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2012

Volume 54, Issue 2, pp. 209-417

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Survey and Review

Desmond J. Higham, Section Editor

SIAM Rev. 54, pp. 209-209 (1 page)

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Dynamical systems is an important and thriving area that focuses on qualitative properties of time-dependent problems. Oscillatory behavior is easy to spot in a physical system and is a natural candidate for investigation. “Mixed-Mode Oscillations with Multiple Time Scales,” by Desroches, Guckenheimer, Krauskopf, Kuehn, Osinga, and Wechselberger, reviews a very specific class of oscillations that are easy to describe and illustrate (see Figure 1 on p. 212). But quantifying and analyzing this type of behavior is an ongoing challenge that has led to the development of new concepts and methodologies. “Mixed mode” refers to the co-presence of small- and large-amplitude oscillations along a trajectory. This article focuses on fast-slow ordinary differential equations (ODEs) where small-amplitude oscillations arise from local phenomena around a special point of the limiting system. Some intricate concepts are introduced through concrete examples, including the well-known Van der Pol equation. It is natural to think geometrically in this field, and the authors have produced some highly creative and, in many cases, stunning graphical illustrations to help the reader along. We are shown how mathematical advances can lead directly to important new insights about complex behavior. Computational experiments are a key investigative tool, and a customized approach, exploiting analytical insights, is often necessary. Indeed, as the treatment moves back and forth between theory and algorithmics, it becomes clear that these two aspects develop best in tandem.
The comprehensive review in sections 2 and 3 is followed by four illustrative case studies. Tables 4–6 then scope the literature on mixed-mode oscillations in three categories: The authors also point to current hot topics, such as dealing with higher dimensional, spatially dependent or stochastic systems and calibrating models to real data. This work will be of interest both to analytically oriented readers looking to get a feel for the key concepts and open problems, and to those with particular applications in mind who wish to learn about the latest tools.

Mixed-Mode Oscillations with Multiple Time Scales

Mathieu Desroches, John Guckenheimer, Bernd Krauskopf, Christian Kuehn, Hinke M. Osinga, and Martin Wechselberger

SIAM Rev. 54, pp. 211-288 (78 pages)

Online Publication Date: May 08, 2012

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Mixed-mode oscillations (MMOs) are trajectories of a dynamical system in which there is an alternation between oscillations of distinct large and small amplitudes. MMOs have been observed and studied for over thirty years in chemical, physical, and biological systems. Few attempts have been made thus far to classify different patterns of MMOs, in contrast to the classification of the related phenomena of bursting oscillations. This paper gives a survey of different types of MMOs, concentrating its analysis on MMOs whose small-amplitude oscillations are produced by a local, multiple-time-scale “mechanism.” Recent work gives substantially improved insight into the mathematical properties of these mechanisms. In this survey, we unify diverse observations about MMOs and establish a systematic framework for studying their properties. Numerical methods for computing different types of invariant manifolds and their intersections are an important aspect of the analysis described in this paper.
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Research Spotlights

Ray Tuminaro, Section Editor

SIAM Rev. 54, pp. 289-289 (1 page)

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Every applied mathematician has used separation of variables. So begins “Synthesis, as Opposed to Separation, of Variables” by A. S. Fokas and E. A. Spence. As the authors note, separation of variables dates back to the 1750s and is a standard staple within any undergraduate partial differential equation course. Its name, separation of variables, captures its essence and at first glance evokes a sense of simplicity. The central idea is to look for solutions which are products of functions of individual variables. In this way, a partial differential equation is converted to a pair of ordinary differential equations. Of course, the devil is always in the details and in this context the details revolve around boundary conditions, initial values, non-self-adjoint operators, and the convergence of series or integrals.
The linearized Korteweg–de Vries (KdV) equation and the Helmholtz equation are both considered. Limitations associated with a classical separation of variables approach are first presented as well as some attempts to address these for Helmholtz operators. For high frequencies, the article concludes that “the angular series expansion obtained by Lord Rayleigh is correct but useless, and the radial series expansion obtained by Watson and Sommerfeld is useful but incorrect!”
The paper describes an alternative transform method that can be used for certain nonseparable and non-self-adjoint problems. This transform is motivated by observations associated with classical approaches and complex variables. Specifically, “the classical approach begins in the complex plane $\ldots,$ abandons the complex plane $\ldots,$ and sometimes returns to the complex plane.” The new method is instead based on remaining in the complex plane while solving the boundary value problem and is more a synthesis of variables (via parametrization) than a separation.
This paper provides a truly interesting perspective on a classical topic, separation of variables, as well as highlighting an intriguing alternative which can address some of its limitations.

