Q&A with Yuri A. Kuznetsov and Hil G. E. Meijer, authors of
Numerical Bifurcation Analysis of Maps: From Theory to Software

Why would you write in the age of Open Access?

Whatever you think of the transition to
Open Access, the issue is that we need to share our knowledge. In this
information age many results are available, but keeping track is getting more
and more a full time job. Now while sharing new results is one side, advancing
methods is just as important. Some of the most-cited papers are those that
simply explain “how-to-do-it-right”, e.g. the numerical recipes. A physical
book is still the best tool to get valuable knowledge for scholars in many
developing countries. Think of slow or censored internet. And writing a book is
an opportunity to reconsider relations between various results and to come up
with a uniform and clear exposition. That is much more work than collecting
several research articles.

Turning to dynamical systems, the field
is rich with powerful analytical results and full of beautiful geometrical
ideas. And this is all driven by true applications, some of theoretical nature,
but also by mechanics, neuroscience, systems biology, economy and so on. While
we find non-linear dynamics fascinating and enjoy the analysis, at the end of
the day, we really want nice figures with numerical results to explain it to
our peers or a friend. So to reach out to applications we need both theory and
numerical tools.

Why
did you decide to write this book now?

Perhaps we could have written this book several years earlier as much of the material was already known and we were already in contact with Cambridge University Press (CUP). But as things go, we then started to work on different projects for some years. Interestingly during this period we had several interdisciplinary collaborations for which we had to develop concrete algorithms. We also observed how people were using our methods and software and helped them out. This has led to better testing and polishing of the algorithms. When we met CUP again at the SIAM Annual meeting in Chicago 2014, they spurred us to revive this project. We then decided to focus on putting all theory and methods together.

What
aspect do you find exciting about this book?

For the theory part we have spent much time filling in gaps to disclose details typically known to experts only. For instance, for every bifurcation we supplement the theoretical picture with a concrete numerical study. Then you can really see how to interpret the analytical results and use them for your own problem. Also, we have described the details of certain resonance bubbles so that their global understanding is nearly complete.

As the numerics is important, we want to highlight that too. As researchers we sometimes do things that seem to be a trick/hack to outsiders, but it can actually be justified. Upon generalization your methods become better and more transparent. The original project was just about bifurcations of maps, and the dynamics itself gets to the background. While reviewing quite papers, we realized we needed to add support to characterize the phase space with Lyapunov exponents and invariant manifolds. This may seem basic, but we are sure the community will be happy with the easily accessible tools.

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