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The scientific objectives of this work are twofold. In problems governed by (very) large DAEs for which each linear stability analysis represents a significant computational cost. In these cases, one cannot afford to perform a large number of such linear stability analyses. Therefore, it becomes central to develop a robust algorithm for the localization of Hopf bifurcations in the framework of a continuation by exploiting the information contained in the Taylor series associated with the complete dynamical system. Thus, once an accurate location of the Hopf bifurcation is performed, three linear stability analyses will allow to decide on the subcritical or supercritical nature of the bifurcation. Moreover, local reduction implementations of the Liapunov-Schmidt type, allowing to apprehend the stability of the periodic solution resulting from the Hopf bifurcation, can also be considered. This technique of local analysis of dynamical systems, which has been applied to chemical reactors, can be generalized to DAE systems.
The implementation of such algorithms in the MANLAB 4 software, developed at the LMA is the first part of this post-doc contract. The second part will consist in their thorough validation on referenced test cases.