Commun. Comput. Phys. doi: 10.4208/cicp.240912.180613a Vol. 15, No. 1, pp. 1-45 January 2014 REVIEW ARTICLE Numerical Bifurcation Methods and their Application to Fluid Dynamics: Analysis beyond Simulation Henk A. Dijkstra 1, ∗ , Fred W. Wubs 2 , Andrew K. Cliffe 3 , Eusebius Doedel 4 , Ioana F. Dragomirescu 5 , Bruno Eckhardt 6 , Alexander Yu. Gelfgat 7 , Andrew L. Hazel 8 , Valerio Lucarini 9,10 , Andy G. Salinger 11 , Erik T. Phipps 11 , Juan Sanchez-Umbria 12 , Henk Schuttelaars 13 , Laurette S. Tuckerman 14 and Uwe Thiele 15 1 Institute for Marine and Atmospheric Research Utrecht, Utrecht University, The Netherlands. 2 Department of Mathematics and Computer Science, University of Groningen, Groningen, The Netherlands. 3 School of Mathematical Sciences, University of Nottingham, Nottingham, UK. 4 Department of Computer Science, Concordia University, Montreal, Canada. 5 National Centre for Engineering Systems of Complex Fluids, University Politehnica of Timisoara, Romania. 6 Fachbereich Physik, Philipps-Universit¨ at Marburg, Marburg, Germany. 7 School of Mechanical Engineering, Faculty of Engineering, Tel-Aviv University, Tel-Aviv, Israel. 8 School of Mathematics, University of Manchester, Manchester, UK. 9 Meteorological Institute, Klimacampus, University of Hamburg, Hamburg, Germany. 10 Department of Mathematics and Statistics, University of Reading, Reading, UK. 11 Sandia National Laboratories, Albuquerque, USA. 12 Departament de Fisica Aplicada, Universitat Politecnica de Catalunya, Barcelona, Spain. 13 Department of Applied Mathematical Analysis, TU Delft, Delft, the Netherlands. 14 PMMH-ESPCI, Paris, France. 15 Department of Mathematical Sciences, Loughborough University, Loughborough, UK. Received 24 September 2012; Accepted (in revised version) 18 June 2013 Available online 26 July 2013 Abstract. 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 ∗ Corresponding author. Email address: H.A.Dijkstra@uu.nl (H. A. Dijkstra) http://www.global-sci.com/ 1 c 2014 Global-Science Press