摘要
We provide an overview of current techniques and typical applications of numerical bifurcation analysis in fluid dynamical problems.Many of these problems are characterized by high-dimensional dynamical systems which undergo transitions as parameters are changed.The computation of the critical conditions associated with these transitions,popularly referred to as‘tipping points’,is important for understanding the transition mechanisms.We describe the two basic classes of methods of numerical bifurcation analysis,which differ in the explicit or implicit use of the Jacobian matrix of the dynamical system.The numerical challenges involved in both methods are mentioned and possible solutions to current bottlenecks are given.To demonstrate that numerical bifurcation techniques are not restricted to relatively low-dimensional dynamical systems,we provide several examples of the application of the modern techniques to a diverse set of fluid mechanical problems.
基金
The workshop and the work of F.W.Wubs and H.A.Dijkstra was partially sponsored by the Netherlands Organization of Scientific Research(NWO)through the NWOCOMPLEXITY project PreKurs
The participation of F.I.Dragomirescu to the workshop was partially supported by a Grant of the Romanian National Authority for Scientific Research,CNCS-UEFISCDI,project number PN-II-RU-PD-2011-3-0153,31/5.10.2011.Sandia National Laboratory is a multiprogram laboratory operated by Sandia Corporation,a Lockheed Martin Company,for the United States Department of Energy’s National Nuclear Security Administration under contract DE-AC04-94AL85000
The work of J.Sanchez-Umbria was supported by projects MTM2010-16930 and 2009-SGR-67.
