view, namely using the extended variational functional. Sufficient and necessary conditions for the existence of a maximal turning point by the approach are proved. Based on these, an algorithm of the quasi-direction of steepest ascent is introduced. Simulation experiments are used to illustrate the behavior of the method and to discuss its advantages and disadvantages in comparison with the alternatives.