Numerical Methods for Bifurcation Problems and Large-Scale Dynamical Systems

This book is based on the combined proceedings of two workshops ("Numerical Methods for Bifurcation Problems" and "Large Scale Dynamical Systems") held as part of the 1997-1998 IMA Academic Year on Emerging Applications of Dynamical Systems. The book consists of 20 papers covering quite a wide area of research. As stated at the back of the book

"Several of the papers in this volume treat computational methods for low- and high-dimensional systems and, in some cases their incorporation into software packages. A few papers treat fundamental theoretical problems, including smooth factorization of matrices, self-organised criticality, and unfolding of singular heteroclinic cycles. Other papers treat applications of dynamical systems computations in various scientific fields, such as biology, chemical engineering, fluid mechanics, and mechanical engineering."

A brief summary of the contents of each paper is given as follows:

1. I found the first paper "Numerical bifurcation techniques for chemical reactor problems" (by V. Balakotaiah and J. Khinast) particularly interesting since it strongly relates to my own area of research. The main thrust of this paper is to "illustrate the computation of various co-dimension one steady-state and dynamic singularities for diffusion-reaction, convection-reaction and diffusion-convection-reaction problems using the shooting technique and sensitivity functions." The examples provided in this article were excellent illustrations of the techniques used.
2. The second paper is titled "Path-following of large bifurcation problems with iterative methods" and was written by K. Bohmer, Z. Mei, A. Schwarzer and R. Sebastian. As the title suggests, it considers detection and numerical continuation of large dynamical systems particularly those obtained from discretization of PDEs. The authors mainly focussed on the reaction-diffusion equation.
3. The paper by T.J. Burns, M.A. Davies and C.J. Evans titled "On the bifurcation from continuous to segmented chip formation in metal cutting" makes up the third article in this book. This paper describes how a nonlinear dynamics approach to modelling the plasticity problem of chip formation during metal cutting is able to provide new insights into this industrial process. In particular, the new models suggest that the transition from continuous to segmented chip formation results from a singular Hopf bifurcation in the flow of the workpiece material as it is deformed by the cutting tool. Anyone (like myself) who has a keen interest in industrial mathematics would find this article to be of particular interest.
4. "Using dynamical systems tools in MATLAB" by W.G. Choe and J. Guckenheimer is the next contribution. The authors discuss guidelines for building an interface between Matlab (which possesses its own rich set of graphics, user interface and numerical operations) and dynamics tools. The authors describe their "...efforts as the first steps towards the construction of a `Dynamics Toolbox' for Matlab...". I am excited at the possibility of having such a toolbox in Matlab. However I feel that this article would be particularly heavy-going for those who are not familiar with Matlab.
5. K. Coughlin's article on "Formation and instabilities of coherent structures in channel flows" which has a strong fluid-dynamics flavour makes up the fifth paper in this book. It discusses the dynamics of intermittent turbulence in channel flows. The author describes how the addition of a small forcing term to the Navier-Stokes equations makes the streamwise vortex solutions become true equilibria. The latter are then studied by direct numerical investigation of the relevant linear equations.
6. The article "Applications of smooth orthogonal factorizations of matrices" by L. Dieci and T. Eirola describes "...instances where computation of smooth orthogonal factorization of matrices.." are encountered in dynamical system problems. The authors also discuss some of the issues that arise during the implementation of their techniques.
7. I found the next article "Continuation of codimension-2 equilibrium bifurcation in CONTENT" (by W. Govaerts, Yu. A. Kuznetsov and B. Sijnave) to be particularly interesting, since I have never used the software package CONTENT and this article gave me the opportunity to learn of its power. (I am an AUTO user.) This article discusses recent extensions to CONTENT which enable the user to compute and continue all co-dimension-2 bifurcations. A brief discussion on the detection of codimension-3 equilibrium bifurcations is also given.
8. A. J. Homburg's article "Inclination-flips in the unfolding of a singular heteroclinic cycle" describes the study of the bifurcations arising from a particular heteroclinic cycle in a two parameter family of three dimensional vector fields.
