Commun. Comput. Phys., 15 (2014), pp. 1-45.

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, Andy G. Salinger 10, Erik T. Phipps 10, Juan Sanchez-Umbria 11, Henk Schuttelaars 12, Laurette S. Tuckerman 13, Uwe Thiele 14

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-Universitat 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; Department of Mathematics and Statistics, University of Reading, Reading, UK.
10 Sandia National Laboratories, Albuquerque, USA.
11 Departament de Fisica Aplicada, Universitat Politecnica de Catalunya, Barcelona, Spain.
12 Department of Applied Mathematical Analysis, TU Delft, Delft, the Netherlands.
13 PMMH-ESPCI, Paris, France.
14 Department of Mathematical Sciences, Loughborough University, Loughborough, UK.

Received 24 September 2012; Accepted (in revised version) 18 June 2013
Available online 26 July 2013


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.

AMS subject classifications: 37H20, 35Q35, 76-02, 37M, 65P30

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Key words: Numerical bifurcation analysis, transitions in fluid flows, high-dimensional dynamical systems.

*Corresponding author.
Email: (H. A. Dijkstra)

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