Synthesis, as Opposed to Separation, of Variables

A. S. Fokas and E. A. Spence

SIAM Rev. 54, pp. 291-324 (34 pages)

Online Publication Date: May 08, 2012

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Every applied mathematician has used separation of variables. For a given boundary value problem (BVP) in two dimensions, the starting point of this powerful method is the separation of the given PDE into two ODEs. If the spectral analysis of either of these ODEs yields an appropriate transform pair, i.e., a transform consistent with the given boundary conditions, then the given BVP can be reduced to a BVP for an ODE. For simple BVPs it is straightforward to choose an appropriate transform and hence the spectral analysis can be avoided. In spite of its enormous applicability, this method has certain limitations. In particular, it requires the given domain, PDE, and boundary conditions to be separable, and also may not be applicable if the BVP is non-self-adjoint. Furthermore, it expresses the solution as either an integral or a series, neither of which are uniformly convergent on the boundary of the domain (for nonvanishing boundary conditions), which renders such expressions unsuitable for numerical computations. This paper describes a recently introduced transform method that can be applied to certain nonseparable and non-self-adjoint problems. Furthermore, this method expresses the solution as an integral in the complex plane that is uniformly convergent on the boundary of the domain. The starting point of the method is to write the PDE as a one-parameter family of equations formulated in a divergence form, and this allows one to consider the variables together. In this sense, the method is based on the “synthesis” as opposed to the “separation” of variables. The new method has already been applied to a plethora of BVPs and furthermore has led to the development of certain novel numerical techniques. However, a large number of related analytical and numerical questions remain open. This paper illustrates the method by applying it to two particular non-self-adjoint BVPs: one for the linearized KdV equation formulated on the half-line, and the other for the Helmholtz equation in the exterior of the disc (the latter is non-self-adjoint due to the radiation condition). The former problem played a crucial role in the development of the new method, whereas the latter problem was instrumental in the full development of the classical transform method. Although the new method can now be presented using only classical techniques, it actually originated in the theory of certain nonlinear PDEs called integrable, whose crucial feature is the existence of a Lax pair formulation. It is shown here that Lax pairs provide the generalization of the divergence formulation from a separable linear to an integrable nonlinear PDE.
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SIGEST

The Editors

SIAM Rev. 54, pp. 325-325 (1 page)

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This issue's SIGEST paper, “Chemical Reactions as $\Gamma$-Limit of Diffusion,” by Mark A. Peletier, Giuseppe Savaré, and Marco Veneroni, is an update of their paper “From Diffusion to Reaction via $\Gamma$-Convergence” from the SIAM Journal on Mathematical Analysis. The work uses advanced ideas from measure theory to connect macroscale chemical reactions, the ones you see in a test tube, to a limit of a molecular (micro)scale diffusion processes.
The macroscale reactions of interest in this paper are “unimolecular,” $A \rightleftharpoons B$. The meaning of this is that $A$ and $B$ are two forms of the same molecule and one can convert to the other. One can express the reaction between the two forms with a reaction diffusion equation, which you may have seen in an elementary course in applied partial differential equations. The microscale or molecular problem is very different. Here we think of $A$ and $B$ as the endpoints of the domain of an continuous “chemical variable.” The chemical variable is the independent variable in the microscale equation for a probability distribution. This microscale equation is the Kramers–Smoluchowski diffusion equation, which is parabolic. The authors show that for reactions like $A \rightleftharpoons B$, the classical reaction-diffusion equations are a singular limit of the microscale equations.
The authors have completely rewritten the introduction for SIAM Review, and the new introduction is a delight. In four pages they walk the reader through much of the technical background and preview the analysis to come. With this background, one can appreciate the discussion of the models and the statements of the two main theorems.
This paper is a fine example of how hard analysis contributes to our understanding of multiscale phenomena.