9. In " Investigating torus bifurcations in the forced Van Der Pol oscillator", B. Krauskopf and H. M. Osinga discuss a new algorithm which they have developed and implemented in the software package DsTool for computing one-dimensional stable and unstable manifolds of (Poincare) maps. The authors use the Van Der Pol oscillator as a test example.
10. The tenth article in the book is titled "Quasiperiodic response to parametric excitations" (by J.M. Lopz and F. Marques). This article describes a technique based on Floquet theory and discrete dynamical systems that can identify the conditions under which a periodically forced system (susceptible to centrifugal instabilities) resonates (resulting in resonance horns). It also can identify higher codimension degenerate bifurcations from space-time resonances.
11. I must confess that I knew very little about self-organized criticality and found the article by J. Lorenz, S. Jackett and W. Qin ("Computation and bifurcation analysis of periodic solutions of large-scale systems") very difficult to read initially. If you are like me, I suggest some background reading of P. Bak's book "How nature works", Springer-Verlag, New York (1996) before attempting to read this article.
12. The efficient computation (based on Newton-Picard method) and bifurcation analysis of periodic solutions of large-scale systems (those arising from discretizing PDEs) is the basis of K. Lust and D. Roose's article ("Computation and bifurcation analysis of periodic solutions of large-scale systems").
13. The article "Multiple equilibria and stability of the North-Atlantic wind-driven ocean circulation" (by M.J. Molemaker and H. A. Dijkstra) describes a numerical method (whose code is dubbed BAGELS by the authors) which is successful in investigating multiple equilibria in an idealized finite element ocean model of the wind-driven ocean circulation in the North-Atlantic -- a large scale dynamical system.
14. The paper by M. Olle ("Numerical exploration of bifurcation phenomena associated with complex instability") discusses numerical methods for path-following and detecting bifurcations in Hamiltonian systems with three or more degrees of freedom.
15. Lattice dynamical systems of unbounded medium (sometimes referred to as coupled map lattices) are the basis of the paper by D.R. Orendovici and Ya.B. Pesin ("Chaos in travelling waves of lattice systems of unbound media"). The authors describe such systems corresponding to some well-known PDEs such as the reaction-diffusion, Swift-Hohenberg, Kuramoto-Sivashinsky and Ginzburg-Landau equations. They also discuss the mechanism for the appearance of spatial and/or temporal chaos associated with various special classes of solutions.
16. The next article by H.G. Othmer, B. Lilly and J.C. Dallon applies dynamical systems computations to biology. Their paper titled "Pattern formation in cellular slime mold" looks at stream formation (which is a prominent feature of aggregation in low-density fields) in slime mold, the relationship between cell-based and continuum descriptions of aggregation, and the origin of streaming in these two models.
17. B. P. Peckham's article "Global parametrization and computation of resonance surfaces for periodically forced oscillators" makes up the 17th article in this book. The title speaks for itself.
18. The paper "Computing invariant tori and circles in dynamical systems" by V. Reichelt discusses several algorithms (based on the discretization of the graph transform) which allow the computation of these tori using continuation strategy. The author shows that these algorithms are faster, more robust and most importantly more accurate than some existing methods. The author illustrates these algorithms on several test examples.
19. R. Seydel's article "A design problem for image processing" is one of my favourites in this book. His paper outlines the role of bifurcation in image processing. I find the whole concept of images being processed by integrating reaction-diffusion equations fascinating and I will make an effort to look up some of the papers in Seydel's bibliography.
20. The final article in the book is " Bifurcation analysis for timesteppers" by L.S. Tuckerman and D. Barkley. The authors discuss a collection of methods used to adapt a pre-existing time-stepping code to enable various bifurcation tasks to be undertaken.

The review of this book was an exhausting and time-consuming precess, since it involved a mini-review of 20 research articles! However, I learnt a lot from this book and will certainly be chasing a few references. This book is an important and useful resource which brings together a collection of papers dealing with bifurcation analysis and large-scale dynamical systems. I highly recommend it for people who are active in this field of research.