Multiple Time Scale Dynamics by Christian Kuehn Springer Applied Mathematical Sciences Series Vol. 191 2015 ISBN: 978-3-319-12315-8
Reviewed by: Steve Schecter Department of Mathematics North Carolina State University

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A standard multiple-time-scale system is the fast-slow system $$ \epsilon\frac{dx}{d\tau}=f(x,y), \quad \frac{dy}{d\tau}=g(x,y), \quad (1) $$ with $x\in\mathbb{R}^m$, $y\in\mathbb{R}^n$, and $0<\epsilon\ll1$. The critical manifold is $$ C_0 = \{(x,y) : f(x,y)=0\}. $$ A common case is the one in which $C_0$ can be written as $x=h(y)$, and for each $y_0$ the equilibrium $x_0=h(y_0)$ is hyperbolically attracting for the system $\frac{dx}{d\tau}=f(x,y_0)$. This is a type of normal hyperbolicity of the critical manifold. In this case one expects solutions to be rapidly attracted to $C_0$ and then drift along it, with the drift given approximately by a solution of $\frac{dy}{d\tau}=g(h(y),y)$.

This situation is commonly encountered in applications. Often one assumes that the fast variables $x$ have reached equilibrium, so that one can reduce to the equation $\frac{dy}{d\tau}=g(h(y),y)$, which has fewer variables. To deal with solutions of the full system, classically one resorts to a matched asymptotic expansion, which uses the fast time $t=\frac{\tau}{\epsilon}$ for the initial part of the solution, and the slow time $\tau$ for the drift along the critical manifold. In terms of the fast time, the system (1) becomes $$ \frac{dx}{dt}=f(x,y), \quad \frac{dy}{dt}=\epsilon g(x,y). $$

The geometric approach to fast-slow systems, called geometric singular perturbation theory, goes back to Fenichel's paper [1]. Fenichel thanks “Lou Howard and Nancy Kopell for introducing me to singular perturbation theory and to its relationship with invariant manifold theory.” In a matched asymptotic expansion, one has a fast solution with parameters and a slow solution with parameters. One tries to choose the parameters to make the two parts fit together. In geometric singular perturbation theory one has a manifold of fast solutions and a manifold of slow solutions. One tries to show that they intersect transversally.

The power of geometric singular perturbation theory increased greatly in subsequent years due to several developments, including:

1. The exchange lemma of Kopell and Chris Jones [3], which deals with solutions near a critical manifold of saddle type: there are solutions that rapidly approach the manifold, drift along it, then rapidly leave it.
2. The blow-up approach to loss of normal hyperbolicity of the critical manifold, due to Krupa and Szmolyan [4], which was in turn inspired by work of Dumortier and Roussarie [5]. Previously loss of normal hyperbolicity had been treated using difficult asymptotic expansions or nonstandard analysis.
3. Analysis of periodic solutions near “folded singularities” of the critical manifold, largely due to Martin Wechselberger and collaborators.

So where does one turn to learn about geometric singular perturbation theory? For many years the standard answer was an expository article by Chris Jones in a conference proceedings [2]. Now we have the book under review, written not by one of the senior experts in the field, but by a young researcher, Christian Kuehn. That would be surprising enough. In fact the book is more ambitious than I have indicated: it introduces other points of view on multiple-time-scale dynamics as well. Here is an approximation of what percentage of the book is devoted to what topics:

• Introduction 2%.
• Geometric singular perturbation theory, including mixed-mode oscillations and bursting 29%.
• Asymptotic methods 10%.
• Numerical methods 8%.
• Delay to bifurcation 2%.
• Chaos 6%.
• Topological methods 3%.
• Uses of the Newton polygon 3%.
• Stochastic systems 6%.
• Extensions, applications, miscellaneous topics 19%.
• Bibliography 12%.

The approach to most topics is to lay out what Kuehn feels are the essentials for someone who is encountering a topic for the first time; not delve too far into technicalities; treat simpler examples; and survey the literature. The literature surveys alone constitute a major contribution, as can be guessed from the size of the bibliography.

The book includes fairly substantial introductions to many topics before delving into their multiple-time-scale aspects, e.g., chaos (secs. 14.3-14.4), stochastic dynamics (secs. 15.1 and 15.6), Conley index (secs. 16.1-16.3), stability of traveling waves (sec. 17.1), conservation laws (sec. 17.3), and nonstandard analysis (sec. 19.5).

Kuehn's approach is quite successful. For example, when I encountered Gevrey asymptotics in a paper I was reading, I turned to this book for an accessible introduction.

Kuehn tried to involve the multiple-time-scale community in the preparation of this book by sending out draft chapters to different people to review. He thanks almost 30 people for providing feedback (conflict-of-interest alert: I am one of them). This is a great approach to producing such a wide-ranging work, and I hope others will imitate it.

With such a book, one's reaction is bound to include both awe that it was actually completed and complaints about some of the choices. I feel an opportunity was missed in the treatment of delay to bifurcation. The simplest situation in which this phenomenon occurs is the two-dimensional system (written in fast time) $$ \frac{dx}{dt}=xf(x,y), \quad \frac{dy}{dt}=\epsilon, $$ in which $f(0,y)$ changes sign from negative to postive as $y$ increases past some value $y_c$. Thus the $y$-axis, which is a line of equilibria for $\epsilon=0$ and remains invariant for all $\epsilon$, changes from normally attracting to normally repelling as $y$ increases past $y_c$. For small $\epsilon > 0$, solutions that start at $(x_0,y_0)$, with $0 < x_0 \ll 1$ and $y_0 < y_c$, are attracted to $y$-axis, drift slowly up it, and are eventually repelled from the $y$-axis near a point $(0,y_1)$ with $y_1>y_c$; this is the delay. The dependence of $y_1$ on $y_0$ is sometimes called the entry-exit function. Kuehn discusses it in the more complicated situation in which the loss of normal hyperbolicity is due to a Hopf bifurcation, but does not treat the simpler case.

A major accomplishment of the book is simply to assemble so many approaches to multiple-time-scale dynamics, and so many applications, in one place. Kuehn succeeds in demonstrating to the reader that there really is a unified mathematical subject here, in which insights from different sides of the subject can be brought to bear on others.

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Bibliography

1. N. Fenichel, Geometric singular perturbation theory for ordinary differential equations, J. Differential Equations 31 (1979), 53-98.
2. C. K. R. T. Jones, Geometric singular perturbation theory, Dynamical systems (Montecatini Terme, June 1322, 1994), Lecture Notes in Mathematics 1609, Springer-Verlag, Berlin, 1995.
3. C. K. R. T. Jones and N. Kopell, Tracking invariant manifolds with differential forms in singularly perturbed systems, J. Differential Equations 108 (1994), 64-88.
4. M. Krupa and P. Szmolyan, Extending geometric singular perturbation theory to nonhyperbolic points-fold and canard points in two dimensions, SIAM J. Math. Anal. 33 (2001), 286-314.
5. F. Dumortier and R. Roussarie, Canard cycles and center manifolds, Mem. Amer. Math. Soc. 121 (1996), no. 577.