Chemical Reactions as $\Gamma$-Limit of Diffusion

Mark A. Peletier, Giuseppe Savaré, and Marco Veneroni

SIAM Rev. 54, pp. 327-352 (26 pages)

Online Publication Date: May 08, 2012

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We study the limit of high activation energy of a special Fokker–Planck equation known as the Kramers–Smoluchowski equation (KS). This equation governs the time evolution of the probability density of a particle performing a Brownian motion under the influence of a chemical potential $H/\varepsilon$. We choose $H$ having two wells corresponding to two chemical states $A$ and $B$. We prove that after a suitable rescaling the solution to the KS converges, in the limit of high activation energy ($\varepsilon\to0$), to the solution of a simpler system modeling the spatial diffusion of $A$ and $B$ combined with the reaction $A\rightleftharpoons B$. With this result we give a rigorous proof of Kramers's formal derivation, and we show how chemical reactions and diffusion processes can be embedded in a common framework. This allows one to derive a chemical reaction as a singular limit of a diffusion process, thus establishing a connection between two worlds often regarded as separate. The proof rests on two main ingredients. One is the formulation of the two disparate equations as evolution equations for measures. The second is a variational formulation of both equations that allows us to use the tools of variational calculus and, specifically, $\Gamma$-convergence.
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Education

Louis F. Rossi, Section Editor

SIAM Rev. 54, pp. 353-353 (1 page)

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The Education section in this issue of SIAM Review features two articles with two unique flavors. The first brings together distinct topics first introduced in the undergraduate curriculum and is suitable for an undergraduate audience. The second is a brief renormalization group tutorial for a graduate curriculum.
In “Modified Gershgorin Disks for Companion Matrices,” author Aaron Melman connects zeros of polynomials with the properties of their corresponding companion matrix. He begins with the basic notion that the zeros of a polynomial are contained within the union of Gershgorin disks of the corresponding companion matrix. While this is an elegant connection, the union of disks can be a fairly large set. The author points out that a careful examination of the structure of the companion matrix and some judicious manipulation can restrict these sets much further. The resulting discussion makes a nice module for an undergraduate course in linear algebra or analysis. This article also includes MATLAB code for rendering the restricted sets where the roots reside.
The second article, “The Renormalization Group: A Perturbation Method for the Graduate Curriculum,” by Eleftherios Kirkinis, provides a gentle introduction to renormalization group (RG) methods for singular perturbation problems. The authors make the case that RG methods provide a more consistent approach to perturbation problems than traditional techniques such as matched asymptotic expansions, WKB, and multiple scales. Readers unfamiliar with RG methods will find a basic introduction to the method along with three detailed examples: A linear oscillator, a nonlinear Duffing oscillator, and wave propagation in a nonlinear medium. The authors compare the RG solutions to those won through more traditional approaches in popular texts, so that instructors can use this article to augment a standard course on perturbation methods or as supplementary material for eager graduate students.
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Modified Gershgorin Disks for Companion Matrices

Aaron Melman

SIAM Rev. 54, pp. 355-373 (19 pages)

Online Publication Date: May 08, 2012

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All the zeros of a polynomial are contained in the union of Gershgorin disks derived from its companion matrix, a consequence of Gershgorin's theorem. However, this theorem does not exploit the structure of the companion matrix. We will use this structure to obtain smaller zero inclusion regions, thereby providing some nonstandard results to accompany and illustrate this frequently covered topic in numerical and matrix analysis.
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The Renormalization Group: A Perturbation Method for the Graduate Curriculum

Eleftherios Kirkinis

SIAM Rev. 54, pp. 374-388 (15 pages)

Online Publication Date: May 08, 2012

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In this paper the renormalization group (RG) method of Chen, Goldenfeld, and Oono [Phys. Rev. Lett., 73 (1994), pp. 1311–1315; Phys. Rev. E, 54 (1996), pp. 376–394] is presented in a pedagogical way to increase its visibility in applied mathematics and to argue favorably for its incorporation into the corresponding graduate curriculum. The method is illustrated by some linear and nonlinear singular perturbation problems.

Book Reviews

Bob O'Malley, Section Editor

SIAM Rev. 54, pp. 391-417 (27 pages)

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Mathematical geoscience is an extremely broad and challenging application of mathematical modeling, surveyed in a new monograph by Andrew Fowler of Oxford and Limerick. Writing a review of the book's 900 pages is clearly a Herculean task, ably taken on in our featured review by Neil Balmforth of the University of British Columbia.
The other reviews in this issue also span a wide spectrum of applied mathematics, including politics, qualitative risk assessment, Fourier series, turbulence and shell models, mathematical inequalities, differential equations, special functions, the aerofoil, porous media, inequalities in probability, probability measures, differential games, and the mathematics of learning. We leave it up to our readers to assimilate and coordinate all of this new material, and we thank our reviewers for their expert opinions and evaluations.
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